From ea6a26025357284a70512dbbbb4aa4d27137a1bb Mon Sep 17 00:00:00 2001 From: Cirdans-Home Date: Fri, 5 Mar 2021 09:04:04 +0100 Subject: [PATCH] Implemented changes to AMG4PSBLAS --- docs/amg4psblas_1.0-guide.pdf | Bin 0 -> 1776230 bytes docs/html/amg4psblaslibrary.png | Bin 0 -> 41208 bytes docs/html/index.html | 4 +- docs/html/userhtml.css | 1 + docs/html/userhtml.html | 4 +- docs/html/userhtml0x.png | Bin 894 -> 905 bytes docs/html/userhtml10x.png | Bin 1930 -> 1938 bytes docs/html/userhtml11x.png | Bin 1284 -> 1299 bytes docs/html/userhtml12x.png | Bin 800 -> 818 bytes docs/html/userhtml13x.png | Bin 6081 -> 6258 bytes docs/html/userhtml14x.png | Bin 5250 -> 5451 bytes docs/html/userhtml15x.png | Bin 1023 -> 1011 bytes docs/html/userhtml16x.png | Bin 2371 -> 2330 bytes docs/html/userhtml17x.png | Bin 8765 -> 8913 bytes docs/html/userhtml18x.png | Bin 4177 -> 4211 bytes docs/html/userhtml19x.png | Bin 646 -> 619 bytes docs/html/userhtml1x.png | Bin 2138 -> 2101 bytes 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+- docs/html/userhtmlsu13.html | 228 +- docs/html/userhtmlsu14.html | 255 +- docs/html/userhtmlsu15.html | 287 +- docs/html/userhtmlsu16.html | 97 +- docs/html/userhtmlsu17.html | 124 +- docs/html/userhtmlsu18.html | 311 +- docs/html/userhtmlsu2.html | 6 +- docs/html/userhtmlsu3.html | 119 +- docs/html/userhtmlsu4.html | 73 +- docs/html/userhtmlsu5.html | 89 +- docs/html/userhtmlsu6.html | 205 +- docs/html/userhtmlsu7.html | 343 +- docs/html/userhtmlsu8.html | 238 +- docs/html/userhtmlsu9.html | 281 +- docs/mld2p4-2.2-guide.pdf | 10752 ----------------------- docs/src/Makefile | 2 +- docs/src/abstract.tex | 32 +- docs/src/bibliography.tex | 48 +- docs/src/building.tex | 77 +- docs/src/distribution.tex | 31 +- docs/src/errors.tex | 10 +- docs/src/figures/amg4psblaslibrary.png | Bin 0 -> 41208 bytes docs/src/gettingstarted.tex | 85 +- docs/src/license.tex | 26 +- docs/src/newobjects.tex | 30 +- docs/src/overview.tex | 100 +- docs/src/userguide.tex | 59 +- docs/src/userinterface.tex | 314 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3 Configuring and Building MLD2P4 +class="cmr-12">Configuring and Building AMG4PSBLAS
 3.1 Auxiliary Methods
7 Adding new smoother and solver objects to MLD2P4 +class="cmr-12">Adding new smoother and solver objects to AMG4PSBLAS
8 Contributors
3 Configuring and Building MLD2P4 +class="cmr-12">Configuring and Building AMG4PSBLAS
 3.1 Auxiliary Methods
7 Adding new smoother and solver objects to MLD2P4 +class="cmr-12">Adding new smoother and solver objects to AMG4PSBLAS
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MLD2P4 (MuAMG4PSBLAS (AltiLevgel Dombrain Decomposic Multion PaiGrallel Pid Preconers -Ps Packsed on PSBLAS) is a package of parallel algebraic multilevel +class="small-caps">n +PSBLAS) is a package of parallel algebraic multilevel preconditioners included in the +PSCToolkit (Parallel Sparse Computation Toolkit) software framework. It is a progress +of a software development project started in 2007, named MLD2P4, which +implemented a multilevel version of some domain decomposition preconditioners of +additive-Schwarz type and was based on a parallel decoupled version of the well known +smoothed aggregation method to generate the multilevel hierarchy of coarser +matrices. In the last years, within the context of the EU-H2020 EoCoE project +(Energy Oriented Center of Excellence), the package was extended including +new algorithms and functionalities for setup and application of new AMG +preconditioners with the final aims of improving efficiency and scalability when tens of preconditioners. The first release of MLD2P4 made available multilevel additive and +class="cmr-12">thousands cores are used and of boosting reliability in dealing with general hybrid Schwarz preconditioners, as well as one-level additive Schwarz preconditioners. +class="cmr-12">symmetric positive definite linear systems. Due to the significant number The package has been extended to include further multilevel cycles and smoothers +class="cmr-12">of changes and the increase in scope, we decided to rename the package as widely used in multigrid methods. In the multilevel case, a purely algebraic approach is +class="cmr-12">AMG4PSBLAS. +

AMG4PSBLAS has been designed to provide scalable and easy-to-use applied to generate coarse-level corrections, so that no geometric background is needed +class="cmr-12">preconditioners in the context of the PSBLAS (Parallel Sparse Basic Linear Algebra concerning the matrix to be preconditioned. The matrix is assumed to be square, real +class="cmr-12">Subprograms) computational framework and can be used in conjuction with the Krylov or complex. -

MLD2P4 has been designed to provide scalable and easy-to-use preconditioners in +class="cmr-12">solvers available in this framework. Our package is based on a completely algebraic the context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms) +class="cmr-12">approach and users level interfaces assume that the system matrix and preconditioners computational framework and can be used in conjuction with the Krylov solvers +class="cmr-12">are represented as PSBLAS distributed sparse matrices. AMG4PSBLAS enables the available in this framework. MLD2P4 enables the user to easily specify different +class="cmr-12">user to easily specify different features of an algebraic multilevel preconditioner, thus features of an algebraic multilevel preconditioner, thus allowing to search for the “best” +class="cmr-12">allowing to experiment with different preconditioners for the problem and parallel preconditioner for the problem at hand. +class="cmr-12">computers at hand.

The package employs object-oriented design techniques in Fortran 2003, with @@ -136,7 +137,7 @@ class="cmr-12">through PSBLAS.

This guide provides a brief description of the functionalities and the user interface of MLD2P4. +class="cmr-12">of AMG4PSBLAS. diff --git a/docs/html/userhtmlli2.html b/docs/html/userhtmlli2.html index 2f15012d..51ce9e89 100644 --- a/docs/html/userhtmlli2.html +++ b/docs/html/userhtmlli2.html @@ -45,7 +45,7 @@ class="cmr-12">Code Distribution
3 Configuring and Building MLD2P4 +class="cmr-12">Configuring and Building AMG4PSBLAS
 3.1 _wrk
7 Adding new smoother and solver objects to MLD2P4 +class="cmr-12">Adding new smoother and solver objects to AMG4PSBLAS
8 ]

Contributors

-

Contributors to version 2:

  • Salvatore Filippone, Cranfield University, UK; +class="cmr-12">Pasqua D’Ambra, IAC-CNR, IT;
  • Pasqua D’Ambra, IAC-CNR, Naples, IT; - - - -
    • -
  • Daniela di Serafino, University of Campania “L. Vanvitelli”, Caserta, IT; -
    • -
  • Ambra Abdullahi Hassan, University of Rome “Tor Vergata”, IT.
-

Contributors to version 1: -

    -
  • Salvatore Filippone; -
    • -
  • Pasqua D’Ambra; -
    • -
  • Daniela di Serafino; +class="cmr-12">Fabio Durastante, University of Pisa and IAC-CNR, IT;
  • Alfredo Buttari, CNRS-IRIT, Toulouse, F.
+class="cmr-12">Salvatore Filippone, University of Rome Tor-Vergata and IAC-CNR, IT; diff --git a/docs/html/userhtmlli4.html b/docs/html/userhtmlli4.html index f68717f4..ef848d13 100644 --- a/docs/html/userhtmlli4.html +++ b/docs/html/userhtmlli4.html @@ -11,549 +11,645 @@

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+class="cmbx-12">up]

References

+class="cmbx-12">References

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[24]   [26]   K. Stüben, An Introduction to Algebraic Multigrid, in A. Schüller, - U. Trottenberg, C. Oosterlee, Multigrid, Academic Press, 2001. +class="cmbx-12">K. Stüben, An Introduction to Algebraic Multigrid, in + A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid, Academic + Press, 2001.

[25]   [27]   R. S. Tuminaro, C. Tong, Parallel Smoothed Aggregation Multigrid: +class="cmbx-12">R. S. Tuminaro, Aggregation Strategies on Massively Parallel Machines, in J. Donnelley, +class="cmbx-12">C. Tong, Parallel Smoothed Aggregation Multigrid: Aggregation editor, Proceedings of SuperComputing 2000, Dallas, 2000. +class="cmbxti-10x-x-120">Strategies on Massively Parallel Machines, in J. Donnelley, + editor, Proceedings of SuperComputing 2000, Dallas, 2000.

[26]   [28]   P. Vaněk, J. Mandel, M. Brezina, Algebraic Multigrid by Smoothed - Aggregation for Second and Fourth Order Elliptic Problems, Computing, 56 - (3) 1996, 179–196. -

-
+class="cmbx-12">P. Vaněk, J. Mandel, M. Brezina, Algebraic Multigrid by + Smoothed Aggregation for Second and Fourth Order Elliptic + Problems, Computing, 56 (3) 1996, 179–196. +

+

+ [29]   P. D’Ambra and P. S. Vassilevski, Adaptive AMG with + coarsening based on compatible weighted matching, Computing + and Visualization in Science, 16, (2013) 59–76. +

+

+ [30]   P. D’Ambra, S. Filippone and P. S. Vassilevski, BootCMatch: + a software package for bootstrap AMG based on graph weighted + matching, ACM Transactions on Mathematical Software, 44, + (2018) 39:1–39:25.

@@ -561,19 +657,19 @@ class="cmr-12">(3) 1996, 179–196.

[[prev] [prev] [prev-tail] [prev-tail] [front] [front] [up]

+class="cmbx-12">up]

diff --git a/docs/html/userhtmlse1.html b/docs/html/userhtmlse1.html index 55f5d6ba..2cfc937a 100644 --- a/docs/html/userhtmlse1.html +++ b/docs/html/userhtmlse1.html @@ -33,39 +33,22 @@ class="cmr-12">1 General Overview

The MuAltiLevgel Dombrain Decomposic Multion PaiGrallel Pid Preconers -Ps Packsed on PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid +class="small-caps">n PSBLAS (AMG) and Domain Decomposition preconditioners (see, e.g., (AMG4PSBLAS) provides parallel Algebraic MultiGrid (AMG) preconditioners (see, +e.g., [3, 24, 2226]), to be used -in the iterative solution of linear systems, +class="cmr-12">), to be used in the iterative solution of linear systems,
@@ -121,25 +98,19 @@ src="userhtml0x.png" alt="Ax = b, " class="math-display" >
(1)
-

+

where A is a square, real or complex, sparse matrix. The name of the package comes -from its original implementation, containing multilevel additive and hybrid Schwarz +class="cmr-12">is a square, real or complex, sparse symmetric positive definite (s.p.d) preconditioners, as well as one-level additive Schwarz preconditioners. The current +class="cmr-12">matrix. +

The preconditioners implemented in AMG4PSBLAS are obtained by combining 3 version extends the original plan by including multilevel cycles and smoothers widely +class="cmr-12">different types of AMG cycles with smoothers and coarsest-level solvers. The V-, W-, used in multigrid methods. -

The multilevel preconditioners implemented in MLD2P4 are obtained by -combining AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and -K-cyclesand a version of a Krylov-type cycle (K-cycle) [3, 2022] are available, which allow to define almost all the preconditioners in +class="cmr-12">are available, which can be the package, including the multilevel hybrid Schwarz ones; a specific cycle is +class="cmr-12">combined with weighted versions of Jacobi, hybrid forward/backward Gauss-Seidel, implemented to obtain multilevel additive Schwarz preconditioners. The Jacobi, +class="cmr-12">block-Jacobi, and additive Schwarz smoothers. An algebraic approach is used to hybrid forward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz +class="cmr-12">generate a hierarchy of coarse-level matrices and operators, without explicitly using methods are available as smoothers. An algebraic approach is used to generate a +class="cmr-12">any information on the geometry of the original problem, e.g., the discretization of a hierarchy of coarse-level matrices and operators, without explicitly using any +class="cmr-12">PDE. To this end, two different coarsening strategies, based on aggregation, are information on the geometry of the original problem, e.g., the discretization of a -PDE. To this end, the smoothed aggregation techniqueavailable: +

    +
  • a decoupled version of the well known smoothed aggregation procedure + proposed in [2, 2628], and already included in the previous versions of the + package [9, 8]; +
    • +
  • the first parallel implementation of a coupled version of Coarsening based + on Compatible Weighted Matching introduced in [29, 30] is applied. -Either exact or approximate solvers can be used on the coarsest-level system. -Specifically, different sparse LU factorizations from external packages, and -native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel, and +class="cmr-12">and described in + details in [10];
+

Either exact or approximate solvers can be used on the coarsest-level system. +Specifically, different sparse LU factorizations from external packages, native block-Jacobi solvers are available. All smoothers can be also exploited as one-level +class="cmr-12">incomplete LU factorizations, weighted Jacobi, hybrid Gauss-Seidel, and block-Jacobi +solvers are available. All the smoothers can be also exploited as one-level preconditioners. -

MLD2P4 is written in Fortran

AMG4PSBLAS is written in Fortran 2003, following an object-oriented design through the exploitation of features such as abstract data type creation, type extension, functional overloading, and dynamic memory management. The +class="cmr-12">extension, functional overloading, and dynamic memory management. The parallel parallel implementation is based on a Single Program Multiple Data (SPMD) +class="cmr-12">implementation is based on a Single Program Multiple Data (SPMD) paradigm. paradigm. Single and double precision implementations of MLD2P4 are available +class="cmr-12">Single and double precision implementations of AMG4PSBLAS are available for both the real and the complex case, which can be used through a single interface. -

MLD2P4 has been designed to implement scalable and easy-to-use multilevel +

AMG4PSBLAS has been designed to implement scalable and easy-to-use multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS) computational framework [1517, 1416]. PSBLAS provides basic linear algebra operators and data management facilities for distributed sparse matrices, as well as parallel Krylov solvers which can be used with the MLD2P4 preconditioners. The choice of PSBLAS has been +class="cmr-12">which can be used with the AMG4PSBLAS preconditioners. The choice of PSBLAS +has been mainly motivated by the need of having a portable and efficient +software infrastructure implementing “de facto” standard parallel sparse linear mainly motivated by the need of having a portable and efficient software infrastructure +class="cmr-12">algebra kernels, to pursue goals such as performance, portability, modularity implementing “de facto” standard parallel sparse linear algebra kernels, to pursue goals +class="cmr-12">ed extensibility in the development of the preconditioner package. On the such as performance, portability, modularity ed extensibility in the development of the +class="cmr-12">other hand, the implementation of AMG4PSBLAS, which was driven by the preconditioner package. On the other hand, the implementation of MLD2P4 has led to +class="cmr-12">need to face the exascale challenge, has led to some important revisions and some revisions and extentions of the original PSBLAS kernels. The inter-process +class="cmr-12">extentions of the PSBLAS infrastructure. The inter-process comunication comunication required by MLD2P4 is encapsulated in the PSBLAS routines; therefore, +class="cmr-12">required by AMG4PSBLAS is encapsulated in the PSBLAS routines; therefore, MLD2P4 can be run on any parallel machine where PSBLAS implementations are +class="cmr-12">AMG4PSBLAS can be run on any parallel machine where PSBLAS implementations available. -

MLD2P4 has a layered and modular software architecture where three main layers +class="cmr-12">are available. +

AMG4PSBLAS has a layered and modular software architecture where three main can be identified. The lower layer consists of the PSBLAS kernels, the middle one +class="cmr-12">layers can be identified. The lower layer consists of the PSBLAS kernels, the middle implements the construction and application phases of the preconditioners, and the +class="cmr-12">one implements the construction and application phases of the preconditioners, and the upper one provides a uniform interface to all the preconditioners. This architecture allows for different levels of use of the package: few black-box layer allow all users to easily build and apply any preconditioner available in MLD2P4; facilities are also available allowing expert users to extend the set of +class="cmr-12">AMG4PSBLAS; facilities are also available allowing expert users to extend the set of smoothers and solvers for building new versions of the preconditioners (see  7). -

We note that the user interface of MLD2P4 2.1 has been extended with respect to -the previous versions in order to separate the construction of the multilevel -hierarchy from the construction of the smoothers and solvers, and to allow for -more flexibility at each level. The software architecture described in [8] has -significantly evolved too, in order to fully exploit the Fortran 2003 features -implemented in PSBLAS 3. However, compatibility with previous versions has been -preserved. -

This guide is organized as follows. General information on the distribution of the source code is reported in Sectionof the package are given in Section 3. A short description of the preconditioners -implemented in MLD2P4 is provided in Section 4, to help the users in choosing among - - - +class="cmr-12">. The basics for building and applying the them. The basics for building and applying the preconditioners with the Krylov solvers +class="cmr-12">preconditioners with the Krylov solvers implemented in PSBLAS are reported implemented in PSBLAS are reported inin Section 5, where the Fortran codes of a few -sample programs are also shown. A reference guide for the user interface routines is +class="cmr-12">, where the Fortran codes of a few sample programs are also shown. + + + provided in SectionA reference guide for the user interface routines is provided in Section 6. Information on the extension of the package through the +class="cmr-12">. +Information on the extension of the package through the addition of new addition of new smoothers and solvers is reported in Sectionsmoothers and solvers is reported in Section 7. The error handling +class="cmr-12">. The error handling mechanism mechanism used by the package is briefly described in Sectionused by the package is briefly described in Section 8. The copyright +class="cmr-12">. The copyright terms terms concerning the distribution and modification of MLD2P4 are reported in +class="cmr-12">concerning the distribution and modification of AMG4PSBLAS are reported in Appendix 2 Code Distribution

MLD2P4 is available from the web site +class="cmr-12">AMG4PSBLAS is available from the web site

https://github.com/sfilippone/mld2p4-2

+class="cmtt-12">https://github.com/psctoolkit/amg4psblas

where contact points for further information can be also found.

into account when treating derived works. class="cmr-12">The library defines a version string with the constant

mld_version_string_ +src=

whose current value is 2.1.01.0.

diff --git a/docs/html/userhtmlse3.html b/docs/html/userhtmlse3.html index 963e7225..cab86ca8 100644 --- a/docs/html/userhtmlse3.html +++ b/docs/html/userhtmlse3.html @@ -1,7 +1,7 @@ -Configuring and Building MLD2P4 +Configuring and Building AMG4PSBLAS @@ -30,9 +30,9 @@ class="cmr-12">]

3 Configuring and Building MLD2P4

+class="cmr-12">Configuring and Building AMG4PSBLAS

In order to build MLD2P4 it is necessary to set up a Makefile with appropriate +class="cmr-12">In order to build AMG4PSBLAS it is necessary to set up a Makefile with appropriate system-dependent variables; this is done by means of the configure distribution also includes the autoconf and automake sources empl script, but usually this is not needed to build the software.

MLD2P4 is implemented almost entirely in Fortran 2003, with some interfaces to +class="cmr-12">AMG4PSBLAS is implemented almost entirely in Fortran 2003, with some external libraries in C; the Fortran compiler must support the Fortran 2003 standard +class="cmr-12">interfaces to external libraries in C; the Fortran compiler must support the plus the extension Fortran 2003 standard plus the extension MOLD= feature, which enhances the usability of ALLOCATE. Many +class="cmr-12">feature, which enhances the usability compilers do this; in particular, this is supported by the GNU Fortran compiler, for +class="cmr-12">of ALLOCATE. Many compilers do this; in particular, this is supported by the GNU which we recommend to use at least version 4.8. The software defines data +class="cmr-12">Fortran compiler, for which we recommend to use at least version 4.8. The software types and interfaces for real and complex data, in both single and double +class="cmr-12">defines data types and interfaces for real and complex data, in both single and double precision.

Building MLD2P4 requires some base libraries (see SectionBuilding AMG4PSBLAS requires some base libraries (see Section 3.1); interfaces to optional third-party libraries, which extend the functionalities of MLD2P4 (see +class="cmr-12">optional third-party libraries, which extend the functionalities of AMG4PSBLAS Section(see Section 3.2), are also available. Many Linux distributions (e.g., Ubuntu, Fedora, +class="cmr-12">), are also available. Many Linux distributions (e.g., Ubuntu, CentOS) provide precompiled packages for the prerequisite and optional software. In +class="cmr-12">Fedora, CentOS) provide precompiled packages for the prerequisite and optional many cases these packages are split between a runtime part and a “developer” part; in +class="cmr-12">software. In many cases these packages are split between a runtime part and a order to build MLD2P4 you need both. A description of the base and optional software +class="cmr-12">“developer” part; in order to build AMG4PSBLAS you need both. A description of used by MLD2P4 is given in the next sections. +class="cmr-12">the base and optional software used by AMG4PSBLAS is given in the next +sections.

Example and test programs

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4 4 Multigrid Background

+class="cmbx-12">Multigrid Background

Multigrid preconditioners, coupled with Krylov iterative solvers, are widely used in the +class="cmbx-12">Multigrid preconditioners, coupled with Krylov iterative solvers, are widely parallel solution of large and sparse linear systems, because of their optimality in the +class="cmbx-12">used in the parallel solution of large and sparse linear systems, because of solution of linear systems arising from the discretization of scalar elliptic Partial +class="cmbx-12">their optimality in the solution of linear systems arising from the Differential Equations (PDEs) on regular grids. Optimality, also known as algorithmic +class="cmbx-12">discretization of scalar elliptic Partial Differential Equations (PDEs) on scalability, is the property of having a computational cost per iteration that depends +class="cmbx-12">regular grids. Optimality, also known as algorithmic scalability, is the linearly on the problem size, and a convergence rate that is independent of the problem +class="cmbx-12">property of having a computational cost per iteration that depends linearly size. +class="cmbx-12">on the problem size, and a convergence rate that is independent of the +problem size.

Multigrid preconditioners are based on a recursive application of a two-grid process +class="cmbx-12">Multigrid preconditioners are based on a recursive application of a +two-grid process consisting of smoother iterations and a coarse-space (or +coarse-level) correction. The smoothers may be either basic iterative consisting of smoother iterations and a coarse-space (or coarse-level) correction. The +class="cmbx-12">methods, such as the Jacobi and Gauss-Seidel ones, or more complex smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel +class="cmbx-12">subspace-correction methods, such as the Schwarz ones. The coarse-space ones, or more complex subspace-correction methods, such as the Schwarz ones. The +class="cmbx-12">correction consists of solving, in an appropriately chosen coarse space, the coarse-space correction consists of solving, in an appropriately chosen coarse space, the +class="cmbx-12">residual equation associated with the approximate solution computed by residual equation associated with the approximate solution computed by the +class="cmbx-12">the smoother, and of using the solution of this equation to correct the smoother, and of using the solution of this equation to correct the previous +class="cmbx-12">previous approximation. The transfer of information between the original approximation. The transfer of information between the original (fine) space and +class="cmbx-12">(fine) space and the coarse one is performed by using suitable restriction the coarse one is performed by using suitable restriction and prolongation +class="cmbx-12">and prolongation operators. The construction of the coarse space and the operators. The construction of the coarse space and the corresponding transfer +class="cmbx-12">corresponding transfer operators is carried out by applying a so-called operators is carried out by applying a so-called coarsening algorithm to the +class="cmbx-12">coarsening algorithm to the system matrix. Two main approaches system matrix. Two main approaches can be used to perform coarsening: +class="cmbx-12">can be used to perform coarsening: the geometric approach, which the geometric approach, which exploits the knowledge of some physical grid +class="cmbx-12">exploits the knowledge of some physical grid associated with the associated with the matrix and requires the user to define transfer operators +class="cmbx-12">matrix and requires the user to define transfer operators from the from the fine to the coarse level and vice versa, and the algebraic approach, +class="cmbx-12">fine to the coarse level and vice versa, and the algebraic approach, which builds the coarse-space correction and the associate transfer operators +class="cmbx-12">which builds the coarse-space correction and the associate transfer using only matrix information. The first approach may be difficult when the +class="cmbx-12">operators using only matrix information. The first approach may be system comes from discretizations on complex geometries; furthermore, ad hoc +class="cmbx-12">difficult when the system comes from discretizations on complex one-level smoothers may be required to get an efficient interplay between +class="cmbx-12">geometries; furthermore, ad hoc one-level smoothers may be required to fine and coarse levels, e.g., when matrices with highly varying coefficients are +class="cmbx-12">get an efficient interplay between fine and coarse levels, e.g., when considered. The second approach performs a fully automatic coarsening and +class="cmbx-12">matrices with highly varying coefficients are considered. The second enforces the interplay between fine and coarse level by suitably choosing the +class="cmbx-12">approach performs a fully automatic coarsening and enforces the coarse space and the coarse-to-fine interpolation (see, e.g., [interplay between fine and coarse level by suitably choosing the coarse +space and the coarse-to-fine interpolation (see, e.g., [3, 3, 24, 26, 22] for +class="cmbx-12">24] for + + + details.) +class="cmbx-12">details.)

MLD2P4 uses a pure algebraic approach, based on the smoothed aggregation +class="cmbx-12">MLD2P4 uses a pure algebraic approach, based on the smoothed algorithm [aggregation algorithm [2, 2, 26], for building the sequence of coarse matrices and transfer +class="cmbx-12">28], for building the sequence of coarse matrices operators, starting from the original one. A decoupled version of this algorithm - - - +class="cmbx-12">and transfer operators, starting from the original one. A decoupled version is implemented, where the smoothed aggregation is applied locally to each +class="cmbx-12">of this algorithm is implemented, where the smoothed aggregation is submatrix [applied locally to each submatrix [25]. A brief description of the AMG preconditioners implemented in +class="cmbx-12">27]. A brief description of the AMG MLD2P4 is given in Sections preconditioners implemented in MLD2P4 is given in Sections 4.1-4.1-4.3. For further details the reader is referred to -[4.3. For +further details the reader is referred to [4, 4, 5, 5, 7, 7, 8]. +class="cmbx-12">8].

We note that optimal multigrid preconditioners do not necessarily correspond to +class="cmbx-12">We note that optimal multigrid preconditioners do not necessarily +correspond to minimum execution times in a parallel setting. Indeed, to minimum execution times in a parallel setting. Indeed, to obtain effective parallel +class="cmbx-12">obtain effective parallel multigrid preconditioners, a tradeoff between the multigrid preconditioners, a tradeoff between the optimality and the cost of +class="cmbx-12">optimality and the cost of building and applying the smoothers and building and applying the smoothers and the coarse-space corrections must be +class="cmbx-12">the coarse-space corrections must be achieved. Effective parallel achieved. Effective parallel preconditioners require algorithmic scalability to +class="cmbx-12">preconditioners require algorithmic scalability to be coupled with be coupled with implementation scalability, i.e., a computational cost per +class="cmbx-12">implementation scalability, i.e., a computational cost per iteration iteration which remains (almost) constant as the number of parallel processors +class="cmbx-12">which remains (almost) constant as the number of parallel processors increases. +class="cmbx-12">increases.

 4.1  4.1 AMG preconditioners +class="cmbx-12">AMG preconditioners
 4.2  4.2 Smoothed Aggregation +class="cmbx-12">Smoothed Aggregation
 4.3  4.3 Smoothers and coarsest-level solvers +class="cmbx-12">Smoothers and coarsest-level solvers
@@ -200,22 +211,22 @@ class="cmr-12">Smoothers and coarsest-level solvers

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5 5 Getting Started

+class="cmbx-12">Getting Started

We describe the basics for building and applying MLD2P4 one-level and multilevel +class="cmbx-12">We describe the basics for building and applying AMG4PSBLAS one-level (i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [13]. The +class="cmbx-12">and multilevel (i.e., AMG) preconditioners with the Krylov solvers included following steps are required: +class="cmbx-12">in PSBLAS [15]. The following steps are required:

  1. Declare the preconditioner data structure. It is a derived data type, - mld_xDeclare the preconditioner data structure. It is a derived data + type, amg_xprec_ type, where x may be , where x may be s, , d, , c or or z, according to the basic +class="cmbx-12">, according to the data type of the sparse matrix (basic data type of the sparse matrix (s = real single precision; = real single precision; d = real double +class="cmbx-12">= precision; real double precision; c = complex single precision; = complex single precision; z = complex double precision). This +class="cmbx-12">= complex data structure is accessed by the user only through the MLD2P4 routines, +class="cmbx-12">double precision). This data structure is accessed by the user only following an object-oriented approach. +class="cmbx-12">through the AMG4PSBLAS routines, following an object-oriented + approach.
  2. Allocate and initialize the preconditioner data structure, according to a +class="cmbxti-10x-x-120">Allocate and initialize the preconditioner data structure, + according to a preconditioner type chosen by the user. This is preconditioner type chosen by the user. This is performed by the routine - performed by the routine init, which also sets defaults for each preconditioner type selected by +class="cmbx-12">, which also sets defaults for each the user. The preconditioner types and the defaults associated with them +class="cmbx-12">preconditioner type selected by the user. The preconditioner types are given in Table and the defaults associated with them are given in Table 1, where the strings used by 1, where + the strings used by init to identify the +class="cmbx-12">to identify the preconditioner types are preconditioner types are also given. Note that these strings are valid also if +class="cmbx-12">also given. Note that these strings are valid also if uppercase letters uppercase letters are substituted by corresponding lowercase ones. +class="cmbx-12">are substituted by corresponding lowercase ones.
  3. Modify the selected preconditioner type, by properly setting preconditioner +class="cmbxti-10x-x-120">Modify the selected preconditioner type, by properly setting parameters. This is performed by the routine preconditioner parameters. This is performed by the routine set. This routine must be +class="cmbx-12">. called only if the user wants to modify the default values of the parameters +class="cmbx-12">This routine must be called only if the user wants to modify associated with the selected preconditioner type, to obtain a variant of that +class="cmbx-12">the default values of the parameters associated with the selected preconditioner. Examples of use of preconditioner type, to obtain a variant of that preconditioner. + Examples of use of set are given in Section are given in Section 5.1; a complete +class="cmbx-12">5.1; a complete list list of all the preconditioner parameters and their allowed and default values +class="cmbx-12">of all the preconditioner parameters and their allowed and default is provided in Section values is provided in Section 6, Tables 6, Tables 2-2-8. +class="cmbx-12">8.
  4. Build the preconditioner for a given matrix. If the selected preconditioner is +class="cmbxti-10x-x-120">Build the preconditioner for a given matrix. If the selected + preconditioner is multilevel, then two steps must be performed, as multilevel, then two steps must be performed, as specified next. +class="cmbx-12">specified next. + + +
    1. Build the aggregation hierarchy for a given matrix. This is performed +class="cmbxti-10x-x-120">Build the aggregation hierarchy for a given matrix. This is by the routine performed by the routine hierarchy_build. - - - +class="cmbx-12">.
    2. Build the preconditioner for a given matrix. This is performed by the +class="cmbxti-10x-x-120">Build the preconditioner for a given matrix. This is routine performed by the routine smoothers_build.
    -

    If the selected preconditioner is one-level, it is built in a single step, performed by +class="cmbx-12">.

+

If the selected preconditioner is one-level, it is built in a single step, the routine performed by the routine bld. +class="cmbx-12">.

  • Apply the preconditioner at each iteration of a Krylov solver. This is performed by +class="cmbxti-10x-x-120">Apply the preconditioner at each iteration of a Krylov solver. This is the method performed by the method apply. When using the PSBLAS Krylov solvers, this step is +class="cmbx-12">. When using the PSBLAS Krylov completely transparent to the user, since solvers, this step is completely transparent to the user, since apply is called by the PSBLAS routine +class="cmbx-12">is + called by the PSBLAS routine implementing the Krylov solver implementing the Krylov solver ((psb_krylov). +class="cmbx-12">).
  • Free the preconditioner data structure. This is performed by the routine free. +class="cmbxti-10x-x-120">Free the preconditioner data structure. This is performed by the This step is complementary to step 1 and should be performed when the +class="cmbx-12">routine free. This step is complementary to step 1 and should be preconditioner is no more used.
    • -

    All the previous routines are available as methods of the preconditioner object. A +class="cmbx-12">performed when the preconditioner is no more used. +

    All the previous routines are available as methods of the preconditioner detailed description of them is given in Section object. A detailed description of them is given in Section 6. Examples showing the basic use of +class="cmbx-12">6. Examples MLD2P4 are reported in Section showing the basic use of AMG4PSBLAS are reported in Section 5.1. +class="cmbx-12">5.1.

    -

    @@ -227,7 +239,7 @@ class="cmr-12">.
    -

    +

    type @@ -296,41 +308,41 @@ to 1. class="hline"> +class="td11">

    Block-Jacobi with ILU(0) on the local blocks.

    smoother

    +class="cmtt-12">class(amg_x_base_smoother_type) +class="cmbx-12">preconditioner. +class="cmtt-12">class(amg_x_base_solver_type)

    string

    der

    No preconditioner

    ’NONE’

    Considered to use the PSBLAS Krylov solvers +class="td11">

    Considered to use the PSBLAS Krylov solvers with no preconditioner.

    Diagonal

    ’DIAG’ or ’JACOBI’

    Diagonal preconditioner. For any zero diagonal +class="td11">

    Diagonal preconditioner. For any zero diagonal entry of the matrix to be preconditioned, the corresponding entry of the preconditioner is set to 1.

    Gauss-Seidel

    ’GS’

    Hybrid Gauss-Seidel (forward), that is, global +class="td11">

    Hybrid Gauss-Seidel (forward), that is, global block Jacobi with Gauss-Seidel as local solver.

    Symmetrized Gauss-Seidel

    ’FBGS’

    Symmetrized +class="td11">

    Symmetrized hybrid Gauss-Seidel,that is, forward Gauss-Seidel followed by backward Gauss-Seidel.

    Block Jacobi

    ’BJAC’

    Block-Jacobi with ILU(0) on the local blocks.

    Additive Schwarz

    ’AS’

    Additive Schwarz (AS), with overlap 1 and +class="td11">

    Additive Schwarz (AS), with overlap 1 and ILU(0) on the local blocks.

    Multilevel

    ’ML’

    V-cycle with one hybrid forward Gauss-Seidel +class="td11">

    V-cycle with one hybrid forward Gauss-Seidel (GS) sweep as pre-smoother and one hybrid backward GS sweep as post-smoother, basic smoothed aggregation as coarsening algorithm, @@ -352,68 +364,73 @@ class="content">Preconditioner types, corresponding strings and default choices.

    -

    Note that the module mld_prec_mod, containing the definition of the preconditioner +

    Note that the module amg_prec_mod, containing the definition of +the preconditioner data type and the interfaces to the routines of data type and the interfaces to the routines of MLD2P4, must be used in any program +class="cmbx-12">AMG4PSBLAS, must be used in any program calling such routines. The calling such routines. The modules modules psb_base_mod, for the sparse matrix and +class="cmbx-12">, for the sparse matrix and communication descriptor communication descriptor data types, and data types, and psb_krylov_mod, for interfacing with the +class="cmbx-12">, for interfacing with the Krylov solvers, Krylov solvers, must be also used (see Section must be also used (see Section 5.1).
    5.1    ).
    -

    Remark 1. Coarsest-level solvers based on the LU factorization, such as those +

    Remark 1. Coarsest-level solvers based on the LU factorization, such as +those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist, implemented in UMFPACK, MUMPS, SuperLU, and SuperLU_Dist, usually lead to +class="cmbx-12">usually lead to smaller numbers of preconditioned Krylov iterations smaller numbers of preconditioned Krylov iterations than inexact solvers, when the +class="cmbx-12">than inexact solvers, when the linear system comes from a standard linear system comes from a standard discretization of basic scalar elliptic PDE +class="cmbx-12">discretization of basic scalar elliptic PDE problems. However, this does not problems. However, this does not necessarily correspond to the smallest execution time +class="cmbx-12">necessarily correspond to the smallest execution time on parallel on parallel computers. +class="cmbx-12">computers. +

    DA MODIFICARE PER INSERIRE TIPO DI AGGREGAZIONE

     5.1  5.1 Examples +class="cmbx-12">Examples

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    +class="cmbx-12">up]

    diff --git a/docs/html/userhtmlse6.html b/docs/html/userhtmlse6.html index b9c69afd..6c84e382 100644 --- a/docs/html/userhtmlse6.html +++ b/docs/html/userhtmlse6.html @@ -11,234 +11,239 @@

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    +class="cmbx-12">up]

    6 6 User Interface

    +class="cmbx-12">User Interface

    The basic user interface of MLD2P4 consists of eight methods. The six methods The basic user interface of AMG4PBLAS consists of eight methods. The six +methods init, -, set, , build, , hierarchy_build, , smoothers_build and apply encapsulate all the +class="cmbx-12">and apply +encapsulate all the functionalities for the setup and the application of any functionalities for the setup and the application of any multilevel and one-level +class="cmbx-12">multilevel and one-level preconditioner implemented in the package. The preconditioner implemented in the package. The method method free deallocates the +class="cmbx-12">deallocates the preconditioner data structure, while descr preconditioner data structure, while descr prints a description of the preconditioner +class="cmbx-12">prints a description of the preconditioner setup by the user. For setup by the user. For backward compatibility, methods are also accessible as +class="cmbx-12">backward compatibility, methods are also accessible as stand-alone stand-alone subroutines. +class="cmbx-12">subroutines.

    For each method, the same user interface is overloaded with respect to the real/ +class="cmbx-12">For each method, the same user interface is overloaded with respect to complex case and the single/double precision; arguments with appropriate data types +class="cmbx-12">the real/ complex case and the single/double precision; arguments with must be passed to the method, i.e., +class="cmbx-12">appropriate data types must be passed to the method, i.e.,

    • the sparse matrix data structure, containing the matrix to be preconditioned, +class="cmbx-12">the sparse matrix data structure, containing the matrix to be must be of type preconditioned, must be of type psb_xxspmat_type with x = with x = s for real single precision, x +class="cmbx-12">for = real single precision, x = d for real double precision, x = for real double precision, x = c for complex single precision, x = z for +class="cmbx-12">for complex double precision; +class="cmbx-12">complex single precision, x = z for complex double precision;
    • the preconditioner data structure must be of type mld_xthe preconditioner data structure must be of type amg_xprec_type, with x +class="cmbx-12">, = with x = s, , d, , c, , z, according to the sparse matrix data structure; +class="cmbx-12">, according to the sparse matrix data structure;
    • the arrays containing the vectors v and w involved in the preconditioner +class="cmbx-12">the arrays containing the vectors v and w involved in application w = the preconditioner application w = B-1v must be of type v must be of type + psb_xxvect_type with x = with x = s, , d, , c, - , z, in a manner completely analogous to the sparse matrix type; +class="cmbx-12">, in a manner completely + analogous to the sparse matrix type;
    • real parameters defining the preconditioner must be declared according to +class="cmbx-12">real parameters defining the preconditioner must be declared the precision of the sparse matrix and preconditioner data structures (see +class="cmbx-12">according to the precision of the sparse matrix and preconditioner Section data structures (see Section 6.2).
    +class="cmbx-12">6.2).

    A description of each method is given in the remainder of this section. +class="cmbx-12">A description of each method is given in the remainder of this section.

     6.1  6.1 Method init +class="cmbx-12">Method init
     6.2  6.2 Method set +class="cmbx-12">Method set
     6.3  6.3 Method hierarchy_build +class="cmbx-12">Method hierarchy_build
     6.4  6.4 Method smoothers_build +class="cmbx-12">Method smoothers_build
     6.5  6.5 Method build +class="cmbx-12">Method build
     6.6  6.6 Method apply +class="cmbx-12">Method apply
     6.7  6.7 Method free +class="cmbx-12">Method free
     6.8  6.8 Method descr +class="cmbx-12">Method descr
     6.9  6.9 Auxiliary Methods +class="cmbx-12">Auxiliary Methods
      6.9.1   6.9.1 Method: dump +class="cmbx-12">Method: dump
      6.9.2   6.9.2 Method: clone +class="cmbx-12">Method: clone
      6.9.3   6.9.3 Method: sizeof +class="cmbx-12">Method: sizeof
      6.9.4   6.9.4 Method: allocate_wrk +class="cmbx-12">Method: allocate_wrk
      6.9.5   6.9.5 Method: free_wrk +class="cmbx-12">Method: free_wrk
    @@ -252,22 +257,22 @@ class="cmr-12">_wrk

    [[next] [next] [prev] [prev] [prev-tail] [prev-tail] [front] [front] [up]

    +class="cmbx-12">up]

    diff --git a/docs/html/userhtmlse7.html b/docs/html/userhtmlse7.html index bb62c447..0a78097d 100644 --- a/docs/html/userhtmlse7.html +++ b/docs/html/userhtmlse7.html @@ -1,7 +1,7 @@ -Adding new smoother and solver objects to MLD2P4 +Adding new smoother and solver objects to AMG4PSBLAS @@ -11,65 +11,71 @@

    [[next] [next] [prev] [prev] [prev-tail] [prev-tail] [tail] [tail] [up]

    +class="cmbx-12">up]

    7 7 Adding new smoother and solver objects to MLD2P4

    +class="cmbx-12">Adding new smoother and solver objects to AMG4PSBLAS

    Developers can add completely new smoother and/or solver classes derived from the +class="cmbx-12">Developers can add completely new smoother and/or solver classes derived base objects in the library (see Remark 2 in Section from the base objects in the library (see Remark 2 in Section 6.2), without recompiling the +class="cmbx-12">6.2), without library itself. +class="cmbx-12">recompiling the library itself.

    To do so, it is necessary first to select the base type to be extended. In our +class="cmbx-12">To do so, it is necessary first to select the base type to be extended. In experience, it is quite likely that the new application needs only the definition of a +class="cmbx-12">our experience, it is quite likely that the new application needs only the “solver” object, which is almost always acting only on the local part of the distributed +class="cmbx-12">definition of a “solver” object, which is almost always acting only on the matrix. The parallel actions required to connect the various solver objects are most +class="cmbx-12">local part of the distributed matrix. The parallel actions required to often already provided by the block-Jacobi or the additive Schwarz smoothers. To +class="cmbx-12">connect the various solver objects are most often already provided by the define a new solver, the developer will then have to define its components and +class="cmbx-12">block-Jacobi or the additive Schwarz smoothers. To define a new solver, the methods, perhaps taking one of the predefined solvers as a starting point, if +class="cmbx-12">developer will then have to define its components and methods, possible. +class="cmbx-12">perhaps taking one of the predefined solvers as a starting point, if +possible.

    Once the new smoother/solver class has been developed, to use it in the context of +class="cmbx-12">Once the new smoother/solver class has been developed, to use +it in the context of the multilevel preconditioners it is necessary the multilevel preconditioners it is necessary to: +class="cmbx-12">to:

    • declare in the application program a variable of the new type; +class="cmbx-12">declare in the application program a variable of the new type;
    • pass that variable as the argument to the pass that variable as the argument to the set routine as in the following: +class="cmbx-12">routine as in the + following:

      @@ -83,55 +89,60 @@ class="cmtt-12"> p%set(solver,info [,ilev,ilmax,pos])

    • link the code implementing the various methods into the application +class="cmbx-12">link the code implementing the various methods into the application executable.
    +class="cmbx-12">executable.

    The new solver object is then dynamically included in the preconditioner structure, and +class="cmbx-12">The new solver object is then dynamically included in the preconditioner acts as a mold to which the preconditioner will conform, even though the MLD2P4 +class="cmbx-12">structure, and acts as a mold to which the preconditioner will conform, library has not been modified to account for this new development. -

    It is possible to define new values for the keyword WHAT in the set routine; if the -library code does not recognize a keyword, it passes it down the composition hierarchy +class="cmbx-12">even though the AMG4PSBLAS library has not been modified to account (levels containing smoothers containing in turn solvers), so that it can be eventually +class="cmbx-12">for this new development. +

    It is possible to define new values for the keyword WHAT in the set caught by the new solver. By the same token, any keyword/value pair that does not +class="cmbx-12">routine; if the library code does not recognize a keyword, it passes it down pertain to a given smoother should be passed down to the contained solver, and +class="cmbx-12">the composition hierarchy (levels containing smoothers containing in turn any keyword/value pair that does not pertain to a given solver is by default +class="cmbx-12">solvers), so that it can be eventually caught by the new solver. By the ignored. +class="cmbx-12">same token, any keyword/value pair that does not pertain to a given +smoother should be passed down to the contained solver, and any +keyword/value pair that does not pertain to a given solver is by default +ignored.

    An example is provided in the source code distribution under the folder +class="cmbx-12">An example is provided in the source code distribution under the folder tests/newslv. In this example we are implementing a new incomplete factorization +class="cmbx-12">. In this example we are implementing a new incomplete variant (which is simply the ILU(0) factorization under a new name). Because of the +class="cmbx-12">factorization variant (which is simply the ILU(0) factorization under a new specifics of this case, it is possible to reuse the basic structure of the ILU solver, with +class="cmbx-12">name). Because of the specifics of this case, it is possible to reuse the basic its L/D/U components and the methods needed to apply the solver; only a few +class="cmbx-12">structure of the ILU solver, with its L/D/U components and the methods, such as the description and most importantly the build, need to be +class="cmbx-12">methods needed to apply the solver; only a few methods, such as the ovverridden (rewritten). +class="cmbx-12">description and most importantly the build, need to be ovverridden +(rewritten).

    The interfaces for the calls shown above are defined using +class="cmbx-12">The interfaces for the calls shown above are defined using

    @@ -144,46 +155,49 @@ id="TBL-21-2">

    class(mld_x_base_smoother_type)

    The user-defined new smoother to be employed in the +class="cmbx-12">The user-defined new smoother to be employed in the preconditioner.

    solver

    class(mld_x_base_solver_type)

    The user-defined new solver to be employed in the preconditioner.

    +class="cmbx-12">The user-defined new solver to be employed in the +preconditioner. +

    The other arguments are defined in the way described in Sec. The other arguments are defined in the way described in Sec. 6.2. As an example, in the -tests/newslv code we define a new object of type mld_d_tlu_solver_type, and we +class="cmbx-12">6.2. As an pass it as follows: +class="cmbx-12">example, in the tests/newslv code we define a new object of type +amg_d_tlu_solver_type, and we pass it as follows:

     
      ! sparse matrix and preconditioner  
      type(psb_dspmat_type) :: a - 
      type(mld_dprec_type)  :: prec - 
      type(mld_d_tlu_solver_type) :: tlusv + 
      type(amg_dprec_type)  :: prec + 
      type(amg_d_tlu_solver_type) :: tlusv  
     
    ......  
      ! @@ -208,22 +222,22 @@ class="cmr-12">pass it as follows:

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    +class="cmbx-12">up]

    diff --git a/docs/html/userhtmlse8.html b/docs/html/userhtmlse8.html index 7f8b5937..87c0ebd9 100644 --- a/docs/html/userhtmlse8.html +++ b/docs/html/userhtmlse8.html @@ -11,50 +11,52 @@

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    +class="cmbx-12">up]

    8 8 Error Handling

    +class="cmbx-12">Error Handling

    The error handling in MLD2P4 is based on the PSBLAS error handling. Error +class="cmbx-12">The error handling in AMG4PSBLAS is based on the PSBLAS error conditions are signaled via an integer argument handling. Error conditions are signaled via an integer argument info; whenever an error condition is +class="cmbx-12">; detected, an error trace stack is built by the library up to the top-level, user-callable +class="cmbx-12">whenever an error condition is detected, an error trace stack is built by the routine. This routine will then decide, according to the user preferences, whether +class="cmbx-12">library up to the top-level, user-callable routine. This routine will then the error should be handled by terminating the program or by returning the +class="cmbx-12">decide, according to the user preferences, whether the error should be error condition to the user code, which will then take action, and whether +class="cmbx-12">handled by terminating the program or by returning the error condition to an error message should be printed. These options may be set by using the +class="cmbx-12">the user code, which will then take action, and whether an error message PSBLAS error handling routines; for further details see the PSBLAS User’s Guide +class="cmbx-12">should be printed. These options may be set by using the PSBLAS error +handling routines; for further details see the PSBLAS User’s Guide [[13]. +class="cmbx-12">15]. @@ -68,22 +70,22 @@ class="cmr-12">.

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    diff --git a/docs/html/userhtmlse9.html b/docs/html/userhtmlse9.html index 5d6436a1..bd832bf7 100644 --- a/docs/html/userhtmlse9.html +++ b/docs/html/userhtmlse9.html @@ -11,41 +11,47 @@

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    +class="cmbx-12">up]

    A A License

    +class="cmbx-12">License

    The MLD2P4 is freely distributable under the following copyright terms: +class="cmbx-12">DA CONTROLLARE E MODIFICARE INCLUDENDO I CREDITS A +MLD2P4 +

    The AMG4PSBLAS is freely distributable under the following copyright +terms:

    - 
                               MLD2P4  version 2.1 - 
      MultiLevel Domain Decomposition Parallel Preconditioners Package + 
                               AMG4PSBLAS  version 1.0 + 
                  Algebraic MultiGrid Preconditioners Package  
                 based on PSBLAS (Parallel Sparse BLAS version 3.5)  
     
      (C) Copyright 2008, 2010, 2012, 2015, 2017  
    - 
      Salvatore Filippone    Cranfield University, Cranfield, UK - 
      Pasqua D’Ambra         IAC-CNR, Naples, IT - 
      Daniela di Serafino    University of Campania L. Vanvitelli, Caserta, IT + 
      Pasqua D’Ambra         IAC-CNR, IT + 
      Fabio Durastante       University of Pisa and IAC-CNR, IT + 
      Salvatore Filippone    University of Rome Tor-Vergata and IAC-CNR, IT  
     
      Redistribution and use in source and binary forms, with or without  
      modification, are permitted provided that the following conditions @@ -72,7 +78,7 @@ class="cmr-12">The MLD2P4 is freely distributable under the following copyright  
      POSSIBILITY OF SUCH DAMAGE.  
    -

    +

    @@ -90,22 +96,22 @@ class="cmr-12">The MLD2P4 is freely distributable under the following copyright

    [[next] [next] [prev] [prev] [prev-tail] [prev-tail] [front] [front] [up]

    +class="cmbx-12">up]

    diff --git a/docs/html/userhtmlsu1.html b/docs/html/userhtmlsu1.html index 55b36829..aed4944e 100644 --- a/docs/html/userhtmlsu1.html +++ b/docs/html/userhtmlsu1.html @@ -36,15 +36,15 @@ class="cmbx-12">BLAS

    [1113, 1214, 1820] Many vendors provide optimized versions of BLAS; if no www.netlib.org/blas) are meant to define the standard behaviour of the BLAS interface, so they are not optimized for any +class="cmr-12">behaviour of the BLAS interface, so they are not optimized for any particular particular plaftorm, and should only be used as a last resort. Note that +class="cmr-12">platform, and should only be used as a last resort. Note that BLAS BLAS computations form a relatively small part of the MLD2P4/PSBLAS +class="cmr-12">computations form a relatively small part of the AMG4PSBLAS/PSBLAS computations; they are however critical when using preconditioners based MPI
    [1719, 2325] A version of MPI is available on most high-performance computing PSBLAS
    [1315, 1517] Parallel Sparse BLAS (PSBLAS) is available from github.com/sfilippone/psblas3; version 3.5.0 (or later) is required. +class="cmtt-12">github.com/sfilippone/psblas3 ?? Mettere tutto sotto psctoolkit; + version 3.5.0 (or later) is required. Indeed, all the prerequisites listed so far Indeed, all the prerequisites listed so far are also prerequisites of PSBLAS.
    +class="cmr-12">are also prerequisites of PSBLAS.

    Please note that the four previous libraries must have Fortran interfaces compatible with MLD2P4; usually this means that they should all be built with the same compiler as +class="cmr-12">AMG4PSBLAS; usually this means that they should all be built with the same MLD2P4. +class="cmr-12">compiler as AMG4PSBLAS. diff --git a/docs/html/userhtmlsu10.html b/docs/html/userhtmlsu10.html index 5fa35dfd..7596f920 100644 --- a/docs/html/userhtmlsu10.html +++ b/docs/html/userhtmlsu10.html @@ -11,20 +11,20 @@

    [[next] [next] [tail] [tail] [up]

    +class="cmbx-12">up]

    6.1 6.1 Method init

    +class="cmbx-12">Method init

    @@ -32,11 +32,11 @@ class="cmr-12">Method init class="cmtt-12">call p%init(icontx,ptype,info)

    This method allocates and initializes the preconditioner This method allocates and initializes the preconditioner p, according to the +class="cmbx-12">, according to the preconditioner type chosen by the user. +class="cmbx-12">preconditioner type chosen by the user.

    Arguments

    +class="cmbx-12">.

    +class="cmbx-12">. +class="cmbx-12">1. +class="td11">

    +class="cmbx-12">. +class="cmbx-12">8for +details.

    icontxt

    integer, intent(in).

    The communication context.

    The communication context.

    ptype

    character(len=*), intent(in).

    The type of preconditioner. Its values are specified in Table 

    The type of preconditioner. Its values are specified in +Table 1.

    Note that the strings are case insensitive.

    Note that the strings are case insensitive.

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    -

    For compatibility with the previous versions of MLD2P4, this method can be also +

    For compatibility with the previous versions of MLD2P4, this method invoked as follows: +class="cmbx-12">can be also invoked as follows:

    -

    -

    +

    call mld_precinit(p,ptype,info)

    +class="cmtt-12"> amg_precinit(p,ptype,info) -

    [

    [next] [next] [front] [front] [up]

    -

    up]

    +

    diff --git a/docs/html/userhtmlsu11.html b/docs/html/userhtmlsu11.html index 22267e04..2f2eb041 100644 --- a/docs/html/userhtmlsu11.html +++ b/docs/html/userhtmlsu11.html @@ -10,48 +10,49 @@ -

    [

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    +class="cmbx-12">up]

    6.2 6.2 Method set

    +class="cmbx-12">Method set
    -

    -

    +

    call p%set(what,val,info [,ilev, ilmax, pos, idx])

    -

    This method sets the parameters defining the preconditioner

    This method sets the parameters defining the preconditioner p. More precisely, the +class="cmbx-12">. More parameter identified by precisely, the parameter identified by what is assigned the value contained in is assigned the value contained in +val. -

    . +

    Arguments @@ -62,415 +63,431 @@ cellspacing="0" cellpadding="0" id="TBL-3-1">

    what

    character(len=*). +class="cmbx-12">.

    The parameter to be set. It can be specified through its name; the +class="td11">

    The parameter to be set. It can be specified through its string is case-insensitive. See Tables name; the string is case-insensitive. See Tables 2-2-8. +class="cmbx-12">8.

    val 

    integer or character(len=*) or real(psb_spk_) or +class="td11">

    integer or character(len=*) or real(psb_spk_) or real(psb_dpk_), , intent(in). +class="cmbx-12">.

    The value of the parameter to be set. The list of allowed values and +class="td11">

    The value of the parameter to be set. The list of allowed the corresponding data types is given in Tables values and the corresponding data types is given in +Tables 2-2-8. When the value -is of type 8. When the value is of type character(len=*), it is also treated as case insensitive. +class="cmbx-12">, it +is also treated as case insensitive.

    info

    integer, intent(out). +class="cmbx-12">.

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details. +class="cmbx-12">8 for +details.

    ilev

    integer, optional, intent(in). +class="cmbx-12">.

    For the multilevel preconditioner, the level at which the +class="td11">

    For the multilevel preconditioner, the level at which the preconditioner parameter has to be set. The levels are numbered +class="cmbx-12">preconditioner parameter has to be set. The levels are in increasing order starting from the finest one, i.e., level 1 is the +class="cmbx-12">numbered in increasing order starting from the finest one, finest level. If i.e., level 1 is the finest level. If ilev is not present, the parameter identified by what +class="cmbx-12">is not present, the is set at all the appropriate levels (see Tables parameter identified by what is set at all the appropriate +levels (see Tables 2-2-8). +class="cmbx-12">8).

    ilmax

    integer, optional, intent(in). +class="cmbx-12">.

    For the multilevel preconditioner, when both

    For the multilevel preconditioner, when both ilev and and +ilmax are -present, the settings are applied at all levels ilev:ilmax. When +class="cmbx-12">are present, the settings are applied at all levels ilev:ilmax. When ilev is present but is present but ilmax is not, then the default is is not, then the +default is ilmax=ilev. +class="cmbx-12">. The levels are numbered in increasing The levels are numbered in increasing order starting from the finest +class="cmbx-12">order starting from the finest one, i.e., level 1 is the finest one, i.e., level 1 is the finest level. +class="cmbx-12">level.

    pos

    charater(len=*), optional, intent(in). +class="cmbx-12">.

    Whether the other arguments apply only to the pre-smoother +class="td11">

    Whether the other arguments apply only to the (pre-smoother (’PRE’) or to the post-smoother () or to the post-smoother (’POST’). If ). +If pos is not present, +class="cmbx-12">is not present, the other arguments are applied the other arguments are applied to both smoothers. If the +class="cmbx-12">to both smoothers. If the preconditioner is one-level or preconditioner is one-level or the parameter identified by the parameter identified by what does +class="cmbx-12">does not concern the not concern the smoothers, smoothers, pos is ignored. +class="cmbx-12">is ignored.

    idx

    integer, optional, intent(in). +class="cmbx-12">.

    An auxiliary input argument that can be passed to the underlying +class="td11">

    An auxiliary input argument that can be passed to the objects. +class="cmbx-12">underlying objects. -

    For compatibility with the previous versions of MLD2P4, this method can be also +

    For compatibility with the previous versions of MLD2P4, this method invoked as follows: +class="cmbx-12">can be also invoked as follows:

    -

    -

    +

    call mld_precset(p,what,val,info)

    -

    However, in this case the optional arguments  amg_precset(p,what,val,info) +

    However, in this case the optional arguments ilev, , ilmax, , pos and idx cannot be -used.
    -

    A variety of preconditioners can be obtained by a suitable setting of the -preconditioner parameters. These parameters can be logically divided into four groups, +class="cmbx-12">and idx i.e., parameters defining +class="cmbx-12">cannot be used.
    +

    A variety of preconditioners can be obtained by a suitable setting of the +preconditioner parameters. These parameters can be logically divided into +four groups, i.e., parameters defining

    1. the type of multilevel cycle and how many cycles must be applied; +class="cmbx-12">the type of multilevel cycle and how many cycles must be applied;
    2. the aggregation algorithm; +class="cmbx-12">the aggregation algorithm;
    3. the coarse-space correction at the coarsest level (for multilevel +class="cmbx-12">the coarse-space correction at the coarsest level (for multilevel preconditioners only); +class="cmbx-12">preconditioners only);
    4. the smoother of the multilevel preconditioners, or the one-level +class="cmbx-12">the smoother of the multilevel preconditioners, or the one-level preconditioner. +class="cmbx-12">preconditioner.
    -

    A list of the parameters that can be set, along with their allowed and default values, is +

    A list of the parameters that can be set, along with their allowed and default given in Tables values, is given in Tables 2-2-8. For a description of the meaning of the parameters, please refer +class="cmbx-12">8. For a description of the meaning of the also to Section parameters, please refer also to Section 4.
    4.
    -

    Remark 2. A smoother is usually obtained by combining two objects: +

    Remark 2. A smoother is usually obtained by combining two objects: a a smoother (smoother (SMOOTHER_TYPE) and a local solver () and a local solver (SUB_SOLVE), as specified +class="cmbx-12">), as specified in in Tables Tables 7-7-8. For example, the block-Jacobi smoother using ILU(0) on the +class="cmbx-12">8. For example, the block-Jacobi smoother using ILU(0) on the blocks is obtained by combining the block-Jacobi smoother object with the +class="cmbx-12">blocks is obtained by combining the block-Jacobi smoother object with the ILU(0) solver object. Similarly, the hybrid Gauss-Seidel smoother (see Note in +class="cmbx-12">ILU(0) solver object. Similarly, the hybrid Gauss-Seidel smoother (see Note Table in Table 7) is obtained by combining the block-Jacobi smoother object with a +class="cmbx-12">7) is obtained by combining the block-Jacobi smoother object with single sweep of the Gauss-Seidel solver object, while the point-Jacobi smoother +class="cmbx-12">a single sweep of the Gauss-Seidel solver object, while the point-Jacobi is the result of combining the block-Jacobi smoother object with a single +class="cmbx-12">smoother is the result of combining the block-Jacobi smoother object sweep of the pointwise-Jacobi solver object. However, for simplicity, shortcuts +class="cmbx-12">with a single sweep of the pointwise-Jacobi solver object. However, are provided to set point-Jacobi, hybrid (forward) Gauss-Seidel, and hybrid +class="cmbx-12">for simplicity, shortcuts are provided to set point-Jacobi, hybrid backward Gauss-Seidel, i.e., the previous smoothers can be defined by setting only -(forward) Gauss-Seidel, and hybrid backward Gauss-Seidel, i.e., the +previous smoothers can be defined by setting only SMOOTHER_TYPE to appropriate values (see Tables to +appropriate values (see Tables 7), i.e., without setting 7), i.e., without setting SUB_SOLVE too. -

    The smoother and solver objects are arranged in a hierarchical manner. When +class="cmbx-12">too. +

    The smoother and solver objects are arranged in a hierarchical manner. +When specifying a smoother object, its parameters, including the specifying a smoother object, its parameters, including the local solver, are set to +class="cmbx-12">local solver, are set to their default values, and when a solver object their default values, and when a solver object is specified, its defaults are also +class="cmbx-12">is specified, its defaults are also set, overriding in both cases any +previous settings even if explicitly specified. Therefore if the user sets + + + set, overriding in both cases any previous settings even if explicitly specified. +class="cmbx-12">a smoother, and wishes to use a solver different from the default Therefore if the user sets a smoother, and wishes to use a solver different from +class="cmbx-12">one, the call to set the solver must come after the call to set the the default one, the call to set the solver must come after the call to set the +class="cmbx-12">smoother. +

    Similar considerations apply to the point-Jacobi, Gauss-Seidel and smoother. -

    Similar considerations apply to the point-Jacobi, Gauss-Seidel and block-Jacobi +class="cmbx-12">block-Jacobi coarsest-level solvers, and shortcuts are available in this case coarsest-level solvers, and shortcuts are available in this case too (see Table too (see Table 5). -
    5).
    - - - -

    Remark 3. In general, a coarsest-level solver cannot be used with both the +

    Remark 3. In general, a coarsest-level solver cannot be used with both replicated and distributed coarsest-matrix layout; therefore, setting the solver after the +class="cmbx-12">the replicated and distributed coarsest-matrix layout; therefore, setting the layout may change the layout. Similarly, setting the layout after the solver may change +class="cmbx-12">solver after the layout may change the layout. Similarly, setting the layout the solver. -

    More precisely, UMFPACK and SuperLU require the coarsest-level matrix to be +class="cmbx-12">after the solver may change the solver. +

    More precisely, UMFPACK and SuperLU require the coarsest-level replicated, while SuperLU_Dist requires it to be distributed. In these cases, setting the +class="cmbx-12">matrix to be replicated, while SuperLU_Dist requires it to be distributed. coarsest-level solver implies that the layout is redefined according to the solver, +class="cmbx-12">In these cases, setting the coarsest-level solver implies that the layout is ovverriding any previous settings. MUMPS, point-Jacobi, hybrid Gauss-Seidel and +class="cmbx-12">redefined according to the solver, ovverriding any previous settings. block-Jacobi can be applied to replicated and distributed matrices, thus their choice +class="cmbx-12">MUMPS, point-Jacobi, hybrid Gauss-Seidel and block-Jacobi can be does not modify any previously specified layout. It is worth noting that, when the +class="cmbx-12">applied to replicated and distributed matrices, thus their choice does not matrix is replicated, the point-Jacobi, hybrid Gauss-Seidel and block-Jacobi +class="cmbx-12">modify any previously specified layout. It is worth noting that, when the solvers reduce to the corresponding local solver objects (see Remark 2). For the +class="cmbx-12">matrix is replicated, the point-Jacobi, hybrid Gauss-Seidel and block-Jacobi point-Jacobi and Gauss-Seidel solvers, these objects correspond to a single +class="cmbx-12">solvers reduce to the corresponding local solver objects (see Remark 2). point-Jacobi sweep and a single Gauss-Seidel sweep, respectively, which are very poor +class="cmbx-12">For the point-Jacobi and Gauss-Seidel solvers, these objects correspond to a solvers. -

    On the other hand, the distributed layout can be used with any solver but +class="cmbxti-10x-x-120">single point-Jacobi sweep and a single Gauss-Seidel sweep, respectively, UMFPACK and SuperLU; therefore, if any of these two solvers has already been +class="cmbx-12">which are very poor solvers. +

    On the other hand, the distributed layout can be used with any solver selected, the coarsest-level solver is changed to block-Jacobi, with the previously +class="cmbx-12">but UMFPACK and SuperLU; therefore, if any of these two solvers chosen solver applied to the local blocks. Likewise, the replicated layout can be +class="cmbx-12">has already been selected, the coarsest-level solver is changed to used with any solver but SuperLu_Dist; therefore, if SuperLu_Dist has been +class="cmbx-12">block-Jacobi, with the previously chosen solver applied to the local previously set, the coarsest-level solver is changed to the default sequential +class="cmbx-12">blocks. Likewise, the replicated layout can be used with any solver solver. -

    Remark 4. The argument but SuperLu_Dist; therefore, if SuperLu_Dist has been previously +set, the coarsest-level solver is changed to the default sequential +solver. +

    Remark 4. The argument idx can be used to allow finer control for those solvers; +class="cmbx-12">can be used to allow finer control for those for instance, by specifying the keyword solvers; for instance, by specifying the keyword MUMPS_IPAR_ENTRY and an appropriate value +class="cmbx-12">and an for appropriate value for idx, it is possible to set any entry in the MUMPS integer control array. See also +class="cmbx-12">, it is possible to set any entry in the MUMPS Sec. integer control array. See also Sec. 7. +class="cmbx-12">7.

    -

    @@ -478,7 +495,7 @@ class="cmr-12">.
    -

    +

    +class="td11">

    1

    what

    		type

    val

    deault

    cots

    ’ML_CYCLE’

    		character(len=*)

    ’VCYCLE’ -

    ’WCYCLE’ -

    ’KCYCLE’ -

    ’MULT’ -

    ’ADD’

    ’VCYCLE’

    Multilevel cycle: V-cycle, W-cycle, K-cycle, +class="td11">

    Multilevel cycle: V-cycle, W-cycle, K-cycle, hybrid Multiplicative Schwarz, and Additive Schwarz. -

    Note that hybrid Multiplicative Schwarz +

    Note that hybrid Multiplicative Schwarz is equivalent to V-cycle and is included for compatibility with previous versions of MLD2P4.

    ’OUTER_SWEEPS’

    		integer

    Any integer -

    number

    Any integer +

    number ≥ 1

    1

    Number of multilevel cycles.

    Number of multilevel cycles.

    -

    @@ -584,7 +601,7 @@ applied.
    -

    +

    +than or equal to this threshold (see Note). +

    +href="userhtmlli4.html#XGrHeJi:16">18]. Parameters defining the aggregation algorithm.

    @@ -774,7 +792,7 @@ class="content">Parameters defining the aggregation algorithm.

    +

    what

    		type

    val

    deault

    cos

    ’MIN_COARSE_SIZE’

    		integer

    Any number -

    Any number +

    > 0

    ⌊40√ -- 3 nn is the dimension of the matrix at the finest level

    Coarse size threshold. The aggregation +class="td11">

    Coarse size threshold. The aggregation stops if the global number of variables of the computed coarsest matrix is lower -than or equal to this threshold (see Note).

    ’MIN_CR_RATIO’

    		real

    Any number -

    Any number +

    > 1

    1.5

    Minimum +class="td11">

    1.5

    Minimum coarsening ratio. The aggregation stops if the ratio between the matrix dimensions at two consecutive levels is lower than or @@ -670,32 +688,32 @@ equal to this threshold (see Note).

    ’MAX_LEVS’

    		integer

    Any integer -

    number

    Any integer +

    number > 1

    20

    Maximum number of levels. The +class="td11">

    20

    Maximum number of levels. The aggregation stops if the number of levels reaches this value (see Note).

    ’PAR_AGGR_ALG’

    		character(len=*)

    ’DEC’, ’SYMDEC’

    ’DEC’

    Parallel aggregation algorithm. -

    Currently, only the decoupled +class="td11">

    Parallel aggregation algorithm. +

    Currently, only the decoupled aggregation (DEC) is available; the SYMDEC @@ -707,36 +725,36 @@ class="cmmi-8">T .

    ’AGGR_TYPE’

    		character(len=*)

    ’SOC1’

    ’SOC1’, ’SOC2’

    Type of aggregation algorithm: currently, +class="td11">

    Type of aggregation algorithm: currently, we implement to measures of strength of connection, the one by Vaněk, Mandel and Brezina [26], and the one by Gratton +href="userhtmlli4.html#XVANEK_MANDEL_BREZINA">28], and the one by Gratton et al [16].

    ’AGGR_PROL’

    		character(len=*)

    ’SMOOTHED’, ’UNSMOOTHED’

    ’SMOOTHED’

    Prolongator used by the aggregation +class="td11">

    Prolongator used by the aggregation algorithm: smoothed or unsmoothed (i.e., tentative prolongator).

    < style="vertical-align:baseline;" id="TBL-6-5-">
    Note. Different thresholds at different levels, such as those used in [26, Section 5.1], can be easily set by invoking the rou-
    +href="userhtmlli4.html#XVANEK_MANDEL_BREZINA">28, Section 5.1], can be easily set by invoking the rou-
    tine Parameters defining the aggregation algorithm (continued).

    @@ -910,7 +928,7 @@ class="content">Parameters defining the aggregation algorithm (continued). -

    +

    what

    		type

    val

    deault

    cots

    ’AGGR_ORD’

    		character(len=*)

    ’NATURAL’ -

    ’DEGREE’

    ’NATURAL’

    Initial ordering of indices for the +class="td11">

    Initial ordering of indices for the aggregation algorithm: either natural ordering or sorted by descending degrees of the nodes in the matrix @@ -838,20 +856,20 @@ graph.

    ’AGGR_THRESH’

    		real(kind_parameter)

    Any real -

    number 

    Any real +

    number ∈ [0,1]

    0.01

    The threshold

    0.01

    The threshold θ in the aggregation algorithm, see (3) in Section 

    ’AGGR_FILTER’

    		character(len=*)

    ’FILTER’ -

    ’NOFILTER’

    ’NOFILTER’

    Matrix used in computing the smoothed +class="td11">

    Matrix used in computing the smoothed prolongator: filtered or unfiltered (see (5) in Section 

    what

    		type

    val

    deault

    cots

    ’COARSE_MAT’

    		character(len=*)

    ’DIST’ -

    ’REPL’

    ’REPL’

    Coarsest matrix layout: distributed among the +class="td11">

    Coarsest matrix layout: distributed among the processes or replicated on each of them.

    ’COARSE_SOLVE’

    		character(len=*)

    ’MUMPS’ -

    ’UMF’ -

    ’SLU’ -

    ’SLUDIST’ -

    ’JACOBI’ -

    ’GS’ -

    ’BJAC’

    See Note.

    Solver used at the coarsest level: sequential LU +class="td11">

    See Note.

    Solver used at the coarsest level: sequential LU from MUMPS, UMFPACK, or SuperLU (plus triangular solve); distributed LU from MUMPS or SuperLU_Dist (plus triangular solve); point-Jacobi, hybrid Gauss-Seidel or block-Jacobi. -

    Note that

    Note that UMF and SLU require the coarsest matrix to be replicated,

    ’COARSE_SUBSOLVE’

    		character(len=*)

    ’ILU’ -

    ’ILUT’ -

    ’MILU’ -

    ’MUMPS’ -

    ’SLU’ -

    ’UMF’

    See Note.

    Solver for the diagonal blocks of the coarse matrix, +class="td11">

    See Note.

    Solver for the diagonal blocks of the coarse matrix, in case the block Jacobi solver is chosen as coarsest-level solver: ILU(p), ILU(Parameters defining the coarse-space correction at the coarsest -

    @@ -1087,7 +1105,7 @@ class="content">Parameters defining the coarse-space correction at the coarsest
    -

    +

    @@ -1194,7 +1212,7 @@ class="content">Parameters defining the coarse-space correction at the co -

    @@ -1202,7 +1220,7 @@ class="content">Parameters defining the coarse-space correction at the co
    -

    +

    what

    		type

    val

    deault

    cots

    ’COARSE_SWEEPS’

    		integer

    Any integer -

    number

    Any integer +

    number > 0

    10

    Number of sweeps when

    10

    Number of sweeps when JACOBI, GS or BJAC @@ -1149,31 +1167,31 @@ is chosen as coarsest-level solver.

    ’COARSE_FILLIN’

    		integer

    Any integer -

    number

    Any integer +

    number ≥ 0

    0

    Fill-in level

    0

    Fill-in level p of the ILU factorizations.

    ’COARSE_ILUTHRS’

    		real(kind_parameter)

    Any real -

    number

    Any real +

    number ≥ 0

    0

    Drop tolerance

    0

    Drop tolerance t in the ILU(p,t) factorization.

    @@ -1416,7 +1434,7 @@ preconditioner. -

    @@ -1424,7 +1442,7 @@ preconditioner.
    -

    +

    what

    		type

    val

    deault

    cots

    ’SMOOTHER_TYPE’

    		character(len=*)

    ’JACOBI’ -

    ’GS’ -

    ’BGS’ -

    ’BJAC’ -

    ’AS’

    ’FBGS’

    Type of smoother used in the multilevel preconditioner: point-Jacobi, hybrid @@ -1274,34 +1292,34 @@ class="cmr-10">(forward) Gauss-Seidel, hybrid backward class="cmr-10">Gauss-Seidel, block-Jacobi, and Additive Schwarz. -

    It is ignored by one-level preconditioners.

    ’SUB_SOLVE’

    		character(len=*)

    ’JACOBI’ -

    ’GS’ -

    ’BGS’ -

    ’ILU’ -

    ’ILUT’ -

    ’MILU’ -

    ’MUMPS’ -

    ’SLU’ -

    ’UMF’

    GS and BGS multilevel class="cmr-10">preconditioners, respectively -

    ILU for block-Jacobi and Additive Schwarz class="cmr-10">one-level preconditioners

    The local solver to be used with the smoother or one-level preconditioner (see @@ -1350,20 +1368,20 @@ class="cmr-10">Note for details on hybrid Gauss-Seidel.

    ’SMOOTHER_SWEEPS’

    		integer

    Any integer -

    number ≥ 0

    1

    Number of sweeps of the smoother or one-level preconditioner. In the multilevel @@ -1382,20 +1400,20 @@ class="cmr-10">respectively.

    ’SUB_OVR’

    		integer

    Any integer -

    number ≥ 0

    1

    Number of overlap layers, for Additive Schwarz only.

    what

    		type

    val

    deault

    cots

    ’SUB_RESTR’

    		character(len=*)

    ’HALO’ -

    ’NONE’

    ’HALO’

    Type of restriction operator, for Additive Schwarz only: for taking into account the class="cmr-10">overlap, NONE for neglecting it. -

    Note that HALO must be chosen for the classical @@ -1501,17 +1519,17 @@ class="cmr-10">variant.

    ’SUB_PROL’

    		character(len=*)

    ’SUM’ -

    ’NONE’

    ’NONE’

    Type of prolongation operator, for Additive Schwarz only: for adding the contributions class="cmr-10">from the overlap, NONE for neglecting them. -

    Note that SUM must be chosen for the classical @@ -1534,20 +1552,20 @@ class="cmr-10">RAS variant.

    ’SUB_FILLIN’

    		integer

    Any integer -

    number ≥ 0

    0

    Fill-in level p of the incomplete LU @@ -1556,23 +1574,23 @@ class="cmr-10">factorizations. <

    ’SUB_ILUTHRS’

    		real(kind_parameter)

    Any real number ≥ 0

    0

    Drop tolerance t in the ILU(p,t) factorization.

    ’MUMPS_LOC_GLOB’

    		character(len=*)

    LOCAL_SOLVER’ -

    GLOBAL_SOLVER’

    GLOBAL_SOLVER’

    Whether MUMPS should be used as a distributed solver, or as a serial solver acting @@ -1603,17 +1621,17 @@ class="cmr-10">only on the part of the matrix local to each class="cmr-10">process.

    ’MUMPS_IPAR_ENTRY’

    		integer

    Any integer number

    0

    Set an entry in the MUMPS integer control array, as chosen via the idx optional argument.

    ’MUMPS_RPAR_ENTRY’

    		real

    Any real number

    0

    Set an entry in the MUMPS real control array, as chosen via the Parameters defining the smoother or the details of the one-level -

    [

    [next] [next] [prev] [prev] [prev-tail] [prev-tail] [front] [front] [up]

    -

    up]

    +

    diff --git a/docs/html/userhtmlsu12.html b/docs/html/userhtmlsu12.html index d703caee..345ac4b2 100644 --- a/docs/html/userhtmlsu12.html +++ b/docs/html/userhtmlsu12.html @@ -10,48 +10,48 @@ -

    [

    [next] [next] [prev] [prev] [prev-tail] [prev-tail] [tail] [tail] [up]

    +class="cmbx-12">up]

    6.3 6.3 Method hierarchy_build

    +class="cmbx-12">Method hierarchy_build
    -

    -

    +

    call p%hierarchy_build(a,desc_a,info)

    -

    This method builds the hierarchy of matrices and restriction/prolongation operators for +

    This method builds the hierarchy of matrices and restriction/prolongation the multilevel preconditioner operators for the multilevel preconditioner p, according to the requirements made by the user +class="cmbx-12">, according to the requirements through the methods made by the user through the methods init and and set. -

    . +

    Arguments

    +class="cmbx-12">. +class="cmbx-12">15]. +class="cmbx-12">. +class="cmbx-12">15]. +class="cmbx-12">. +class="cmbx-12">8for +details.

    a

    type(psb_xxspmat_type), intent(in).

    The sparse matrix structure containing the local part of the matrix +class="td11">

    The sparse matrix structure containing the local part to be preconditioned. Note that x must be chosen according to +class="cmbx-12">of the matrix to be preconditioned. Note that x must the real/complex, single/double precision version of MLD2P4 under +class="cmbx-12">be chosen according to the real/complex, single/double use. See the PSBLAS User’s Guide for details [precision version of MLD2P4 under use. See the PSBLAS +User’s Guide for details [13].

    desc_a

    type(psb_desc_type), intent(in).

    The communication descriptor of

    The communication descriptor of a. See the PSBLAS User’s Guide +class="cmbx-12">. See the PSBLAS for details [User’s Guide for details [13].

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    @@ -133,23 +137,23 @@ class="td11">
    -

    [

    [next] [next] [prev] [prev] [prev-tail] [prev-tail] [front] [front] [up]

    -

    up]

    +

    diff --git a/docs/html/userhtmlsu13.html b/docs/html/userhtmlsu13.html index 70382613..d23e9e36 100644 --- a/docs/html/userhtmlsu13.html +++ b/docs/html/userhtmlsu13.html @@ -10,54 +10,58 @@ -

    [

    [next] [next] [prev] [prev] [prev-tail] [prev-tail] [tail] [tail] [up]

    +class="cmbx-12">up]

    6.4 6.4 Method smoothers_build

    +class="cmbx-12">Method smoothers_build
    -

    -

    +

    call p%smoothers_build(a,desc_a,p,info[,amold,vmold,imold])

    -

    This method builds the smoothers and the coarsest-level solvers for the multilevel +

    This method builds the smoothers and the coarsest-level solvers for the preconditioner multilevel preconditioner p, according to the requirements made by the user through the methods -, according to the requirements made +by the user through the methods init and and set, and based on the aggregation hierarchy produced by a previous call to -, and based on the +aggregation hierarchy produced by a previous call to hierarchy_build (see Section (see +Section 6.3). -

    6.3). +

    Arguments

    +class="cmbx-12">. +class="cmbx-12">15]. +class="cmbx-12">. +class="cmbx-12">15]. +class="cmbx-12">. +class="cmbx-12">8for +details. +class="cmbx-12">. +class="cmbx-12">15]. +class="cmbx-12">. +class="cmbx-12">this allows e.g. running on GPUs. +class="cmbx-12">. +class="cmbx-12">components; this allows e.g. running on GPUs.

    a

    type(psb_xxspmat_type), intent(in).

    The sparse matrix structure containing the local part of the matrix +class="td11">

    The sparse matrix structure containing the local part +of the matrix to be preconditioned. Note that x must to be preconditioned. Note that x must be chosen according to +class="cmbx-12">be chosen according to the real/complex, single/double the real/complex, single/double precision version of MLD2P4 under +class="cmbx-12">precision version of MLD2P4 under use. See the PSBLAS use. See the PSBLAS User’s Guide for details [User’s Guide for details [13].

    desc_a

    type(psb_desc_type), intent(in).

    The communication descriptor of

    The communication descriptor of a. See the PSBLAS User’s Guide +class="cmbx-12">. See the PSBLAS for details [User’s Guide for details [13].

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    amold

    class(psb_xx_base_sparse_mat), intent(in), optional.

    The desired dynamic type for internal matrix components; this +class="td11">

    The desired dynamic type for internal matrix components; allows e.g. running on GPUs; it needs not be the same on all +class="cmbx-12">this allows e.g. running on GPUs; it needs not be the same processes. See the PSBLAS User’s Guide for details [on all processes. See the PSBLAS User’s Guide for details +[13].

    vmold

    class(psb_xx_base_vect_type), intent(in), optional.

    The desired dynamic type for internal vector components; this +class="td11">

    The desired dynamic type for internal vector components; allows e.g. running on GPUs.

    imold

    class(psb_i_base_vect_type), intent(in), optional.

    The desired dynamic type for internal integer vector components; +class="td11">

    The desired dynamic type for internal integer vector this allows e.g. running on GPUs.

    @@ -198,23 +207,26 @@ class="td11">
    -

    [

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    up]

    +

    diff --git a/docs/html/userhtmlsu14.html b/docs/html/userhtmlsu14.html index 6e534a23..88966c45 100644 --- a/docs/html/userhtmlsu14.html +++ b/docs/html/userhtmlsu14.html @@ -10,236 +10,243 @@ -

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    6.5 6.5 Method build

    +class="cmbx-12">Method build
    -

    -

    +

    call p%build(a,desc_a,info[,amold,vmold,imold])

    -

    This method builds the preconditioner

    This method builds the preconditioner p according to the requirements made by the +class="cmbx-12">according to the requirements user through the methods made by the user through the methods init and and set (see Sections (see Sections 6.3 and 6.3 +and 6.4 for multilevel +class="cmbx-12">6.4 for multilevel preconditioners). It is mostly provided for backward preconditioners). It is mostly provided for backward compatibility; indeed, it is +class="cmbx-12">compatibility; indeed, it is internally implemented by invoking the internally implemented by invoking the two previous methods two previous methods hierarchy_build and -and smoothers_build, whose nomenclature would however be somewhat unnatural when +class="cmbx-12">, whose +nomenclature would however be somewhat unnatural when dealing with dealing with simple one-level preconditioners. -

    simple one-level preconditioners. +

    Arguments + + +

    +class="cmbx-12">. +class="cmbx-12">15]. +class="cmbx-12">. +class="cmbx-12">15]. +class="cmbx-12">. +class="cmbx-12">8for +details. +class="cmbx-12">. +class="cmbx-12">15]. +class="cmbx-12">. +class="cmbx-12">this allows e.g. running on GPUs. +class="cmbx-12">. +class="cmbx-12">components; this allows e.g. running on GPUs.

    a

    type(psb_xxspmat_type), intent(in).

    The sparse matrix structure containing the local part of the matrix +class="td11">

    The sparse matrix structure containing the local part to be preconditioned. Note that x must be chosen according to +class="cmbx-12">of the matrix to be preconditioned. Note that x must the real/complex, single/double precision version of MLD2P4 under +class="cmbx-12">be chosen according to the real/complex, single/double use. See the PSBLAS User’s Guide for details [precision version of MLD2P4 under use. See the PSBLAS +User’s Guide for details [13].

    desc_a

    type(psb_desc_type), intent(in).

    The communication descriptor of

    The communication descriptor of a. See the PSBLAS User’s Guide +class="cmbx-12">. See the PSBLAS for details [User’s Guide for details [13].

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    amold

    class(psb_xx_base_sparse_mat), intent(in), optional.

    The desired dynamic type for internal matrix components; this +class="td11">

    The desired dynamic type for internal matrix components; allows e.g. running on GPUs; it needs not be the same on all +class="cmbx-12">this allows e.g. running on GPUs; it needs not be the same processes. See the PSBLAS User’s Guide for details [on all processes. See the PSBLAS User’s Guide for details +[13].

    vmold

    class(psb_xx_base_vect_type), intent(in), optional.

    The desired dynamic type for internal vector components; this +class="td11">

    The desired dynamic type for internal vector components; allows e.g. running on GPUs.

    imold

    class(psb_i_base_vect_type), intent(in), optional.

    The desired dynamic type for internal integer vector components; +class="td11">

    The desired dynamic type for internal integer vector this allows e.g. running on GPUs.

    - - - -

    For compatibility with the previous versions of MLD2P4, this method can be also +

    For compatibility with the previous versions of MLD2P4, this method invoked as follows: +class="cmbx-12">can be also invoked as follows:

    -

    -

    +

    call mld_precbld(p,what,val,info[,amold,vmold,imold])

    -

    The method can be used to build multilevel preconditioners too. +class="cmtt-12"> amg_precbld(p,what,val,info[,amold,vmold,imold])

    +

    The method can be used to build multilevel preconditioners too. -

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    +

    diff --git a/docs/html/userhtmlsu15.html b/docs/html/userhtmlsu15.html index 4630b72a..71fdaa57 100644 --- a/docs/html/userhtmlsu15.html +++ b/docs/html/userhtmlsu15.html @@ -10,37 +10,37 @@ -

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    6.6 6.6 Method apply

    +class="cmbx-12">Method apply
    -

    -

    +

    call p%apply(x,y,desc_a,info [,trans,work])

    -

    This method computes

    This method computes y = op-1) x, where , where B is a previously built preconditioner, +class="cmbx-12">is a previously built stored into preconditioner, stored into p, and , and op denotes the preconditioner itself or its transpose, according to the +class="cmbx-12">denotes the preconditioner itself or its value of transpose, according to the value of trans. Note that, when MLD2P4 is used with a Krylov solver from PSBLAS, -. Note that, when MLD2P4 is +used with a Krylov solver from PSBLAS, p%apply is called within the PSBLAS method is called within the +PSBLAS method psb_krylov and hence it is completely +class="cmbx-12">and hence it is completely transparent to the transparent to the user. -

    user. +

    Arguments + + +

    +class="cmbx-12">. +class="cmbx-12">under use. +class="cmbx-12">. +class="cmbx-12">under use. +class="cmbx-12">. +class="cmbx-12">be preconditioned. +class="cmbx-12">. +class="cmbx-12">8for +details. +class="cmr-12">). +class="cmbx-12">. +class="cmbx-12">precision version of MLD2P4 under use.

    x

    type

    type(kind_parameterkind_parameter), dimension(:), intent(in).

    The local part of the vector

    The local part of the vector x. Note that type and kind_parameter +class="cmbx-12">. Note that type +and kind_parameter must be chosen according to the must be chosen according to the real/complex, single/double +class="cmbx-12">real/complex, single/double precision version of MLD2P4 precision version of MLD2P4 under use.

    y

    type

    type(kind_parameterkind_parameter), dimension(:), intent(out).

    The local part of the vector

    The local part of the vector y. Note that type and kind_parameter +class="cmbx-12">. Note that type +and kind_parameter must be chosen according to the must be chosen according to the real/complex, single/double +class="cmbx-12">real/complex, single/double precision version of MLD2P4 precision version of MLD2P4 under use.

    desc_a

    type(psb_desc_type), intent(in).

    The communication descriptor associated to the matrix to be +class="td11">

    The communication descriptor associated to the matrix to preconditioned.

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    trans

    character(len=1), optional, intent(in).

    If

    If trans = = ’N’,’n’ then then op(B-1) = ) = B-1; if ; if trans = = ’T’,’t’ then then op(B) = B-T (transpose of (transpose of B-1); if ); if trans = ’C’,’c’ then +class="cmbx-12">= ’C’,’c’ then op(B) = B-C (conjugate transpose of (conjugate transpose of B-1).

    work

    type

    type(kind_parameterkind_parameter), dimension(:), optional, target.

    Workspace. Its size should be at least

    Workspace. Its size should be at +least 4 * psb_cd_get_local_ - psb_cd_get_local_ cols(desc_a) (see the PSBLAS User’s Guide). Note that type +class="cmbx-12">(see the PSBLAS +User’s Guide). Note that type and kind_parameter must and kind_parameter must be chosen according to the real/complex, +class="cmbx-12">be chosen according to the real/complex, single/double single/double precision version of MLD2P4 under use.

    - - - -

    For compatibility with the previous versions of MLD2P4, this method can be also +

    For compatibility with the previous versions of MLD2P4, this method invoked as follows: +class="cmbx-12">can be also invoked as follows:

    -

    -

    +

    call mld_precaply(p,what,val,info)

    +class="cmtt-12"> amg_precaply(p,what,val,info) -

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    6.7 6.7 Method free

    +class="cmbx-12">Method free
    -

    -

    +

    call p%free(p,info)

    -

    This method deallocates the preconditioner data structure

    This method deallocates the preconditioner data structure p. -

    . +

    Arguments

    +class="cmbx-12">. +class="cmbx-12">for details.

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for +class="cmbx-12">8 details.

    -

    For compatibility with the previous versions of MLD2P4, this method can be also +

    For compatibility with the previous versions of MLD2P4, this method invoked as follows: +class="cmbx-12">can be also invoked as follows:

    -

    -

    +

    call mld_precfree(p,info)

    +class="cmtt-12"> amg_precfree(p,info) -

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    +

    diff --git a/docs/html/userhtmlsu17.html b/docs/html/userhtmlsu17.html index 06df7c92..d3f1bbda 100644 --- a/docs/html/userhtmlsu17.html +++ b/docs/html/userhtmlsu17.html @@ -10,50 +10,50 @@ -

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    6.8 6.8 Method descr

    +class="cmbx-12">Method descr
    -

    -

    +

    call p%descr(info, [iout])

    -

    This method prints a description of the preconditioner

    This method prints a description of the preconditioner p to the standard output or to a +class="cmbx-12">to the standard file. It must be called after output or to a file. It must be called after hierachy_build and and smoothers_build, or build, have +class="cmbx-12">, been called. -

    or build, have been called. +

    Arguments

    +class="cmbx-12">. +class="cmbx-12">8for +details. +class="cmbx-12">. +class="cmbx-12">be printed; the default is the standard output.

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    iout

    integer, intent(in), optional.

    The id of the file where the preconditioner description will be +class="td11">

    The id of the file where the preconditioner description will printed; the default is the standard output.

    -

    For compatibility with the previous versions of MLD2P4, this method can be also +

    For compatibility with the previous versions of MLD2P4, this method invoked as follows: +class="cmbx-12">can be also invoked as follows:

    -

    -

    +

    call mld_precdescr(p,info amg_precdescr(p,info [,iout])

    -

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    up]

    +

    diff --git a/docs/html/userhtmlsu18.html b/docs/html/userhtmlsu18.html index fe5a234e..92a43874 100644 --- a/docs/html/userhtmlsu18.html +++ b/docs/html/userhtmlsu18.html @@ -10,43 +10,43 @@ -

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    +class="cmbx-12">up]

    6.9 6.9 Auxiliary Methods

    -

    Various functionalities are implemented as additional methods of the preconditioner +class="cmbx-12">Auxiliary Methods +

    Various functionalities are implemented as additional methods of the object. -

    +class="cmbx-12">preconditioner object. +

    6.9.1 6.9.1 Method: dump
    +class="cmbx-12">Method: dump
    -

    -

    +

    call p%dump(info[,istart,iend,prefix,head,ac,rp,smoother,solver,global_num])

    -

    Dump on file. -

    Dump on file. +

    Arguments

    +class="cmbx-12">. +class="cmbx-12">8for +details. +class="cmbx-12">. +class="cmbx-12">15].

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    amold

    class(psb_xx_base_sparse_mat), intent(in), optional.

    The desired dynamic type for internal matrix components; this +class="td11">

    The desired dynamic type for internal matrix components; allows e.g. running on GPUs; it needs not be the same on all +class="cmbx-12">this allows e.g. running on GPUs; it needs not be the same processes. See the PSBLAS User’s Guide for details [on all processes. See the PSBLAS User’s Guide for details +[13].

    -

    +

    6.9.2 6.9.2 Method: clone
    +class="cmbx-12">Method: clone
    -

    -

    +

    call p%clone(pout,info)

    -

    Create a (deep) copy of the preconditioner object. -

    Create a (deep) copy of the preconditioner object. +

    Arguments

    +class="cmbx-12">. +class="cmbx-12">single/double precision version of MLD2P4 under use. +class="cmbx-12">. +class="cmbx-12">8for +details.

    pout

    type(mld_x

    type(amg_xprec_type), intent(out).

    The copy of the preconditioner data structure. Note that x must +class="td11">

    The copy of the preconditioner data structure. Note be chosen according to the real/complex, single/double precision +class="cmbx-12">that x must be chosen according to the real/complex, version of MLD2P4 under use.

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    -

    +

    6.9.3 6.9.3 Method: sizeof
    +class="cmbx-12">Method: sizeof
    -

    -

    +

    sz =  p%sizeof()

    -

    Return memory footprint in bytes. -

    +

    Return memory footprint in bytes. +

    6.9.4 6.9.4 Method: allocate_wrk
    +class="cmbx-12">Method: allocate_wrk
    -

    -

    +

    call p%allocate_wrk(info[, vmold])

    -

    Allocate internal work vectors. Each application of the preconditioner uses a number of +

    Allocate internal work vectors. Each application of the preconditioner work vectors which are allocated internally as necessary; therefore allocation and +class="cmbx-12">uses a number of work vectors which are allocated internally as deallocation of memory occurs multiple times during the execution of a Krylov method. +class="cmbx-12">necessary; therefore allocation and deallocation of memory occurs In most cases this strategy is perfectly acceptable, but on some platforms, most +class="cmbx-12">multiple times during the execution of a Krylov method. In most cases notably GPUs, memory allocation is a slow operation, and the default behaviour would +class="cmbx-12">this strategy is perfectly acceptable, but on some platforms, most lead to a slowdown. This method allows to trade space for time by preallocating +class="cmbx-12">notably GPUs, memory allocation is a slow operation, and the default the internal workspace outside of the invocation of a Krylov method. When +class="cmbx-12">behaviour would lead to a slowdown. This method allows to trade using GPUs or other specialized devices, the vmold argument is also necessary -to ensure the internal work vectors are of the appropriate dynamic type to +class="cmbx-12">space for time by preallocating the internal workspace outside of the exploit the accelerator hardware; when allocation occurs internally this is +class="cmbx-12">invocation of a Krylov method. When using GPUs or other specialized +devices, the vmold argument is also necessary to ensure the internal +work vectors are of the appropriate dynamic type to exploit the +accelerator hardware; when allocation occurs internally this is taken taken care of based on the dynamic type of the care of based on the dynamic type of the x argument to the argument to the apply method. -

    method. +

    Arguments

    +class="cmbx-12">. +class="cmbx-12">8for +details. +class="cmbx-12">. +class="cmbx-12">this allows e.g. running on GPUs.

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    vmold

    class(psb_xx_base_vect_type), intent(in), optional.

    The desired dynamic type for internal vector components; this +class="td11">

    The desired dynamic type for internal vector components; allows e.g. running on GPUs.

    -

    +

    6.9.5 6.9.5 Method: free_wrk
    +class="cmbx-12">Method: free_wrk
    -

    -

    +

    call p%free_wrk(info)

    -

    Deallocate internal work vectors. -

    Deallocate internal work vectors. +

    Arguments

    +class="cmbx-12">. +class="cmbx-12">8for +details.

    info

    integer, intent(out).

    Error code. If no error, 0 is returned. See Section 

    Error code. If no error, 0 is returned. See Section 8 for details.

    @@ -318,19 +331,19 @@ class="td11">

    [[prev] [prev] [prev-tail] [prev-tail] [front] [front] [up]

    +class="cmbx-12">up]

    diff --git a/docs/html/userhtmlsu2.html b/docs/html/userhtmlsu2.html index edfab22b..67c015dc 100644 --- a/docs/html/userhtmlsu2.html +++ b/docs/html/userhtmlsu2.html @@ -44,7 +44,7 @@ class="cmbx-12">UMFPACK

    [911] A sparse LU factorization package included in the SuiteSparse SuperLU
    [1012] A sparse LU factorization package available from _Dist
    [1921] A sparse LU factorization package available from the same 3.3 Configuration options

    In order to build MLD2P4, the first step is to use the In order to build AMG4PSBLAS, the first step is to use the configure script in the main +class="cmr-12">script in the directory to generate the necessary makefile. -

    As a minimal example consider the following: +class="cmr-12">main directory to generate the necessary makefile. +

    DA RISCRIVERE +

    As a minimal example consider the following:

    ./configure --with-psblas=PSB-INSTALL-DIR
    -

    which assumes that the various MPI compilers and support libraries are available in +

    which assumes that the various MPI compilers and support libraries are the standard directories on the system, and specifies only the PSBLAS install directory +class="cmbx-12">available in the standard directories on the system, and specifies only (note that the latter directory must be specified with an absolute path). The full set of +class="cmbx-12">the PSBLAS install directory (note that the latter directory must options may be looked at by issuing the command be specified with an absolute path). The full set of options may +be looked at by issuing the command ./configure --help, which +class="cmbx-12">, which produces: +class="cmbx-12">produces: @@ -237,18 +241,17 @@ class="cmr-12">produces:  
     
    Report bugs to <https://github.com/sfilippone/mld2p4-2/issues>.

    -

    -

    For instance, if a user has built and installed PSBLAS 3.5 under the /opt directory +

    +

    For instance, if a user has built and installed PSBLAS 3.5 under the +/opt directory and is using the SuiteSparse package (which includes and is using the SuiteSparse package (which includes UMFPACK), then MLD2P4 -might be configured with: +class="cmbx-12">UMFPACK), then MLD2P4 might be configured with: @@ -256,72 +259,74 @@ class="cmr-12">might be configured with:  ./configure --with-psblas=/opt/psblas-3.5/ \  
     --with-umfpackincdir=/usr/include/suitesparse/

    -

    Once the configure script has completed execution, it will have generated the file -

    Once the configure script has completed execution, it will have generated +the file Make.inc which will then be used by all Makefiles in the directory tree; this file will be +class="cmbx-12">which will then be used by all Makefiles in the directory copied in the install directory under the name tree; this file will be copied in the install directory under the name +Make.inc.MLD2P4. -

    To use the MUMPS solver package, the user has to add the appropriate options to +class="cmbx-12">. +

    To use the MUMPS solver package, the user has to add the appropriate the configure script; by default we are looking for the libraries -ldmumps -lsmumps +class="cmbx-12">options to the configure script; by default we are looking for the libraries -ldmumps -lsmumps  -lzmumps -lcmumps -mumps_common -lpord. MUMPS often uses additional +class="cmbx-12">. MUMPS often packages such as ScaLAPACK, ParMETIS, SCOTCH, as well as enabling OpenMP; in +class="cmbx-12">uses additional packages such as ScaLAPACK, ParMETIS, SCOTCH, as such cases it is necessary to add linker options with the --with-extra-libs configure +class="cmbx-12">well as enabling OpenMP; in such cases it is necessary to add linker options option. -

    To build the library the user will now enter +class="cmbx-12">with the --with-extra-libs configure option. +

    To build the library the user will now enter

    make
    -

    followed (optionally) by +

    followed (optionally) by

    make install
    -

    +

    -

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    up]

    +

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    +class="cmbx-12">up]

    3.4 3.4 Bug reporting

    -

    If you find any bugs in our codes, please report them through our issues page +class="cmbx-12">Bug reporting +

    If you find any bugs in our codes, please report them through our issues on
    page on
    https://github.com/sfilippone/mld2p4-2/issues
    To enable us to track the bug, please provide a log from the failing application, the +href="https://github.com/psctoolkit/amg4psblas/issues" class="url" >https://github.com/psctoolkit/amg4psblas/issues
    +

    To enable us to track the bug, please provide a log from the failing test conditions, and ideally a self-contained test program reproducing the +class="cmbx-12">application, the test conditions, and ideally a self-contained test program issue. +class="cmbx-12">reproducing the issue. -

    [

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    -

    up]

    +

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    +class="cmbx-12">up]

    3.5 3.5 Example and test programs

    -

    The package contains the Example and test programs +

    The package contains the examples and and tests directories; both of them are +class="cmbx-12">directories; both of them are further divided into further divided into fileread and and pdegen subdirectories. Their purpose is as +class="cmbx-12">subdirectories. Their purpose is as follows: +class="cmbx-12">follows:

    examples
    contains a set of simple example programs with a predefined choice of +class="cmbx-12">contains a set of simple example programs with a predefined preconditioners, selectable via integer values. These are intended to get an +class="cmbx-12">choice of preconditioners, selectable via integer values. These acquaintance with the multilevel preconditioners available in MLD2P4. +class="cmbx-12">are intended to get an acquaintance with the multilevel + preconditioners available in AMG4PSBLAS.
    tests
    contains a set of more sophisticated examples that will allow the user, via +class="cmbx-12">contains a set of more sophisticated examples that will allow the the input files in the user, via the input files in the runs subdirectories, to experiment with the full range +class="cmbx-12">subdirectories, to experiment of preconditioners implemented in the package.
    -

    The with the full range of preconditioners implemented in the package. +

    The fileread directories contain sample programs that read sparse matrices from files, +class="cmbx-12">directories contain sample programs that read sparse matrices according to the Matrix Market or the Harwell-Boeing storage format; the pdegen +class="cmbx-12">from files, according to the Matrix Market or the Harwell-Boeing storage programs generate matrices in full parallel mode from the discretization of a sample +class="cmbx-12">format; the pdegen programs generate matrices in full parallel mode from partial differential equation. +class="cmbx-12">the discretization of a sample partial differential equation. @@ -80,19 +83,19 @@ class="cmr-12">partial differential equation.

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    4.1 4.1 AMG preconditioners

    +class="cmbx-12">AMG preconditioners

    In order to describe the AMG preconditioners available in MLD2P4, we consider a +class="cmbx-12">In order to describe the AMG preconditioners available in MLD2P4, we linear system +class="cmbx-12">consider a linear system
    Ax = b, + + +
    		(2)

    where where A = (aℝn×n is a nonsingular sparse matrix; for ease of presentation we +class="cmbx-12">is a nonsingular sparse matrix; for ease of assume presentation we assume A has a symmetric sparsity pattern. - - - +class="cmbx-12">has a symmetric sparsity pattern.

    Let us consider as finest index space the set of row (column) indices of Let us consider as finest index space the set of row (column) indices of A, +class="cmbx-12">, i.e., Ω = i.e., Ω = {1, ,…,n}. Any algebraic multilevel preconditioners implemented in +class="cmbx-12">. Any algebraic multilevel preconditioners implemented in MLD2P4 generates a hierarchy of index spaces and a corresponding hierarchy of +class="cmbx-12">MLD2P4 generates a hierarchy of index spaces and a corresponding matrices, +class="cmbx-12">hierarchy of matrices,

    1 2 nlev 1 2 nlev -Ω ≡ Ω ⊃ Ω ⊃ ...⊃ Ω , A ≡ A, A ,...,A , +src=

    by using the information contained in by using the information contained in A, without assuming any knowledge of +class="cmbx-12">, without assuming any knowledge the geometry of the problem from which of the geometry of the problem from which A originates. A vector space originates. A vector space ℝnk is +class="cmmi-6">k associated with Ωis associated with Ωk, where , where n k is the size of Ωis the size of Ωk. For all . For all k < nlev, a restriction +class="cmbx-12">, a operator and a prolongation one are built, which connect two levels k and +class="cmbx-12">restriction operator and a prolongation one are built, which connect two levels k + 1: +class="cmbx-12">and k + 1:

    P k ∈ ℝnk×nk+1, Rk ∈ ℝnk+1×nk ; +src=

    the matrix the matrix Ak+1 is computed by using the previous operators according to the +class="cmr-8">+1 is computed by using the previous operators according to Galerkin approach, i.e., +class="cmbx-12">the Galerkin approach, i.e.,

    k+1 k k k -A = R A P . +src=

    In the current implementation of MLD2P4 we have In the current implementation of MLD2P4 we have Rk = (Pk)T A smoother with +class="cmbx-12">A smoother iteration matrix with iteration matrix Mk is set up at each level is set up at each level k < nlev, and a solver is set up at the +class="cmbx-12">, and a solver is set coarsest level, so that they are ready for application (for example, setting up a solver +class="cmbx-12">up at the coarsest level, so that they are ready for application (for example, based on the setting up a solver based on the LU factorization means computing and storing the factorization means computing and +storing the L and and U factors). The +class="cmbx-12">factors). The construction of the hierarchy of AMG construction of the hierarchy of AMG components described so far corresponds to the +class="cmbx-12">components described so far corresponds to the so-called build phase of the so-called build phase of the preconditioner. +class="cmbx-12">preconditioner.

    @@ -178,9 +183,9 @@ class="cmr-12">so-called build phase of the preconditioner. cellpadding="0" border="0" cellspacing="0" class="tabbing">

    Application phase of a V-cycle preconditioner.

    The components produced in the build phase may be combined in several ways to +class="cmbx-12">The components produced in the build phase may be combined in obtain different multilevel preconditioners; this is done in the application phase, i.e., in +class="cmbx-12">several ways to obtain different multilevel preconditioners; this is done in the computation of a vector of type the application phase, i.e., in the computation of a vector of type w = B-1v, where , +where B denotes the preconditioner, +class="cmbx-12">denotes the preconditioner, usually within an iteration of a Krylov usually within an iteration of a Krylov solver [solver [21]. An example of such a combination, +class="cmbx-12">23]. An example of such a combination, known as V-cycle, is given in known as V-cycle, is given in Figure Figure 1. In this case, a single iteration of the same +class="cmbx-12">1. In this case, a single iteration of the same smoother is used before smoother is used before and after the the recursive call to the V-cycle (i.e., in the +class="cmbx-12">and after the the recursive call to the V-cycle (i.e., in the pre-smoothing pre-smoothing and post-smoothing phases); however, different choices can be +class="cmbx-12">and post-smoothing phases); however, different choices can be performed. performed. Other cycles can be defined; in MLD2P4, we implemented the +class="cmbx-12">Other cycles can be defined; in MLD2P4, we implemented the standard standard V-cycle and W-cycle [V-cycle and W-cycle [3], and a version of the K-cycle described +class="cmbx-12">3], and a version of the K-cycle described in [in [20]. +class="cmbx-12">22].

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    4.2 4.2 Smoothed Aggregation

    +class="cmbx-12">Smoothed Aggregation

    In order to define the prolongator In order to define the prolongator Pk, used to compute the coarse-level matrix , used to compute the coarse-level +matrix Ak+1, +class="cmbx-12">, MLD2P4 uses the smoothed aggregation algorithm described MLD2P4 uses the smoothed aggregation algorithm described in [in [2, 2, 26]. The basic idea +class="cmbx-12">28]. The basic idea of this algorithm is to build a coarse set of indices of this algorithm is to build a coarse set of indices ΩΩk+1 by suitably grouping the -indices of Ω+1 by suitably grouping the indices of Ωk into disjoint subsets (aggregates), and to define the coarse-to-fine space +class="cmbx-12">into disjoint subsets transfer operator (aggregates), and to define the coarse-to-fine space transfer operator Pk by applying a suitable smoother to a simple piecewise constant +class="cmbx-12">by prolongation operator, with the aim of improving the quality of the coarse-space +class="cmbx-12">applying a suitable smoother to a simple piecewise constant prolongation correction. +class="cmbx-12">operator, with the aim of improving the quality of the coarse-space +correction.

    Three main steps can be identified in the smoothed aggregation procedure: +class="cmbx-12">Three main steps can be identified in the smoothed aggregation +procedure:

    1. aggregation of the indices of Ωaggregation of the indices of Ωk to obtain Ωto obtain Ωk+1; +class="cmbx-12">;
    2. construction of the prolongator construction of the prolongator Pk; +class="cmbx-12">;
    3. application of application of Pk and and Rk = (Pk)T to build to build Ak+1.
    +class="cmbx-12">.

    In order to perform the coarsening step, the smoothed aggregation algorithm +class="cmbx-12">In order to perform the coarsening step, the smoothed aggregation described in [algorithm described in [26] is used. In this algorithm, each index 28] is used. In this algorithm, each index j ∈ Ωk+1 corresponds +class="cmr-8">+1 to an aggregate Ωcorresponds to an aggregate Ωjk of Ωof Ωk, consisting of a suitably chosen index , consisting of a suitably chosen index +i ∈ Ωk and +class="cmbx-12">and indices that are (usually) contained in a strongly-coupled indices that are (usually) contained in a strongly-coupled neighborood of neighborood of i, -i.e., +class="cmbx-12">, i.e.,

    { ∘ -------} -Ωkj ⊂ N ki (θ) = r ∈ Ωk : |akir| > θ |akiiakrr| ∪ {i} , +src=
    		(3)
    @@ -154,53 +161,57 @@ class="cmr-12">(3)

    for a given threshold for a given threshold θ ∈ [0, 1] (see [1] (see [26] for the details). Since this algorithm has a +class="cmbx-12">28] for the details). Since this algorithm +has a sequential nature, a decoupled version of it is applied, where each sequential nature, a decoupled version of it is applied, where each processor +class="cmbx-12">processor independently executes the algorithm on the set of indices independently executes the algorithm on the set of indices assigned to it in the initial +class="cmbx-12">assigned to it in the initial data distribution. This version is embarrassingly data distribution. This version is embarrassingly parallel, since it does not require any +class="cmbx-12">parallel, since it does not require any data communication. On the other data communication. On the other hand, it may produce some nonuniform aggregates +class="cmbx-12">hand, it may produce some nonuniform aggregates and is strongly and is strongly dependent on the number of processors and on the initial +class="cmbx-12">dependent on the number of processors and on the initial partitioning of partitioning of the matrix the matrix A. Nevertheless, this parallel algorithm has been chosen +class="cmbx-12">. Nevertheless, this parallel algorithm has been chosen for for MLD2P4, since it has been shown to produce good results in practice +class="cmbx-12">MLD2P4, since it has been shown to produce good results in practice [[5, 5, 7, 7, 25]. +class="cmbx-12">27].

    The prolongator The prolongator Pk is built starting from a tentative prolongator is built starting from a tentative prolongator +Pk ∈ ×nk+1, -defined as +class="cmbx-12">, defined as
    { k -P¯k = (¯pkij), p¯kij = 1 if i ∈ Ω j, - 0 otherwise, +src=
    		(4)

    where Ωwhere Ωjk is the aggregate of Ωis the aggregate of Ωk corresponding to the index corresponding to the index j ∈ Ωk+1. . Pk is obtained +class="cmbx-12">is by applying to obtained by applying to Pk a smoother a smoother Sk ∈ k×nk: +class="cmbx-12">:

    Pk = Sk ¯Pk,
    in +src="userhtml15x.png" alt="Pk = Sk ¯P k, " class="math-display" > +in order to remove nonsmooth components from the range of the order to remove nonsmooth components from the range of the prolongator, and hence +class="cmbx-12">prolongator, and hence to improve the convergence properties of the to improve the convergence properties of the multilevel method [multilevel method [2, 2, 24]. A simple -choice for 26]. A simple choice for Sk is the damped Jacobi smoother: +class="cmbx-12">is the damped Jacobi +smoother: + + +
    k k k -1 k -S = I - ω (D ) A F, +src=

    where where Dk is the diagonal matrix with the same diagonal entries as is the diagonal matrix with the same diagonal entries as Ak, A +class="cmbx-12">, AF k = (a ijk) is -the filtered matrix defined as +class="cmr-12">) is the filtered matrix defined as
    { k k ∑ -a¯kij = aij if j ∈ N i (θ), (j ⁄= i), ¯akii = akii - (akij - ¯akij), - 0 otherwise, j⁄=i +src=
    		(5)

    and and ωk is an approximation of 4is an approximation of 4∕(3ρk), where ), where ρk is the spectral radius of (is the spectral radius of +(Dk)1A F k -[k [2]. In MLD2P4 this approximation is obtained by using 2]. In MLD2P4 this approximation is obtained by using ∥AA +F k∥ ∞ as an estimate of ∞ +as an estimate of ρk. +class="cmbx-12">. Note that for systems coming from uniformly Note that for systems coming from uniformly elliptic problems, filtering the matrix elliptic problems, filtering the matrix Ak +class="cmmi-8">k has little or no effect, and has little or no effect, and Ak can be used instead of can be used instead of A F k. The latter choice is the +class="cmbx-12">. The latter choice is the default in default in MLD2P4. +class="cmbx-12">MLD2P4.

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    4.3 4.3 Smoothers and coarsest-level solvers

    +class="cmbx-12">Smoothers and coarsest-level solvers

    The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods, +class="cmbx-12">The smoothers implemented in MLD2P4 include the Jacobi and a hybrid version of the forward and backward Gauss-Seidel methods, and the additive +class="cmbx-12">block-Jacobi methods, a hybrid version of the forward and backward Schwarz (AS) ones (see, e.g., [Gauss-Seidel methods, and the additive Schwarz (AS) ones (see, e.g., +[21, 23, 22]). +class="cmbx-12">24]).

    The hybrid Gauss-Seidel version is considered because the original Gauss-Seidel +class="cmbx-12">The hybrid Gauss-Seidel version is considered because the original method is inherently sequential. At each iteration of the hybrid version, each parallel +class="cmbx-12">Gauss-Seidel method is inherently sequential. At each iteration of the process uses the most recent values of its own local variables and the values +class="cmbx-12">hybrid version, each parallel process uses the most recent values of its own of the non-local variables computed at the previous iteration, obtained by +class="cmbx-12">local variables and the values of the non-local variables computed at the exchanging data with other processes before the beginning of the current +class="cmbx-12">previous iteration, obtained by exchanging data with other processes before iteration. +class="cmbx-12">the beginning of the current iteration.

    In the AS methods, the index space ΩIn the AS methods, the index space Ωk is divided into is divided into m k subsets Ωsubsets Ωik of +class="cmbx-12">of size size nk,i, possibly overlapping. For each , possibly overlapping. For each i we consider the restriction operator +class="cmbx-12">we consider the restriction operator Rik k,i×nk that maps a vector that maps a vector xk to the vector to the vector x ik made of the components of +class="cmbx-12">made of the components of xk with indices in Ω +class="cmbx-12">with indices in Ω ik, and the prolongation operator P +class="cmbx-12">, and the prolongation operator Pik = (ik)T . These +class="cmbx-12">. These operators are then used to build A operators are then used to build Aik = P ik, which is the restriction of +class="cmbx-12">, which is the restriction of Ak to the index space Ω +class="cmbx-12">to the index space Ω ik. The classical AS preconditioner M +class="cmbx-12">. The classical AS preconditioner MASk is defined -as +class="cmbx-12">is defined as

    m∑k -(M AkS )-1 = Pki (Aki)-1Rki, - i=1 +src=

    where where Aik is supposed to be nonsingular. We observe that an approximate inverse of +class="cmbx-12">is supposed to be nonsingular. We observe that an approximate inverse of Aik is usually considered instead of (is usually considered instead of (A ik)-1. The setup of . The setup of M ASk during the multilevel +class="cmmi-8">k build phase involves +class="cmbx-12">during the multilevel build phase involves

    • the definition of the index subspaces Ωthe definition of the index subspaces Ωik and of the corresponding operators +class="cmbx-12">and of the corresponding operators Rik (and (and P ik); +class="cmbx-12">);
    • the computation of the submatrices the computation of the submatrices Aik; +class="cmbx-12">;
    • the computation of their inverses (usually approximated through some form +class="cmbx-12">the computation of their inverses (usually approximated through of incomplete factorization).
    +class="cmbx-12">some form of incomplete factorization).

    The computation of The computation of zk = ASkwk, with , with wk ∈ ℝnk, during the multilevel application +class="cmbx-12">, during the multilevel phase, requires +class="cmbx-12">application phase, requires

    • the restriction of the restriction of wk to the subspaces to the subspaces ℝnk,i, i.e. , i.e. w iikwk; +class="cmbx-12">;
    • the computation of the vectors the computation of the vectors zik w ik; +class="cmbx-12">;
    • the prolongation and the sum of the previous vectors, i.e. the prolongation and the sum of the previous vectors, i.e. zk k = ∑ @@ -293,43 +301,47 @@ class="cmmi-12">z ik.
    +class="cmbx-12">.

    Variants of the classical AS method, which use modifications of the restriction and +class="cmbx-12">Variants of the classical AS method, which use modifications of the restriction prolongation operators, are also implemented in MLD2P4. Among them, the Restricted +class="cmbx-12">and prolongation operators, are also implemented in MLD2P4. Among AS (RAS) preconditioner usually outperforms the classical AS preconditioner in terms +class="cmbx-12">them, the Restricted AS (RAS) preconditioner usually outperforms of convergence rate and of computation and communication time on parallel +class="cmbx-12">the classical AS preconditioner in terms of convergence rate and of distributed-memory computers, and is therefore the most widely used among the AS +class="cmbx-12">computation and communication time on parallel distributed-memory preconditioners [computers, and is therefore the most widely used among the AS +preconditioners [6]. +class="cmbx-12">6].

    Direct solvers based on sparse LU factorizations, implemented in the third-party +class="cmbx-12">Direct solvers based on sparse LU factorizations, implemented in libraries reported in Section the third-party libraries reported in Section 3.2, can be applied as coarsest-level solvers by +class="cmbx-12">3.2, can be applied as +coarsest-level solvers by MLD2P4. Native inexact solvers based on MLD2P4. Native inexact solvers based on incomplete LU factorizations, as well as +class="cmbx-12">incomplete LU factorizations, as well as Jacobi, hybrid (forward) Jacobi, hybrid (forward) Gauss-Seidel, and block Jacobi preconditioners are +class="cmbx-12">Gauss-Seidel, and block Jacobi preconditioners are also available. Direct also available. Direct solvers usually lead to more effective preconditioners in +class="cmbx-12">solvers usually lead to more effective preconditioners in terms of terms of algorithmic scalability; however, this does not guarantee parallel +class="cmbx-12">algorithmic scalability; however, this does not guarantee parallel efficiency. +class="cmbx-12">efficiency. @@ -337,19 +349,19 @@ class="cmr-12">efficiency.

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    5.1 5.1 Examples

    -

    The code reported in Figure Examples +

    The code reported in Figure 2 shows how to set and apply the default multilevel +class="cmbx-12">2 shows how to set and apply the default preconditioner available in the real double precision version of MLD2P4 (see Table multilevel preconditioner available in the real double precision version of +AMG4PSBLAS (see Table 1). +class="cmbx-12">1). This preconditioner is chosen by simply This preconditioner is chosen by simply specifying specifying ’ML’ as the second argument of -as the second argument of P%init (a call to (a call to P%set is not needed) and is applied with the CG solver provided by +class="cmbx-12">is not PSBLAS (the matrix of the system to be solved is assumed to be positive definite). As +class="cmbx-12">needed) and is applied with the CG solver provided by PSBLAS (the previously observed, the modules psb_base_mod, mld_prec_mod and psb_krylov_mod +class="cmbx-12">matrix of the system to be solved is assumed to be positive definite). As must be used by the example program. -

    The part of the code concerning the reading and assembling of the sparse matrix +class="cmbx-12">previously observed, the modules psb_base_mod, amg_prec_mod and +psb_krylov_mod must be used by the example program. +

    The part of the code concerning the reading and assembling of the and the right-hand side vector, performed through the PSBLAS routines for sparse +class="cmbx-12">sparse matrix and the right-hand side vector, performed through the matrix and vector management, is not reported here for brevity; the statements +class="cmbx-12">PSBLAS routines for sparse matrix and vector management, is not concerning the deallocation of the PSBLAS data structure are neglected too. The +class="cmbx-12">reported here for brevity; the statements concerning the deallocation of the complete code can be found in the example program file mld_dexample_ml.f90, +class="cmbx-12">PSBLAS data structure are neglected too. The complete code can be found in the directory in the example program file amg_dexample_ml.f90, in the directory +examples/fileread of the MLD2P4 implementation (see -Section of the AMG4PSBLAS implementation (see Section 3.5). A sample test problem along with the relevant input data is available in +class="cmbx-12">3.5). +A sample test problem along with the relevant input data is available in + + + examples/fileread/runs. For details on the use of the PSBLAS routines, see the +class="cmbx-12">. For details on the use of the PSBLAS routines, see PSBLAS User’s Guide [the PSBLAS User’s Guide [13]. -

    The setup and application of the default multilevel preconditioner for the real single +class="cmbx-12">15]. +

    The setup and application of the default multilevel preconditioner for precision and the complex, single and double precision, versions are obtained +class="cmbx-12">the real single precision and the complex, single and double precision, with straightforward modifications of the previous example (see Section versions are obtained with straightforward modifications of the previous +example (see Section 6 for - - - +class="cmbx-12">6 for details). If these versions are installed, the details). If these versions are installed, the corresponding codes are available in -corresponding codes are available in examples/fileread/. -

    . +

    @@ -117,13 +122,13 @@ class="cmr-12">.
    -

    +

      use psb_base_mod - 
      use mld_prec_mod + 
      use amg_prec_mod  
      use psb_krylov_mod  
    ... ...  
    ! @@ -132,7 +137,7 @@ class="cmr-12">.  
    ! sparse matrix descriptor  
      type(psb_desc_type)   :: desc_A  
    ! preconditioner - 
      type(mld_dprec_type)  :: P + 
      type(amg_dprec_type)  :: P  
    ! right-hand side and solution vectors  
      type(psb_d_vect_type) :: b, x  
    ... ... @@ -174,7 +179,7 @@ class="cmr-12">.  
      call psb_exit(ictxt)  
      stop
    -

    +

    @@ -185,80 +190,86 @@ class="content">setup and application of the default multilevel preconditioner ( -

    -

    Different versions of the multilevel preconditioner can be obtained by changing the +

    +

    Different versions of the multilevel preconditioner can be obtained by +changing the default values of the preconditioner parameters. The code default values of the preconditioner parameters. The code reported in Figure reported in Figure 3 shows +class="cmbx-12">3 shows how to set a V-cycle preconditioner which how to set a V-cycle preconditioner which applies 1 block-Jacobi sweep as pre- and +class="cmbx-12">applies 1 block-Jacobi sweep as pre- and post-smoother, and solves the post-smoother, and solves the coarsest-level system with 8 block-Jacobi sweeps. Note +class="cmbx-12">coarsest-level system with 8 block-Jacobi sweeps. Note that the ILU(0) that the ILU(0) factorization (plus triangular solve) is used as local solver for the +class="cmbx-12">factorization (plus triangular solve) is used as local solver for the block-Jacobi sweeps, since this is the default associated with block-Jacobi and set +class="cmbx-12">block-Jacobi sweeps, since this is the default associated with block-Jacobi by and set by P%init. Furthermore, specifying block-Jacobi as coarsest-level solver implies that +class="cmbx-12">. Furthermore, specifying block-Jacobi as coarsest-level the coarsest-level matrix is distributed among the processes. Figure solver implies that the coarsest-level matrix is distributed among the +processes. Figure 4 shows how to set +class="cmbx-12">4 shows how to set a W-cycle preconditioner which a W-cycle preconditioner which applies 2 hybrid Gauss-Seidel sweeps as pre- and +class="cmbx-12">applies 2 hybrid Gauss-Seidel sweeps as pre- and post-smoother, and solves post-smoother, and solves the coarsest-level system with the multifrontal LU +class="cmbx-12">the coarsest-level system with the multifrontal LU factorization factorization implemented in MUMPS. It is specified that the coarsest-level matrix is +class="cmbx-12">implemented in MUMPS. It is specified that the coarsest-level matrix is distributed, since MUMPS can be used on both replicated and distributed matrices, +class="cmbx-12">distributed, since MUMPS can be used on both replicated and distributed and by default it is used on replicated ones. The code fragments shown in +class="cmbx-12">matrices, and by default it is used on replicated ones. The code fragments Figures shown in Figures 3 and 3 and 4 are included in the example program file mld_dexample_ml.f90 -too. -

    Finally, Figure 4 are included in the example program file +amg_dexample_ml.f90 too. +

    Finally, Figure 5 shows the setup of a one-level additive Schwarz preconditioner, +class="cmbx-12">5 shows the setup of a one-level additive Schwarz i.e., RAS with overlap 2. Note also that a Krylov method different from CG +class="cmbx-12">preconditioner, i.e., RAS with overlap 2. Note also that a Krylov method must be used to solve the preconditioned system, since the preconditione in +class="cmbx-12">different from CG must be used to solve the preconditioned system, since nonsymmetric. The corresponding example program is available in the file -mld_dexample_1lev.f90. -

    For all the previous preconditioners, example programs where the sparse matrix and +class="cmbx-12">the preconditione in nonsymmetric. The corresponding example program is the right-hand side are generated by discretizing a PDE with Dirichlet boundary +class="cmbx-12">available in the file amg_dexample_1lev.f90. +

    For all the previous preconditioners, example programs where the sparse conditions are also available in the directory matrix and the right-hand side are generated by discretizing a PDE with +Dirichlet boundary conditions are also available in the directory +examples/pdegen. -

    . +

    @@ -270,7 +281,7 @@ class="cmr-12">.
    -

    +

    ... ...  
    ! build a V-cycle preconditioner with 1 block-Jacobi sweep (with @@ -284,7 +295,7 @@ class="cmr-12">.  
      call P%smoothers_build(A,desc_A,info)  
    ... ...
    -

    +


    Figure 3: setup of a multilevel preconditioner
    @@ -292,8 +303,8 @@ class="content">setup of a multilevel preconditioner

    -

    +

    @@ -305,7 +316,7 @@ class="content">setup of a multilevel preconditioner

    +

    ... ...  
    ! build a W-cycle preconditioner with 2 hybrid Gauss-Seidel sweeps @@ -321,7 +332,7 @@ class="content">setup of a multilevel preconditioner

    +


    Figure 4: setup of a multilevel preconditioner
    @@ -329,8 +340,8 @@ class="content">setup of a multilevel preconditioner

    -

    +

    @@ -342,7 +353,7 @@ class="content">setup of a multilevel preconditioner

    +

    ... ...  
    ! set RAS with overlap 2 and ILU(0) on the local blocks @@ -353,7 +364,7 @@ class="content">setup of a multilevel preconditioner

    +


    Figure 5: setup of a one-level Schwarz preconditioner.
    @@ -361,7 +372,7 @@ class="content">setup of a one-level Schwarz preconditioner.

    +

    @@ -369,13 +380,13 @@ class="content">setup of a one-level Schwarz preconditioner.

    [[front] [front] [up]

    +class="cmbx-12">up]

    diff --git a/docs/mld2p4-2.2-guide.pdf b/docs/mld2p4-2.2-guide.pdf deleted file mode 100644 index dada05df..00000000 --- a/docs/mld2p4-2.2-guide.pdf +++ /dev/null @@ -1,10752 +0,0 @@ -%PDF-1.5 -%ÐÔÅØ -145 0 obj -<< -/Length 1210 ->> -stream -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -BT -/F17 24.7871 Tf 394.538 617.737 Td [(MLD2P4)]TJ -229.059 -27.023 Td [(User's)-375(and)-375(Reference)-375(Guide)]TJ -ET -q -1 0 0 1 93.6 572.468 cm -0 0 412.451 4.981 re f -Q -BT -/F19 14.3462 Tf 197.154 547.49 Td [(A)-350(guide)-350(for)-350(the)-350(MultiL)50(evel)-350(Domain)-350(De)50(c)50(omp)50(osition)]TJ -10.534 -17.256 Td [(Par)50(al)-50(lel)-350(Pr)50(e)50(c)50(onditioners)-350(Package)-350(b)50(ase)50(d)-350(on)-350(PSBLAS)]TJ -0 g 0 G -0 g 0 G -/F17 11.9552 Tf 218.644 -80.484 Td [(P)31(asqua)-375(D'Am)31(bra)]TJ/F37 11.9552 Tf -22.655 -13.948 Td [(IA)27(C-CNR,)-326(Naples,)-326(Italy)]TJ/F17 11.9552 Tf 11.494 -29.39 Td [(Daniela)-375(di)-375(Sera\014no)]TJ/F37 11.9552 Tf -181.63 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[(MLD2P4)-229(has)-230(b)-27(e)-1(en)-229(designed)-229(to)-230(implemen)28(t)-230(scalable)-229(and)-229(easy-to-use)-230(m)28(ultilev)28(el)-230(precon-)]TJ -16.937 -13.549 Td [(ditioners)-349(in)-349(the)-350(con)28(text)-349(of)-349(the)-349(PSBLAS)-349(\050P)27(arallel)-349(Sparse)-349(BLAS\051)-349(computational)-349(frame-)]TJ 0 -13.549 Td [(w)28(ork)-360([)]TJ -1 0 0 rg 1 0 0 RG - [(15)]TJ -0 g 0 G - [(,)]TJ -1 0 0 rg 1 0 0 RG - [-360(14)]TJ -0 g 0 G - [(].)-524(PSBLAS)-359(pro)28(vides)-360(basic)-360(linear)-360(algebra)-360(op)-27(erators)-360(and)-360(data)-360(managemen)28(t)]TJ 0 -13.549 Td [(facilities)-414(for)-414(distributed)-414(sparse)-414(matrices,)-434(as)-414(w)27(ell)-414(as)-414(parallel)-414(Krylo)28(v)-414(solv)28(ers)-414(whic)28(h)-414(can)]TJ 0 -13.55 Td [(b)-28(e)-401(used)-402(with)-401(the)-401(MLD2P4)-401(prec)-1(on)1(ditioners.)-649(The)-401(c)27(hoice)-401(of)-401(PSBLAS)-401(has)-402(b)-28(een)-401(mainly)]TJ 0 -13.549 Td 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[(directories)-500(con)28(tain)-500(sample)-499(programs)-500(that)-500(read)-499(sparse)-500(matrices)-500(from)]TJ -75.508 -13.55 Td [(\014les,)-295(according)-285(to)-285(the)-285(Matrix)-285(Mark)28(et)-285(or)-285(the)-285(Harw)27(ell-Bo)-27(eing)-286(storage)-285(format;)-301(the)]TJ/F45 10.9091 Tf 378.088 0 Td [(pdegen)]TJ/F15 10.9091 Tf -378.088 -13.549 Td [(programs)-416(generate)-415(matrices)-416(in)-416(full)-415(parallel)-416(mo)-28(de)-416(from)-415(the)-416(discretization)-416(of)-415(a)-416(sample)]TJ 0 -13.549 Td [(partial)-333(di\013eren)28(tial)-334(equation.)]TJ -0 g 0 G -0 g 0 G -ET - -endstream -endobj -344 0 obj -<< -/Length 8161 ->> -stream -0 g 0 G -0 0 1 rg 0 0 1 RG -BT -/F42 10.9091 Tf 93.6 740.002 Td [(4)]TJ -0 g 0 G - [-378(Mul)67(tigrid)-378(Ba)22(ck)22(gr)23(ound)]TJ/F15 10.9091 Tf 401.542 0 Td [(11)]TJ -0 g 0 G -/F17 14.3462 Tf -401.542 -35.866 Td [(4)-1125(Multigrid)-375(Bac)31(kground)]TJ/F15 10.9091 Tf 0 -25.431 Td [(Multigrid)-467(preconditioners,)-500(coupled)-468(with)-467(Krylo)28(v)-467(iterativ)28(e)-467(solv)27(ers,)-500(are)-467(widely)-467(use)-1(d)-467(in)]TJ 0 -13.549 Td [(the)-374(par)1(allel)-374(solution)-373(of)-374(large)-374(and)-373(sparse)-374(linear)-373(systems,)-384(b)-28(ecause)-373(of)-374(their)-373(optimalit)28(y)-374(in)]TJ 0 -13.55 Td [(the)-411(solution)-411(of)-411(linear)-411(systems)-411(arising)-411(from)-411(the)-412(d)1(isc)-1(r)1(e)-1(tization)-411(of)-411(scalar)-411(elliptic)-411(P)28(artial)]TJ 0 -13.549 Td [(Di\013eren)28(tial)-331(Equations)-330(\050PDEs\051)-331(on)-330(regular)-331(grids.)-443(Optimalit)28(y)83(,)-331(also)-331(kn)1(o)27(wn)-330(as)-331(algorithmic)]TJ 0 -13.549 Td [(scalabilit)28(y)83(,)-404(is)-389(the)-390(prop)-28(ert)28(y)-390(of)-390(ha)28(ving)-389(a)-390(computational)-390(cost)-390(p)-27(er)-390(iteration)-390(that)-390(dep)-27(ends)]TJ 0 -13.549 Td 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[(\012)]TJ/F25 10.9091 Tf 10.909 0 Td [(\033)]TJ/F15 10.9091 Tf 11.515 0 Td [(\012)]TJ/F20 7.9701 Tf 7.879 4.505 Td [(2)]TJ/F25 10.9091 Tf 7.763 -4.505 Td [(\033)]TJ/F22 10.9091 Tf 11.515 0 Td [(:)-167(:)-166(:)]TJ/F25 10.9091 Tf 15.757 0 Td [(\033)]TJ/F15 10.9091 Tf 11.516 0 Td [(\012)]TJ/F23 7.9701 Tf 7.878 4.505 Td [(nl)-12(ev)]TJ/F22 10.9091 Tf 16.597 -4.505 Td [(;)-1167(A)]TJ/F20 7.9701 Tf 23.939 4.505 Td [(1)]TJ/F25 10.9091 Tf 7.763 -4.505 Td [(\021)]TJ/F22 10.9091 Tf 11.515 0 Td [(A;)-167(A)]TJ/F20 7.9701 Tf 21.212 4.505 Td [(2)]TJ/F22 10.9091 Tf 4.732 -4.505 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(A)]TJ/F23 7.9701 Tf 32.424 4.505 Td [(nl)-12(ev)]TJ/F22 10.9091 Tf 16.597 -4.505 Td [(;)]TJ/F15 10.9091 Tf -328.044 -23.49 Td [(b)28(y)-282(using)-282(the)-282(information)-282(con)28(tained)-282(in)]TJ/F22 10.9091 Tf 180.373 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)-292(without)-282(assuming)-282(an)27(y)-282(kno)28(wledge)-282(of)-282(the)-282(geom-)]TJ -188.555 -13.55 Td [(etry)-295(of)-294(the)-295(problem)-294(from)-295(whic)28(h)]TJ/F22 10.9091 Tf 150.888 0 Td [(A)]TJ/F15 10.9091 Tf 11.395 0 Td [(originates.)-431(A)-295(v)28(ector)-295(space)]TJ/F34 10.9091 Tf 126.348 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F15 10.9091 Tf 8.285 -2.553 Td [(is)-295(asso)-27(ciate)-1(d)-294(with)-295(\012)]TJ/F23 7.9701 Tf 94.368 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(,)]TJ -409.421 -13.549 Td [(where)]TJ/F22 10.9091 Tf 31.714 0 Td [(n)]TJ/F23 7.9701 Tf 6.548 -1.777 Td [(k)]TJ/F15 10.9091 Tf 8.925 1.777 Td [(is)-349(the)-349(size)-348(of)-349(\012)]TJ/F23 7.9701 Tf 71.401 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(.)-491(F)84(or)-349(all)]TJ/F22 10.9091 Tf 43.449 0 Td [(k)-335(<)-304(nl)-19(ev)]TJ/F15 10.9091 Tf 41.907 0 Td [(,)-353(a)-348(restriction)-349(op)-28(erator)-349(and)-348(a)-349(prolongation)]TJ -209.064 -13.549 Td [(one)-333(are)-334(built,)-333(whic)28(h)-333(connect)-334(t)28(w)28(o)-334(l)1(e)-1(v)28(els)]TJ/F22 10.9091 Tf 188.273 0 Td [(k)]TJ/F15 10.9091 Tf 9.659 0 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(k)]TJ/F15 10.9091 Tf 8.447 0 Td [(+)-222(1:)]TJ/F22 10.9091 Tf -101.285 -23.491 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F25 10.9091 Tf 8.15 -4.504 Td [(2)]TJ/F34 10.9091 Tf 10.303 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 4.504 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F26 7.9701 Tf 4.573 1.405 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F21 5.9776 Tf 4.075 0 Td [(+1)]TJ/F22 10.9091 Tf 10.239 -3.099 Td [(;)-1167(R)]TJ/F23 7.9701 Tf 24.125 4.504 Td [(k)]TJ/F25 10.9091 Tf 8.15 -4.504 Td [(2)]TJ/F34 10.9091 Tf 10.303 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 4.504 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F21 5.9776 Tf 4.075 0 Td [(+1)]TJ/F26 7.9701 Tf 9.741 1.405 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.405 Td [(k)]TJ/F15 10.9091 Tf 5.071 -3.099 Td [(;)]TJ -283.114 -23.491 Td [(the)-282(matrix)]TJ/F22 10.9091 Tf 53.146 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 14.392 -3.959 Td [(is)-282(computed)-281(b)27(y)-281(using)-282(the)-282(previou)1(s)-282(op)-28(erators)-282(according)-281(to)-282(the)-282(Galerkin)]TJ -80.341 -13.549 Td [(approac)28(h,)-333(i.e.,)]TJ/F22 10.9091 Tf 165.163 -13.549 Td [(A)]TJ/F23 7.9701 Tf 8.182 4.504 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 14.349 -4.504 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(A)]TJ/F23 7.9701 Tf 8.181 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(:)]TJ/F15 10.9091 Tf -244.258 -19.516 Td [(In)-476(the)-477(curren)28(t)-476(implemen)28(tation)-477(of)-476(MLD2P4)-476(w)28(e)-477(ha)28(v)28(e)]TJ/F22 10.9091 Tf 261.479 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.958 Td [(k)]TJ/F15 10.9091 Tf 10.749 -3.958 Td [(=)-516(\050)]TJ/F22 10.9091 Tf 18.358 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.958 Td [(\051)]TJ/F23 7.9701 Tf 4.243 3.958 Td [(T)]TJ/F15 10.9091 Tf 11.801 -3.958 Td [(A)-476(smo)-28(other)-476(with)]TJ -328.636 -13.55 Td [(iteration)-441(matrix)]TJ/F22 10.9091 Tf 82.101 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.928 -3.959 Td [(is)-441(set)-441(up)-440(at)-441(eac)28(h)-441(lev)28(el)]TJ/F22 10.9091 Tf 113.517 0 Td [(k)-488(<)-457(nl)-20(ev)]TJ/F15 10.9091 Tf 45.251 0 Td [(,)-468(and)-440(a)-441(solv)28(er)-441(is)-441(set)-441(up)-440(at)-441(the)]TJ -262.57 -13.549 Td [(coarsest)-369(lev)28(el,)-378(so)-368(that)-369(they)-369(are)-369(ready)-368(for)-369(application)-369(\050for)-368(example,)-378(setting)-369(up)-368(a)-369(solv)28(er)]TJ 0 -13.549 Td [(based)-292(on)-291(the)]TJ/F22 10.9091 Tf 62.938 0 Td [(LU)]TJ/F15 10.9091 Tf 19.243 0 Td [(factorization)-292(means)-291(computing)-292(and)-291(storing)-292(the)]TJ/F22 10.9091 Tf 224.422 0 Td [(L)]TJ/F15 10.9091 Tf 10.605 0 Td [(and)]TJ/F22 10.9091 Tf 20.757 0 Td [(U)]TJ/F15 10.9091 Tf 11.819 0 Td [(factors\051.)-431(The)]TJ -349.784 -13.549 Td [(construction)-345(of)-344(the)-345(hierarc)28(h)27(y)-344(of)-345(AMG)-345(comp)-28(on)1(e)-1(n)28(ts)-345(d)1(e)-1(scrib)-27(ed)-345(so)-345(far)-345(corresp)-27(onds)-345(to)-345(the)]TJ 0 -13.549 Td [(so-called)-333(build)-334(ph)1(as)-1(e)-333(of)-333(the)-334(p)1(rec)-1(on)1(ditioner.)]TJ 16.936 -13.55 Td [(The)-359(com)-1(p)-27(onen)28(ts)-360(pro)-28(du)1(c)-1(ed)-359(in)-359(the)-360(build)-359(phase)-359(ma)27(y)-359(b)-28(e)-359(com)28(bined)-360(in)-359(sev)28(eral)-360(w)28(a)28(ys)-360(to)]TJ -16.936 -13.549 Td [(obtain)-306(di\013eren)28(t)-306(m)27(ultilev)28(el)-306(preconditioners;)-315(this)-306(is)-306(done)-307(in)-306(the)-306(application)-306(phase,)-312(i)1(.e)-1(.)1(,)-312(in)]TJ 0 -13.549 Td [(the)-391(computation)-391(of)-391(a)-391(v)27(ector)-391(of)-391(t)28(yp)-28(e)]TJ/F22 10.9091 Tf 178.081 0 Td [(w)]TJ/F15 10.9091 Tf 12.185 0 Td [(=)]TJ/F22 10.9091 Tf 12.566 0 Td [(B)]TJ/F26 7.9701 Tf 8.822 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -3.959 Td [(v)]TJ/F15 10.9091 Tf 5.68 0 Td [(,)-406(where)]TJ/F22 10.9091 Tf 39.631 0 Td [(B)]TJ/F15 10.9091 Tf 13.089 0 Td [(denotes)-391(the)-391(preconditioner,)]TJ -281.372 -13.549 Td [(usually)-334(with)1(in)-334(an)-334(iteration)-333(of)-334(a)-334(Kr)1(ylo)27(v)-333(solv)27(er)-333([)]TJ -1 0 0 rg 1 0 0 RG - [(21)]TJ -0 g 0 G - [(].)-446(An)-333(example)-334(of)-333(s)-1(u)1(c)27(h)-333(a)-334(com)28(bination,)]TJ 0 -13.549 Td [(kno)28(wn)-448(as)-448(V-cycle,)-476(is)-448(giv)28(en)-448(in)-447(Figure)]TJ -0 0 1 rg 0 0 1 RG - [-448(1)]TJ -0 g 0 G - [(.)-788(In)-447(this)-448(case,)-476(a)-448(single)-448(iteration)-448(of)-447(the)-448(same)]TJ 0 -13.549 Td [(smo)-28(other)-333(is)-332(used)-333(b)-28(efore)-333(and)-332(after)-333(the)-333(the)-333(recursiv)28(e)-333(call)-333(to)-332(the)-333(V-cycle)-333(\050i.e.,)-333(in)-333(th)1(e)-333(pre-)]TJ 0 -13.55 Td [(smo)-28(othing)-409(an)1(d)-409(p)-28(ost-smo)-28(othing)-409(ph)1(as)-1(es\051;)-446(ho)28(w)27(ev)28(er,)-427(di\013eren)27(t)-408(c)27(hoices)-409(can)-408(b)-28(e)-409(p)-28(erformed.)]TJ 0 -13.549 Td [(Other)-405(cycles)-405(c)-1(an)-405(b)-27(e)-406(de\014ned;)-441(in)-405(MLD2P4,)-423(w)28(e)-405(implemen)27(ted)-405(the)-405(standard)-405(V-cycle)-405(and)]TJ 0 -13.549 Td [(W-cycle)-333([)]TJ -1 0 0 rg 1 0 0 RG - [(3)]TJ -0 g 0 G - [(],)-334(and)-333(a)-333(v)28(e)-1(r)1(s)-1(ion)-333(of)-333(the)-333(K-cyc)-1(l)1(e)-334(describ)-28(ed)-333(in)-333([)]TJ -1 0 0 rg 1 0 0 RG - [(20)]TJ -0 g 0 G - [(].)]TJ/F17 11.9552 Tf 0 -29.053 Td [(4.2)-1125(Smo)-31(othed)-375(Aggregation)]TJ/F15 10.9091 Tf 0 -20.595 Td [(In)-374(order)-374(to)-374(de\014ne)-374(the)-375(prolongator)]TJ/F22 10.9091 Tf 167.339 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(,)-384(used)-375(to)-374(compute)-374(the)-374(coarse-lev)28(el)-375(matrix)]TJ/F22 10.9091 Tf 204.32 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.959 Td [(,)]TJ -409.421 -13.549 Td [(MLD2P4)-319(uses)-320(the)-319(smo)-28(othed)-319(aggregation)-319(algorithm)-320(describ)-27(ed)-320(in)-319([)]TJ -1 0 0 rg 1 0 0 RG - [(2)]TJ -0 g 0 G - [(,)]TJ -1 0 0 rg 1 0 0 RG - [-319(26)]TJ -0 g 0 G - [(].)-440(The)-319(basic)-320(idea)]TJ 0 -13.549 Td [(of)-270(this)-271(algorithm)-270(is)-271(to)-270(build)-270(a)-271(coarse)-270(se)-1(t)-270(of)-270(indices)-271(\012)]TJ/F23 7.9701 Tf 243.122 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 14.27 -3.959 Td [(b)28(y)-271(suitabl)1(y)-271(grouping)-270(the)-271(indices)]TJ -262.013 -13.549 Td [(of)-350(\012)]TJ/F23 7.9701 Tf 20.481 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.934 -3.959 Td [(in)28(to)-350(disjoin)28(t)-350(sub)1(s)-1(ets)-349(\050aggregates\051,)-354(and)-350(to)-349(de\014ne)-350(the)-349(coarse)-1(-to-\014n)1(e)-350(space)-350(transfer)]TJ -0 g 0 G -0 g 0 G -ET - -endstream -endobj -377 0 obj -<< -/Length 11453 ->> -stream -0 g 0 G -0 0 1 rg 0 0 1 RG -BT -/F42 10.9091 Tf 93.6 740.002 Td [(4)]TJ -0 g 0 G - [-378(Mul)67(tigrid)-378(Ba)22(ck)22(gr)23(ound)]TJ/F15 10.9091 Tf 401.542 0 Td [(13)]TJ -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G - -292.247 -42.833 Td [(pro)-28(cedure)-333(V-cycle)]TJ/F28 10.9091 Tf 85.818 8.837 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.837 Td [(k)-32(;)-166(A)]TJ/F23 7.9701 Tf 19.053 3.959 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.959 Td [(;)-167(b)]TJ/F23 7.9701 Tf 9.53 3.959 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.959 Td [(;)-167(u)]TJ/F23 7.9701 Tf 11.093 3.959 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.878 Td [(\001)]TJ/F15 10.9091 Tf -134.945 -28.055 Td [(if)-333(\050)]TJ/F22 10.9091 Tf 14.242 0 Td [(k)]TJ/F25 10.9091 Tf 9.053 0 Td [(6)]TJ/F15 10.9091 Tf 0 0 Td [(=)]TJ/F22 10.9091 Tf 11.516 0 Td [(nl)-20(ev)]TJ/F15 10.9091 Tf 20.776 0 Td [(\051)-333(then)]TJ/F22 10.9091 Tf -44.678 -16.393 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.958 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F15 10.9091 Tf 7.544 -3.958 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.958 Td [(k)]TJ/F28 10.9091 Tf 6.938 4.878 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.958 Td [(k)]TJ/F25 10.9091 Tf 7.544 -3.958 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.958 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.958 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.878 Td [(\001)]TJ/F22 10.9091 Tf -122.12 -25.229 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 14.349 -3.958 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F28 10.9091 Tf 13.137 4.878 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.958 Td [(k)]TJ/F25 10.9091 Tf 7.544 -3.958 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.958 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.958 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.878 Td [(\001)]TJ/F22 10.9091 Tf -114.095 -25.229 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 14.349 -3.958 Td [(=)-333(V-cycle)]TJ/F28 10.9091 Tf 47.273 8.836 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(k)]TJ/F15 10.9091 Tf 8.447 0 Td [(+)-222(1)]TJ/F22 10.9091 Tf 16.363 0 Td [(;)-167(A)]TJ/F23 7.9701 Tf 13.03 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F22 10.9091 Tf 11.319 -3.958 Td [(;)-167(b)]TJ/F23 7.9701 Tf 9.53 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F22 10.9091 Tf 11.319 -3.958 Td [(;)]TJ/F15 10.9091 Tf 4.848 0 Td [(0)]TJ/F28 10.9091 Tf 5.455 8.836 Td [(\001)]TJ/F22 10.9091 Tf -167.044 -25.229 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.958 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F15 10.9091 Tf 7.544 -3.958 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F22 10.9091 Tf 11.319 -3.958 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F22 10.9091 Tf -85.934 -20.352 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.959 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F15 10.9091 Tf 7.544 -3.959 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ/F28 10.9091 Tf 6.938 4.878 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.837 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.959 Td [(k)]TJ/F25 10.9091 Tf 7.544 -3.959 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.959 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.959 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.878 Td [(\001)]TJ/F15 10.9091 Tf -133.029 -25.22 Td [(else)]TJ/F22 10.9091 Tf 10.909 -18.375 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.959 Td [(=)]TJ/F28 10.9091 Tf 11.515 8.836 Td [(\000)]TJ/F22 10.9091 Tf 5 -8.836 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F28 10.9091 Tf 5.12 4.877 Td [(\001)]TJ/F26 7.9701 Tf 5 -2.497 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 6.551 -6.339 Td [(b)]TJ/F23 7.9701 Tf 4.682 3.959 Td [(k)]TJ/F15 10.9091 Tf -77.94 -20.343 Td [(endif)]TJ 0 -16.393 Td [(return)]TJ/F22 10.9091 Tf 33.394 0 Td [(u)]TJ/F23 7.9701 Tf 6.245 3.959 Td [(k)]TJ/F15 10.9091 Tf -50.548 -20.343 Td [(end)]TJ -ET -q -1 0 0 1 195.871 714.896 cm -[]0 d 0 J 0.398 w 0 0 m 207.909 0 l S -Q -q -1 0 0 1 196.071 504.814 cm -[]0 d 0 J 0.398 w 0 0 m 0 210.082 l S -Q -q -1 0 0 1 403.581 504.814 cm -[]0 d 0 J 0.398 w 0 0 m 0 210.082 l S -Q -q -1 0 0 1 195.871 504.814 cm -[]0 d 0 J 0.398 w 0 0 m 207.909 0 l S -Q -0 g 0 G -BT -/F15 10.9091 Tf 165.28 486.08 Td [(Figure)-333(1:)-445(Application)-333(phase)-333(of)-333(a)-334(V-cycle)-333(preconditioner.)]TJ -0 g 0 G -0 g 0 G - -71.68 -45.436 Td [(op)-28(erator)]TJ/F22 10.9091 Tf 42.856 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 7.611 -3.958 Td [(b)28(y)-229(apply)1(ing)-229(a)-228(suitable)-229(smo)-28(ot)1(he)-1(r)-228(to)-228(a)-229(simple)-228(piecewise)-229(constan)28(t)-228(prolongation)]TJ -58.986 -13.55 Td [(op)-28(erator,)-333(with)-333(the)-334(aim)-333(of)-333(impro)28(ving)-334(the)-333(qualit)28(y)-333(of)-334(the)-333(coarse-space)-334(correction.)]TJ 16.937 -13.787 Td [(Three)-333(main)-334(steps)-333(can)-333(b)-28(e)-334(i)1(den)27(ti\014ed)-333(in)-333(the)-334(smo)-27(othed)-334(aggregation)-333(pro)-28(cedure:)]TJ -0 g 0 G - -3.603 -23.466 Td [(1.)]TJ -0 g 0 G - [-500(aggregation)-333(of)-334(t)1(he)-334(indices)-333(of)-333(\012)]TJ/F23 7.9701 Tf 160.091 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.959 Td [(to)-333(obtain)-334(\012)]TJ/F23 7.9701 Tf 55.151 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.959 Td [(;)]TJ -0 g 0 G - -239.939 -23.467 Td [(2.)]TJ -0 g 0 G - [-500(construction)-333(of)-333(the)-334(prolongator)]TJ/F22 10.9091 Tf 166.515 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(;)]TJ -0 g 0 G - -180.154 -23.466 Td [(3.)]TJ -0 g 0 G - [-500(application)-333(of)]TJ/F22 10.9091 Tf 82.727 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.958 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.958 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.757 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.958 Td [(\051)]TJ/F23 7.9701 Tf 4.242 3.958 Td [(T)]TJ/F15 10.9091 Tf 10.241 -3.958 Td [(to)-333(build)]TJ/F22 10.9091 Tf 41.212 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.958 Td [(.)]TJ -243.343 -23.467 Td [(In)-407(order)-407(to)-407(p)-28(erform)-407(the)-407(coarsening)-407(s)-1(tep,)-425(the)-407(smo)-28(othed)-407(aggregation)-407(algorithm)-407(de-)]TJ -16.937 -13.549 Td [(scrib)-28(ed)-478(in)-478([)]TJ -1 0 0 rg 1 0 0 RG - [(26)]TJ -0 g 0 G - [(])-479(is)-478(used.)-879(In)-479(this)-478(algorithm,)-514(eac)27(h)-478(index)]TJ/F22 10.9091 Tf 271.801 0 Td [(j)]TJ/F25 10.9091 Tf 10.783 0 Td [(2)]TJ/F15 10.9091 Tf 12.939 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 16.537 -3.959 Td [(corresp)-28(onds)-478(to)-478(an)]TJ -324.561 -13.549 Td [(aggregate)-374(\012)]TJ/F23 7.9701 Tf 57.445 3.958 Td [(k)]TJ 0 -7.014 Td [(j)]TJ/F15 10.9091 Tf 9.201 3.056 Td [(of)-374(\012)]TJ/F23 7.9701 Tf 20.747 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.958 Td [(,)-384(consisting)-374(of)-374(a)-374(s)-1(u)1(itably)-374(c)27(hosen)-374(index)]TJ/F22 10.9091 Tf 187.709 0 Td [(i)]TJ/F25 10.9091 Tf 7.529 0 Td [(2)]TJ/F15 10.9091 Tf 11.044 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.201 -3.958 Td [(and)-374(indices)-374(that)-374(are)]TJ -315.875 -13.55 Td [(\050usually\051)-333(con)28(tained)-334(in)-333(a)-333(strongly-coupled)-334(n)1(e)-1(igh)28(b)-27(oro)-28(o)-28(d)-333(of)]TJ/F22 10.9091 Tf 274.122 0 Td [(i)]TJ/F15 10.9091 Tf 3.758 0 Td [(,)-333(i.e.,)]TJ -194.032 -31.214 Td [(\012)]TJ/F23 7.9701 Tf 7.879 4.505 Td [(k)]TJ 0 -7.202 Td [(j)]TJ/F25 10.9091 Tf 8.15 2.697 Td [(\032)-278(N)]TJ/F23 7.9701 Tf 22.073 4.505 Td [(k)]TJ -1.607 -7.202 Td [(i)]TJ/F15 10.9091 Tf 6.727 2.697 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(\022)]TJ/F15 10.9091 Tf 5.425 0 Td [(\051)-278(=)]TJ/F28 10.9091 Tf 18.788 15.382 Td [(\032)]TJ/F22 10.9091 Tf 8.181 -15.382 Td [(r)]TJ/F25 10.9091 Tf 8.255 0 Td [(2)]TJ/F15 10.9091 Tf 10.303 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 4.505 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.505 Td [(:)]TJ/F25 10.9091 Tf 6.061 0 Td [(j)]TJ/F22 10.9091 Tf 3.03 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 4.505 Td [(k)]TJ 0 -7.202 Td [(ir)]TJ/F25 10.9091 Tf 7.438 2.697 Td [(j)]TJ/F22 10.9091 Tf 6.061 0 Td [(>)-278(\022)]TJ/F28 10.9091 Tf 16.939 13.424 Td [(q)]TJ -ET -q -1 0 0 1 348.097 261.221 cm -[]0 d 0 J 0.436 w 0 0 m 32.469 0 l S -Q -BT -/F25 10.9091 Tf 348.097 247.579 Td [(j)]TJ/F22 10.9091 Tf 3.031 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 3.758 Td [(k)]TJ 0 -7.014 Td [(ii)]TJ/F22 10.9091 Tf 6.264 3.256 Td [(a)]TJ/F23 7.9701 Tf 5.767 3.152 Td [(k)]TJ 0 -5.849 Td [(r)-29(r)]TJ/F25 10.9091 Tf 8.611 2.697 Td [(j)]TJ/F28 10.9091 Tf 3.031 15.382 Td [(\033)]TJ/F25 10.9091 Tf 10.606 -15.382 Td [([)-222(f)]TJ/F22 10.9091 Tf 15.151 0 Td [(i)]TJ/F25 10.9091 Tf 3.758 0 Td [(g)]TJ/F22 10.9091 Tf 7.273 0 Td [(;)]TJ -0 g 0 G -/F15 10.9091 Tf 74.757 0 Td [(\0503\051)]TJ -0 g 0 G - -398.512 -31.214 Td [(for)-431(a)-430(giv)27(en)-430(threshold)]TJ/F22 10.9091 Tf 106.493 0 Td [(\022)]TJ/F25 10.9091 Tf 10.225 0 Td [(2)]TJ/F15 10.9091 Tf 12.074 0 Td [([0)]TJ/F22 10.9091 Tf 8.485 0 Td [(;)]TJ/F15 10.9091 Tf 4.848 0 Td [(1])-431(\050see)-431([)]TJ -1 0 0 rg 1 0 0 RG - [(26)]TJ -0 g 0 G - [(])-430(for)-431(the)-431(details\051.)-736(Since)-431(this)-431(algorithm)-430(has)-431(a)]TJ -142.125 -13.549 Td [(sequen)28(tial)-382(n)1(ature,)-394(a)-381(decoupled)-381(v)28(e)-1(r)1(s)-1(i)1(on)-382(of)-381(it)-381(is)-382(appli)1(e)-1(d,)-393(where)-381(eac)28(h)-382(pro)-27(ces)-1(sor)-381(indep)-28(en-)]TJ 0 -13.549 Td [(den)28(tly)-433(exe)-1(cutes)-433(the)-433(algorithm)-433(on)-434(the)-433(set)-433(of)-434(indices)-433(assigned)-433(to)-434(it)-433(in)-433(the)-433(initial)-434(data)]TJ 0 -13.549 Td [(distribution.)-442(Thi)1(s)-326(v)28(ersion)-325(is)-326(em)28(barrassingly)-325(parallel,)-327(since)-325(it)-326(do)-27(es)-326(not)-325(require)-325(an)27(y)-325(data)]TJ 0 -13.55 Td [(comm)28(unication.)-522(On)-359(the)-359(other)-359(hand,)-365(it)-360(ma)28(y)-359(pro)-28(du)1(c)-1(e)-359(some)-359(non)28(uniform)-359(aggregates)-359(and)]TJ 0 -13.549 Td [(is)-295(strongly)-295(dep)-27(enden)27(t)-294(on)-295(the)-295(n)28(um)28(b)-28(er)-295(of)-295(pro)-27(cess)-1(or)1(s)-295(and)-295(on)-295(the)-295(initial)-294(partitioning)-295(of)-295(the)]TJ 0 -13.549 Td [(matrix)]TJ/F22 10.9091 Tf 35.775 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(.)-524(Nev)28(e)-1(rt)1(hele)-1(ss,)-366(this)-360(parallel)-360(algorithm)-360(has)-360(b)-27(ee)-1(n)-359(c)27(hosen)-360(for)-359(MLD2P4,)-367(since)-360(it)]TJ -43.957 -13.549 Td [(has)-333(b)-28(een)-333(s)-1(h)1(o)27(wn)-333(to)-333(pro)-28(duce)-333(go)-28(o)-28(d)-333(results)-334(in)-333(practice)-333([)]TJ -1 0 0 rg 1 0 0 RG - [(5)]TJ -0 g 0 G - [(,)]TJ -1 0 0 rg 1 0 0 RG - [-333(7)]TJ -0 g 0 G - [(,)]TJ -1 0 0 rg 1 0 0 RG - [-334(25)]TJ -0 g 0 G - [(].)]TJ -0 g 0 G -0 g 0 G -ET - -endstream -endobj -392 0 obj -<< -/Length 15462 ->> -stream -0 g 0 G -BT -/F15 10.9091 Tf 86.4 740.002 Td [(14)]TJ/F42 10.9091 Tf 203.265 0 Td [(MLD2P4)-378(User)67('s)-378(and)-378(Ref)1(erence)-378(Guide)]TJ -0 g 0 G -/F15 10.9091 Tf -186.329 -35.866 Td [(The)-375(prolongator)]TJ/F22 10.9091 Tf 82.192 0 Td [(P)]TJ/F23 7.9701 Tf 8.518 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.216 -3.959 Td [(is)-375(built)-376(starting)-375(from)-375(a)-376(ten)28(tativ)28(e)-376(prolongator)]TJ 222.868 2.758 Td [(\026)]TJ/F22 10.9091 Tf -2.441 -2.758 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F25 10.9091 Tf 8.915 -3.959 Td [(2)]TJ/F34 10.9091 Tf 11.069 0 Td [(R)]TJ/F23 7.9701 Tf 7.878 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.139 -1.406 Td [(k)]TJ/F26 7.9701 Tf 4.573 1.406 Td [(\002)]TJ/F23 7.9701 Tf 6.586 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F21 5.9776 Tf 4.075 0 Td [(+1)]TJ/F15 10.9091 Tf 10.24 -2.553 Td [(,)]TJ -409.421 -13.549 Td [(de\014ned)-333(as)]TJ 115.609 -14.79 Td [(\026)]TJ/F22 10.9091 Tf -2.442 -2.757 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.504 Td [(=)-278(\050)-85(\026)]TJ/F22 10.9091 Tf 15.758 0 Td [(p)]TJ/F23 7.9701 Tf 5.489 4.504 Td [(k)]TJ 0 -7.201 Td [(ij)]TJ/F15 10.9091 Tf 7.265 2.697 Td [(\051)]TJ/F22 10.9091 Tf 4.242 0 Td [(;)]TJ/F15 10.9091 Tf 16.684 0 Td [(\026)]TJ/F22 10.9091 Tf -0.926 0 Td [(p)]TJ/F23 7.9701 Tf 5.489 4.504 Td [(k)]TJ 0 -7.201 Td [(ij)]TJ/F15 10.9091 Tf 10.295 2.697 Td [(=)]TJ/F28 10.9091 Tf 11.515 15.382 Td [(\032)]TJ/F15 10.9091 Tf 13.163 -8.325 Td [(1)-1913(if)]TJ/F22 10.9091 Tf 35.721 0 Td [(i)]TJ/F25 10.9091 Tf 6.788 0 Td [(2)]TJ/F15 10.9091 Tf 10.303 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(k)]TJ 0 -7.014 Td [(j)]TJ/F22 10.9091 Tf 5.12 3.055 Td [(;)]TJ/F15 10.9091 Tf -65.811 -14.089 Td [(0)-1913(otherwise)]TJ/F22 10.9091 Tf 71.266 0 Td [(;)]TJ -0 g 0 G -/F15 10.9091 Tf 108.435 7.032 Td [(\0504\051)]TJ -0 g 0 G - -398.511 -26.829 Td [(where)-357(\012)]TJ/F23 7.9701 Tf 39.687 3.959 Td [(k)]TJ 0 -7.015 Td [(j)]TJ/F15 10.9091 Tf 9.02 3.056 Td [(is)-357(the)-358(aggregate)-357(of)-358(\012)]TJ/F23 7.9701 Tf 100.235 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.019 -3.959 Td [(corresp)-28(onding)-357(to)-358(the)-357(index)]TJ/F22 10.9091 Tf 132.69 0 Td [(j)]TJ/F25 10.9091 Tf 8.586 0 Td [(2)]TJ/F15 10.9091 Tf 10.742 0 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.318 -3.959 Td [(.)]TJ/F22 10.9091 Tf 8.669 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.02 -3.959 Td [(is)-357(obtained)]TJ -360.006 -15.095 Td [(b)28(y)-333(applying)-334(to)]TJ 75.471 2.757 Td [(\026)]TJ/F22 10.9091 Tf -2.441 -2.757 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.959 Td [(a)-333(smo)-28(other)]TJ/F22 10.9091 Tf 56.758 0 Td [(S)]TJ/F23 7.9701 Tf 7.318 3.959 Td [(k)]TJ/F25 10.9091 Tf 8.15 -3.959 Td [(2)]TJ/F34 10.9091 Tf 10.303 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F26 7.9701 Tf 4.573 1.406 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F15 10.9091 Tf 5.071 -2.553 Td [(:)]TJ/F22 10.9091 Tf -29.64 -22.041 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.504 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(S)]TJ/F23 7.9701 Tf 7.318 4.504 Td [(k)]TJ/F15 10.9091 Tf 7.561 -1.746 Td [(\026)]TJ/F22 10.9091 Tf -2.441 -2.758 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(;)]TJ/F15 10.9091 Tf -231.841 -22.041 Td [(in)-255(order)-255(to)-255(remo)27(v)28(e)-255(nonsmo)-28(oth)-255(comp)-28(onen)28(ts)-255(from)-255(the)-256(r)1(ange)-256(of)-255(the)-255(prolongator,)-271(and)-255(hence)]TJ 0 -13.549 Td [(to)-293(impro)27(v)28(e)-293(the)-294(con)28(v)28(ergence)-294(prop)-27(erties)-294(of)-293(the)-294(m)28(ultilev)28(el)-294(metho)-27(d)-294([)]TJ -1 0 0 rg 1 0 0 RG - [(2)]TJ -0 g 0 G - [(,)]TJ -1 0 0 rg 1 0 0 RG - [-293(24)]TJ -0 g 0 G - [(].)-431(A)-294(simple)-293(c)28(hoice)]TJ 0 -13.549 Td [(for)]TJ/F22 10.9091 Tf 16.697 0 Td [(S)]TJ/F23 7.9701 Tf 7.318 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.959 Td [(is)-333(the)-334(damp)-27(ed)-334(Jacobi)-333(smo)-28(other:)]TJ/F22 10.9091 Tf 118.068 -22.041 Td [(S)]TJ/F23 7.9701 Tf 7.318 4.505 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.505 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(I)]TJ/F25 10.9091 Tf 8.076 0 Td [(\000)]TJ/F22 10.9091 Tf 10.909 0 Td [(!)]TJ/F23 7.9701 Tf 7.182 4.505 Td [(k)]TJ/F15 10.9091 Tf 5.119 -4.505 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(D)]TJ/F23 7.9701 Tf 9.335 4.505 Td [(k)]TJ/F15 10.9091 Tf 5.119 -4.505 Td [(\051)]TJ/F26 7.9701 Tf 4.243 4.505 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.733 -4.505 Td [(A)]TJ/F23 7.9701 Tf 8.181 4.505 Td [(k)]TJ 0 -7.202 Td [(F)]TJ/F22 10.9091 Tf 7.034 2.697 Td [(;)]TJ/F15 10.9091 Tf -258.582 -22.04 Td [(where)]TJ/F22 10.9091 Tf 32.045 0 Td [(D)]TJ/F23 7.9701 Tf 9.335 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.255 -3.958 Td [(is)-379(the)-379(diagonal)-379(matrix)-379(with)-379(the)-379(s)-1(ame)-379(diagonal)-379(en)28(tries)-379(as)]TJ/F22 10.9091 Tf 276.813 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.958 Td [(,)]TJ/F22 10.9091 Tf 7.291 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ 0 -7.191 Td [(F)]TJ/F15 10.9091 Tf 10.896 3.233 Td [(=)-354(\050)-14(\026)]TJ/F22 10.9091 Tf 16.59 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 3.958 Td [(k)]TJ 0 -7.014 Td [(ij)]TJ/F15 10.9091 Tf 7.266 3.056 Td [(\051)-379(is)]TJ -396.74 -13.55 Td [(the)-333(\014ltered)-334(matrix)-333(de\014ned)-333(as)]TJ 48.968 -26.039 Td [(\026)]TJ/F22 10.9091 Tf -0.156 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 4.505 Td [(k)]TJ 0 -7.202 Td [(ij)]TJ/F15 10.9091 Tf 10.296 2.697 Td [(=)]TJ/F28 10.9091 Tf 11.515 15.382 Td [(\032)]TJ/F22 10.9091 Tf 13.163 -8.324 Td [(a)]TJ/F23 7.9701 Tf 5.766 3.959 Td [(k)]TJ 0 -7.015 Td [(ij)]TJ/F15 10.9091 Tf 17.228 3.056 Td [(if)]TJ/F22 10.9091 Tf 10 0 Td [(j)]TJ/F25 10.9091 Tf 8.147 0 Td [(2)-278(N)]TJ/F23 7.9701 Tf 20.862 3.959 Td [(k)]TJ -1.608 -7.015 Td [(i)]TJ/F15 10.9091 Tf 6.728 3.056 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(\022)]TJ/F15 10.9091 Tf 5.424 0 Td [(\051)]TJ/F22 10.9091 Tf 4.243 0 Td [(;)]TJ/F15 10.9091 Tf -81.032 -14.09 Td [(0)-1608(otherwise)]TJ/F22 10.9091 Tf 67.934 0 Td [(;)]TJ/F15 10.9091 Tf 27.153 7.032 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(j)]TJ/F25 10.9091 Tf 8.147 0 Td [(6)]TJ/F15 10.9091 Tf 0 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(i)]TJ/F15 10.9091 Tf 3.758 0 Td [(\051)]TJ/F22 10.9091 Tf 4.243 0 Td [(;)]TJ/F15 10.9091 Tf 26.822 0 Td [(\026)]TJ/F22 10.9091 Tf -0.156 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 4.505 Td [(k)]TJ 0 -7.202 Td [(ii)]TJ/F15 10.9091 Tf 9.294 2.697 Td [(=)]TJ/F22 10.9091 Tf 11.516 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 4.505 Td [(k)]TJ 0 -7.202 Td [(ii)]TJ/F25 10.9091 Tf 8.689 2.697 Td [(\000)]TJ/F28 10.9091 Tf 10.909 10.364 Td [(X)]TJ/F23 7.9701 Tf 1.202 -23.717 Td [(j)]TJ/F26 7.9701 Tf 3.884 0 Td [(6)]TJ/F20 7.9701 Tf 0 0 Td [(=)]TJ/F23 7.9701 Tf 6.586 0 Td [(i)]TJ/F15 10.9091 Tf 4.085 13.353 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 4.505 Td [(k)]TJ 0 -7.202 Td [(ij)]TJ/F25 10.9091 Tf 9.69 2.697 Td [(\000)]TJ/F15 10.9091 Tf 11.065 0 Td [(\026)]TJ/F22 10.9091 Tf -0.156 0 Td [(a)]TJ/F23 7.9701 Tf 5.766 4.505 Td [(k)]TJ 0 -7.202 Td [(ij)]TJ/F15 10.9091 Tf 7.265 2.697 Td [(\051)]TJ/F22 10.9091 Tf 4.243 0 Td [(;)]TJ -0 g 0 G -/F15 10.9091 Tf 39.721 0 Td [(\0505\051)]TJ -0 g 0 G - -398.512 -34.975 Td [(and)]TJ/F22 10.9091 Tf 20.68 0 Td [(!)]TJ/F23 7.9701 Tf 7.182 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.224 -3.959 Td [(is)-285(an)-284(appro)28(ximation)-285(of)-284(4)]TJ/F22 10.9091 Tf 114.629 0 Td [(=)]TJ/F15 10.9091 Tf 5.455 0 Td [(\0503)]TJ/F22 10.9091 Tf 9.697 0 Td [(\032)]TJ/F23 7.9701 Tf 5.64 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(\051,)-294(where)]TJ/F22 10.9091 Tf 41.497 0 Td [(\032)]TJ/F23 7.9701 Tf 5.64 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.224 -3.959 Td [(is)-285(the)-284(sp)-28(ectral)-285(r)1(adius)-285(of)-284(\050)]TJ/F22 10.9091 Tf 117.582 0 Td [(D)]TJ/F23 7.9701 Tf 9.335 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(\051)]TJ/F26 7.9701 Tf 4.242 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -3.959 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ 0 -7.192 Td [(F)]TJ/F15 10.9091 Tf 10.138 3.233 Td [([)]TJ -1 0 0 rg 1 0 0 RG - [(2)]TJ -0 g 0 G - [(].)]TJ -397.906 -13.549 Td [(In)-288(MLD2P4)-288(this)-287(appro)27(ximation)-287(is)-288(obtained)-288(b)28(y)-288(using)]TJ/F25 10.9091 Tf 250.336 0 Td [(k)]TJ/F22 10.9091 Tf 5.455 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.959 Td [(k)]TJ 0 -7.192 Td [(F)]TJ/F25 10.9091 Tf 7.034 3.233 Td [(k)]TJ/F26 7.9701 Tf 5.454 -1.636 Td [(1)]TJ/F15 10.9091 Tf 12.107 1.636 Td [(as)-288(an)-288(estimate)-288(of)]TJ/F22 10.9091 Tf 82.683 0 Td [(\032)]TJ/F23 7.9701 Tf 5.641 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.959 Td [(.)-429(Note)]TJ -382.01 -13.549 Td [(that)-373(for)-373(systems)-374(coming)-373(from)-373(uniformly)-373(elliptic)-373(problems,)-384(\014)1(ltering)-374(the)-373(matrix)]TJ/F22 10.9091 Tf 379.26 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.191 -3.959 Td [(has)]TJ -396.633 -13.549 Td [(little)-306(or)-306(no)-306(e\013ect,)-312(and)]TJ/F22 10.9091 Tf 106.178 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ/F15 10.9091 Tf 8.459 -3.958 Td [(can)-306(b)-28(e)-306(used)-306(instead)-306(of)]TJ/F22 10.9091 Tf 108.331 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ 0 -7.191 Td [(F)]TJ/F15 10.9091 Tf 7.033 3.233 Td [(.)-435(The)-306(latter)-306(c)27(hoice)-306(is)-306(the)-306(default)-306(in)]TJ -246.365 -13.55 Td [(MLD2P4.)]TJ/F17 11.9552 Tf 0 -28.825 Td [(4.3)-1125(Smo)-31(others)-375(and)-375(coarsest-lev)31(el)-375(solv)31(ers)]TJ/F15 10.9091 Tf 0 -20.594 Td [(The)-294(smo)-28(others)-294(implemen)28(ted)-295(i)1(n)-295(MLD2P4)-294(include)-294(the)-294(Jacobi)-294(and)-294(blo)-28(c)28(k-Jacobi)-294(metho)-28(ds,)]TJ 0 -13.549 Td [(a)-344(h)28(ybrid)-343(v)28(e)-1(r)1(s)-1(ion)-343(of)-344(the)-343(forw)28(ard)-344(and)-344(b)1(ac)27(kw)28(ard)-344(Gau)1(s)-1(s-Seidel)-343(metho)-28(ds,)-346(and)-344(the)-343(additiv)28(e)]TJ 0 -13.55 Td [(Sc)28(h)28(w)27(arz)-333(\050AS\051)-333(ones)-334(\050see,)-333(e.g.,)-333([)]TJ -1 0 0 rg 1 0 0 RG - [(21)]TJ -0 g 0 G - [(,)]TJ -1 0 0 rg 1 0 0 RG - [-334(22)]TJ -0 g 0 G - [(]\051.)]TJ 16.936 -13.549 Td [(The)-481(h)28(ybrid)-480(Gauss-Seidel)-481(v)28(ersion)-481(is)-480(considered)-481(b)-27(ecause)-481(the)-481(origin)1(al)-481(Gauss-Seidel)]TJ -16.936 -13.549 Td [(metho)-28(d)-364(is)-365(inheren)28(tly)-364(sequen)27(tial.)-537(A)27(t)-364(eac)28(h)-365(iteration)-364(of)-365(the)-364(h)28(ybrid)-365(v)28(ersion,)-372(eac)27(h)-364(parallel)]TJ 0 -13.549 Td [(pro)-28(cess)-452(uses)-452(the)-451(mos)-1(t)-451(recen)27(t)-451(v)55(alues)-452(of)-452(i)1(ts)-452(o)27(wn)-451(lo)-28(cal)-452(v)56(ariables)-452(and)-452(the)-452(v)56(alues)-452(of)-452(the)]TJ 0 -13.549 Td [(non-lo)-28(cal)-418(v)55(ari)1(ables)-419(computed)-418(at)-419(the)-418(previous)-418(iteration,)-440(obtained)-418(b)28(y)-419(exc)28(hanging)-418(data)]TJ 0 -13.55 Td [(with)-333(other)-334(pr)1(o)-28(cesses)-334(b)-28(efore)-333(the)-333(b)-28(eginning)-333(of)-334(th)1(e)-334(curren)28(t)-333(iteration.)]TJ 16.936 -13.549 Td [(In)-372(the)-372(AS)-372(me)-1(th)1(o)-28(ds,)-382(the)-372(index)-372(space)-373(\012)]TJ/F23 7.9701 Tf 185.618 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.18 -3.959 Td [(is)-372(divided)-372(in)28(to)]TJ/F22 10.9091 Tf 72.847 0 Td [(m)]TJ/F23 7.9701 Tf 9.578 -1.777 Td [(k)]TJ/F15 10.9091 Tf 9.18 1.777 Td [(subsets)-372(\012)]TJ/F23 7.9701 Tf 46.06 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 9.18 3.055 Td [(of)-372(size)]TJ/F22 10.9091 Tf 33.939 0 Td [(n)]TJ/F23 7.9701 Tf 6.548 -1.777 Td [(k)-27(;i)]TJ/F15 10.9091 Tf 10.355 1.777 Td [(,)]TJ -409.421 -13.549 Td [(p)-28(ossibly)-432(o)28(v)28(erlapping.)-740(F)83(or)-432(eac)28(h)]TJ/F22 10.9091 Tf 155.272 0 Td [(i)]TJ/F15 10.9091 Tf 8.47 0 Td [(w)28(e)-432(consider)-432(the)-432(restriction)-432(op)-28(erator)]TJ/F22 10.9091 Tf 178.985 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(k)]TJ -0.084 -7.014 Td [(i)]TJ/F25 10.9091 Tf 10.027 3.055 Td [(2)]TJ/F34 10.9091 Tf 12.095 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)-21(;i)]TJ/F26 7.9701 Tf 9.506 1.406 Td [(\002)]TJ/F23 7.9701 Tf 6.587 0 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)]TJ/F15 10.9091 Tf -407.38 -16.102 Td [(that)-366(maps)-366(a)-365(v)27(ector)]TJ/F22 10.9091 Tf 95.448 0 Td [(x)]TJ/F23 7.9701 Tf 6.235 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.11 -3.959 Td [(to)-366(the)-366(v)28(ector)]TJ/F22 10.9091 Tf 65.942 0 Td [(x)]TJ/F23 7.9701 Tf 6.235 3.959 Td [(k)]TJ 0 -7.015 Td [(i)]TJ/F15 10.9091 Tf 9.11 3.056 Td [(made)-366(of)-366(the)-365(com)-1(p)-27(onen)28(ts)-366(of)]TJ/F22 10.9091 Tf 134.56 0 Td [(x)]TJ/F23 7.9701 Tf 6.234 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.111 -3.959 Td [(with)-366(indices)-366(in)]TJ -341.985 -13.549 Td [(\012)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(k)]TJ 0 -7.015 Td [(i)]TJ/F15 10.9091 Tf 5.12 3.056 Td [(,)-301(and)-292(the)-293(prolongation)-292(op)-28(erator)]TJ/F22 10.9091 Tf 152.201 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ -1.515 -7.015 Td [(i)]TJ/F15 10.9091 Tf 9.665 3.056 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.758 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(k)]TJ -0.084 -7.015 Td [(i)]TJ/F15 10.9091 Tf 5.204 3.056 Td [(\051)]TJ/F23 7.9701 Tf 4.242 3.959 Td [(T)]TJ/F15 10.9091 Tf 6.605 -3.959 Td [(.)-431(These)-293(op)-27(erators)-293(are)-293(then)-292(used)-293(to)-292(build)]TJ/F22 10.9091 Tf -221.961 -13.549 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.557 3.056 Td [(=)]TJ/F22 10.9091 Tf 11.923 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.958 Td [(k)]TJ -0.084 -7.014 Td [(i)]TJ/F22 10.9091 Tf 5.204 3.056 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ/F22 10.9091 Tf 5.119 -3.958 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ -1.515 -7.014 Td [(i)]TJ/F15 10.9091 Tf 6.635 3.056 Td [(,)-361(whic)27(h)-355(is)-356(the)-356(restriction)-355(of)]TJ/F22 10.9091 Tf 133.83 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.001 -3.958 Td [(to)-356(the)-355(index)-356(space)-356(\012)]TJ/F23 7.9701 Tf 99.523 3.958 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.119 3.056 Td [(.)-512(The)-355(class)-1(ical)-355(AS)]TJ -324.744 -13.55 Td [(preconditioner)]TJ/F22 10.9091 Tf 72.182 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ -1.19 -7.192 Td [(AS)]TJ/F15 10.9091 Tf 16.074 3.233 Td [(is)-333(de\014ned)-334(as)]TJ 38.283 -31.315 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 4.504 Td [(k)]TJ -1.19 -7.201 Td [(AS)]TJ/F15 10.9091 Tf 12.437 2.697 Td [(\051)]TJ/F26 7.9701 Tf 4.242 4.504 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F15 10.9091 Tf 7.763 -4.504 Td [(=)]TJ/F23 7.9701 Tf 13.362 14.072 Td [(m)]TJ/F24 5.9776 Tf 7.49 -1.406 Td [(k)]TJ/F28 10.9091 Tf -9.337 -2.303 Td [(X)]TJ/F23 7.9701 Tf 1.027 -23.45 Td [(i)]TJ/F20 7.9701 Tf 2.883 0 Td [(=1)]TJ/F22 10.9091 Tf 13.665 13.087 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.504 Td [(k)]TJ -1.515 -7.201 Td [(i)]TJ/F15 10.9091 Tf 6.635 2.697 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 4.504 Td [(k)]TJ 0 -7.201 Td [(i)]TJ/F15 10.9091 Tf 5.12 2.697 Td [(\051)]TJ/F26 7.9701 Tf 4.243 4.504 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -4.504 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.504 Td [(k)]TJ -0.084 -7.201 Td [(i)]TJ/F22 10.9091 Tf 5.204 2.697 Td [(;)]TJ -0 g 0 G -0 g 0 G -ET - -endstream -endobj -401 0 obj -<< -/Length 7392 ->> -stream -0 g 0 G -0 0 1 rg 0 0 1 RG -BT -/F42 10.9091 Tf 93.6 740.002 Td [(4)]TJ -0 g 0 G - [-378(Mul)67(tigrid)-378(Ba)22(ck)22(gr)23(ound)]TJ/F15 10.9091 Tf 401.542 0 Td [(15)]TJ -0 g 0 G - -401.542 -35.866 Td [(where)]TJ/F22 10.9091 Tf 31.037 0 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.247 3.055 Td [(is)-287(supp)-27(osed)-287(to)-287(b)-27(e)-287(nonsingular.)-429(W)83(e)-286(observ)28(e)-287(that)-287(an)-286(appro)28(ximate)-287(in)28(v)28(e)-1(rse)-286(of)]TJ/F22 10.9091 Tf 351.685 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf -407.332 -10.494 Td [(is)-342(usually)-342(considered)-342(instead)-342(of)-342(\050)]TJ/F22 10.9091 Tf 156.503 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.119 3.055 Td [(\051)]TJ/F26 7.9701 Tf 4.243 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F15 10.9091 Tf 4.733 -3.959 Td [(.)-470(The)-342(setup)-342(of)]TJ/F22 10.9091 Tf 72.445 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ -1.19 -7.192 Td [(AS)]TJ/F15 10.9091 Tf 16.168 3.233 Td [(during)-342(the)-342(m)28(ultilev)28(el)-342(build)]TJ -284.562 -13.549 Td [(phase)-333(in)27(v)28(olv)28(es)]TJ -0 g 0 G -/F25 10.9091 Tf 16.364 -22.516 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-383(de\014ni)1(tion)-383(of)-383(the)-382(index)-383(subspaces)-382(\012)]TJ/F23 7.9701 Tf 188.861 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 9.294 3.055 Td [(and)-383(of)-382(the)-383(corresp)-27(onding)-383(op)-28(erators)]TJ/F22 10.9091 Tf 173.536 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(k)]TJ -0.085 -7.014 Td [(i)]TJ/F15 10.9091 Tf -379.974 -10.494 Td [(\050and)]TJ/F22 10.9091 Tf 25.455 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ -1.516 -7.014 Td [(i)]TJ/F15 10.9091 Tf 6.635 3.055 Td [(\051;)]TJ -0 g 0 G -/F25 10.9091 Tf -50.002 -22.515 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-333(computation)-334(of)-333(the)-333(submatrices)]TJ/F22 10.9091 Tf 173.788 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.958 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.12 3.056 Td [(;)]TJ -0 g 0 G -/F25 10.9091 Tf -197.999 -22.516 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-424(computation)-425(of)-424(their)-424(in)27(v)28(erses)-424(\050usually)-425(appro)28(ximated)-424(through)-424(some)-425(form)-424(of)]TJ 0 -13.549 Td [(incomplete)-333(factorization\051.)]TJ -27.273 -22.516 Td [(The)-410(computation)-411(of)]TJ/F22 10.9091 Tf 101.006 0 Td [(z)]TJ/F23 7.9701 Tf 5.553 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.551 -3.959 Td [(=)]TJ/F22 10.9091 Tf 12.916 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ -1.19 -7.192 Td [(AS)]TJ/F22 10.9091 Tf 12.437 3.233 Td [(w)]TJ/F23 7.9701 Tf 8.104 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.959 Td [(,)-430(with)]TJ/F22 10.9091 Tf 33.407 0 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.959 Td [(k)]TJ/F25 10.9091 Tf 9.551 -3.959 Td [(2)]TJ/F34 10.9091 Tf 11.704 0 Td [(R)]TJ/F23 7.9701 Tf 7.878 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.139 -1.406 Td [(k)]TJ/F15 10.9091 Tf 5.071 -2.553 Td [(,)-430(dur)1(ing)-411(the)-410(m)28(ultilev)27(el)-410(application)]TJ -246.122 -13.549 Td [(phase,)-333(requires)]TJ -0 g 0 G -/F25 10.9091 Tf 16.364 -22.516 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-333(restriction)-334(of)]TJ/F22 10.9091 Tf 83.455 0 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.959 Td [(to)-333(the)-334(subspaces)]TJ/F34 10.9091 Tf 82 0 Td [(R)]TJ/F23 7.9701 Tf 7.879 3.959 Td [(n)]TJ/F24 5.9776 Tf 5.138 -1.406 Td [(k)-21(;i)]TJ/F15 10.9091 Tf 10.004 -2.553 Td [(,)-333(i.e.)]TJ/F22 10.9091 Tf 24.243 0 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.959 Td [(k)]TJ -0.293 -7.014 Td [(i)]TJ/F15 10.9091 Tf 8.443 3.055 Td [(=)]TJ/F22 10.9091 Tf 11.516 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(k)]TJ -0.084 -7.014 Td [(i)]TJ/F22 10.9091 Tf 5.204 3.055 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(;)]TJ -0 g 0 G -/F25 10.9091 Tf -294.966 -22.515 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-333(computation)-334(of)-333(the)-333(v)28(ectors)]TJ/F22 10.9091 Tf 150.697 0 Td [(z)]TJ/F23 7.9701 Tf 5.553 3.959 Td [(k)]TJ -0.48 -7.015 Td [(i)]TJ/F15 10.9091 Tf 8.63 3.056 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.758 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ 0 -7.015 Td [(i)]TJ/F15 10.9091 Tf 5.119 3.056 Td [(\051)]TJ/F26 7.9701 Tf 4.243 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.733 -3.959 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.959 Td [(k)]TJ -0.293 -7.015 Td [(i)]TJ/F15 10.9091 Tf 5.413 3.056 Td [(;)]TJ -0 g 0 G -/F25 10.9091 Tf -233.153 -22.516 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-333(prolongation)-333(and)-334(the)-333(sum)-333(of)-334(the)-333(previous)-333(v)28(e)-1(ctors,)-333(i.e.)]TJ/F22 10.9091 Tf 277.849 0 Td [(z)]TJ/F23 7.9701 Tf 5.553 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.959 Td [(=)]TJ/F28 10.9091 Tf 11.515 8.182 Td [(P)]TJ/F23 7.9701 Tf 11.515 -3.018 Td [(m)]TJ/F24 5.9776 Tf 7.491 -1.406 Td [(k)]TJ/F23 7.9701 Tf -7.491 -7.014 Td [(i)]TJ/F20 7.9701 Tf 2.883 0 Td [(=1)]TJ/F22 10.9091 Tf 13.137 3.256 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ -1.515 -7.014 Td [(i)]TJ/F22 10.9091 Tf 6.635 3.055 Td [(z)]TJ/F23 7.9701 Tf 5.553 3.959 Td [(k)]TJ -0.48 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.6 3.055 Td [(.)]TJ -382.187 -22.515 Td [(V)83(arian)28(ts)-291(of)-291(the)-291(classical)-291(AS)-291(metho)-28(d,)-299(whic)28(h)-291(use)-291(mo)-28(di\014cations)-291(of)-291(the)-291(restriction)-291(and)-291(pro-)]TJ 0 -13.549 Td [(longation)-244(op)-28(erators,)-262(are)-244(also)-244(implemen)28(ted)-244(in)-244(MLD2P4.)-415(Among)-244(them,)-262(the)-244(Restricted)-244(AS)]TJ 0 -13.55 Td [(\050RAS\051)-375(preconditi)1(one)-1(r)-374(usually)-375(outp)-27(erforms)-375(the)-375(classical)-375(AS)-374(preconditioner)-375(in)-374(terms)-375(of)]TJ 0 -13.549 Td [(con)28(v)28(erge)-1(n)1(c)-1(e)-337(rate)-337(and)-338(of)-337(computation)-338(and)-337(comm)28(unication)-338(time)-337(on)-338(parallel)-337(distributed-)]TJ 0 -13.549 Td [(memory)-309(computers,)-314(and)-310(is)-309(therefore)-309(the)-310(most)-309(widely)-309(used)-309(am)-1(on)1(g)-310(the)-309(AS)-309(precondition-)]TJ 0 -13.549 Td [(ers)-333([)]TJ -1 0 0 rg 1 0 0 RG - [(6)]TJ -0 g 0 G - [(].)]TJ 16.937 -13.549 Td [(Direct)-427(solv)28(ers)-426(based)-427(on)-426(s)-1(p)1(arse)-427(LU)-427(factori)1(z)-1(ation)1(s)-1(,)-449(implemen)27(ted)-426(in)-427(the)-426(third-part)28(y)]TJ -16.937 -13.55 Td [(libraries)-374(rep)-27(orted)-374(in)-374(Section)]TJ -0 0 1 rg 0 0 1 RG - [-373(3.2)]TJ -0 g 0 G - [(,)-384(can)-374(b)-27(e)-374(applied)-374(as)-373(coarsest-lev)27(el)-373(solv)27(ers)-373(b)27(y)-373(MLD2P4.)]TJ 0 -13.549 Td 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[(W)83(e)-270(describ)-28(e)-270(the)-270(basic)-1(s)-270(for)-270(building)-270(and)-271(ap)1(plying)-271(MLD2P4)-270(one-lev)28(el)-271(and)-270(m)28(ultilev)28(el)-271(\050i.e.,)]TJ 0 -13.549 Td [(AMG\051)-283(preconditioners)-283(with)-283(the)-283(Krylo)27(v)-283(solv)28(ers)-283(included)-283(in)-283(PSBLAS)-283([)]TJ -1 0 0 rg 1 0 0 RG - [(13)]TJ -0 g 0 G - [(].)-428(The)-283(follo)28(wing)]TJ 0 -13.549 Td [(steps)-333(are)-334(required:)]TJ -0 g 0 G - 13.333 -23.442 Td [(1.)]TJ -0 g 0 G -/F18 10.9091 Tf 13.94 0 Td [(De)51(clar)51(e)-370(the)-371(pr)51(e)51(c)51(onditioner)-370(data)-371(structur)51(e)]TJ/F15 10.9091 Tf 195.447 0 Td [(.)-487(It)-347(is)-347(a)-348(deriv)28(ed)-347(data)-348(t)28(yp)-28(e,)]TJ/F45 10.9091 Tf 131.812 0 Td [(mld_)]TJ/F18 10.9091 Tf 22.909 0 Td [(x)]TJ/F45 10.9091 Tf 6.374 0 Td [(prec_)]TJ -356.542 -13.55 Td [(type)]TJ/F15 10.9091 Tf 22.908 0 Td [(,)-322(where)]TJ/F18 10.9091 Tf 37.923 0 Td [(x)]TJ/F15 10.9091 Tf 9.849 0 Td [(ma)28(y)-319(b)-28(e)]TJ/F45 10.9091 Tf 38.163 0 Td [(s)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)]TJ/F45 10.9091 Tf 6.538 0 Td [(d)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)]TJ/F45 10.9091 Tf 6.538 0 Td [(c)]TJ/F15 10.9091 Tf 9.202 0 Td [(or)]TJ/F45 10.9091 Tf 13.203 0 Td [(z)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)-321(ac)-1(cord)1(ing)-319(to)-319(th)1(e)-319(basic)-319(data)-318(t)28(yp)-28(e)-319(of)-318(the)-319(sparse)]TJ -161.505 -13.549 Td [(matrix)-372(\050)]TJ/F45 10.9091 Tf 40.153 0 Td [(s)]TJ/F15 10.9091 Tf 9.79 0 Td [(=)-372(real)-373(single)-372(precision;)]TJ/F45 10.9091 Tf 114.22 0 Td [(d)]TJ/F15 10.9091 Tf 9.789 0 Td [(=)-372(real)-373(double)-372(precision;)]TJ/F45 10.9091 Tf 119.008 0 Td [(c)]TJ/F15 10.9091 Tf 9.79 0 Td [(=)-372(complex)-373(single)]TJ -302.75 -13.549 Td [(precision;)]TJ/F45 10.9091 Tf 48.807 0 Td [(z)]TJ/F15 10.9091 Tf 9.518 0 Td [(=)-347(c)-1(omp)1(le)-1(x)-347(double)-347(precision\051.)-487(This)-348(data)-347(structure)-348(is)-347(accessed)-348(b)28(y)-347(the)]TJ -58.325 -13.549 Td [(user)-333(only)-334(thr)1(ough)-334(the)-333(MLD2P4)-333(routines,)-334(fol)1(lo)27(wing)-333(an)-333(ob)-56(ject-orien)28(ted)-333(approac)28(h.)]TJ -0 g 0 G - -13.94 -23.442 Td [(2.)]TJ -0 g 0 G -/F18 10.9091 Tf 13.94 0 Td [(A)26(l)-52(l)1(o)51(c)51(ate)-444(and)-444(initialize)-444(the)-444(pr)51(e)52(c)51(onditioner)-444(data)-444(structur)51(e,)-465(ac)51(c)51(or)51(ding)-444(to)-444(a)-444(pr)51(e)52(c)51(on-)]TJ 0 -13.549 Td [(ditioner)-407(typ)52(e)-407(chosen)-407(by)-406(the)-407(user)]TJ/F15 10.9091 Tf 154.603 0 Td [(.)-604(This)-386(is)-387(p)-27(erformed)-387(b)28(y)-386(the)-387(routine)]TJ/F45 10.9091 Tf 172.699 0 Td [(init)]TJ/F15 10.9091 Tf 22.909 0 Td [(,)-400(whic)28(h)]TJ -350.211 -13.55 Td [(also)-288(sets)-288(d)1(e)-1(f)1(aults)-288(for)-288(eac)28(h)-288(preconditioner)-287(t)27(yp)-27(e)-288(selected)-288(b)28(y)-288(the)-287(use)-1(r)1(.)-430(The)-287(precondi-)]TJ 0 -13.549 Td [(tioner)-273(t)28(yp)-28(es)-273(and)-273(the)-273(d)1(e)-1(f)1(aults)-273(asso)-28(ciated)-273(with)-273(them)-273(are)-273(giv)28(en)-273(in)-273(T)83(abl)1(e)]TJ -0 0 1 rg 0 0 1 RG - [-273(1)]TJ -0 g 0 G - [(,)-285(where)-273(the)]TJ 0 -13.549 Td [(strings)-268(used)-268(b)27(y)]TJ/F45 10.9091 Tf 73.232 0 Td [(init)]TJ/F15 10.9091 Tf 25.835 0 Td [(to)-268(iden)28(tify)-268(the)-269(preconditioner)-268(t)28(yp)-28(es)-268(are)-268(also)-268(giv)27(en.)-422(Note)-269(that)]TJ -99.067 -13.549 Td [(these)-370(strings)-370(are)-371(v)56(alid)-370(also)-370(if)-370(upp)-28(ercase)-370(letters)-371(are)-370(substituted)-370(b)28(y)-370(corresp)-28(onding)]TJ 0 -13.549 Td [(lo)28(w)28(ercas)-1(e)-333(ones.)]TJ -0 g 0 G - -13.94 -23.442 Td [(3.)]TJ -0 g 0 G -/F18 10.9091 Tf 13.94 0 Td [(Mo)51(dify)-340(the)-341(sele)51(c)1(t)-1(e)52(d)-341(pr)51(e)51(c)51(onditi)1(o)-1(ner)-340(typ)51(e,)-344(by)-340(pr)51(op)51(erly)-340(setting)-341(pr)51(e)52(c)51(onditioner)-341(p)51(ar)51(am-)]TJ 0 -13.55 Td [(eters.)]TJ/F15 10.9091 Tf 31.58 0 Td [(This)-354(is)-353(p)-28(erformed)-354(b)28(y)-353(the)-354(routine)]TJ/F45 10.9091 Tf 160.936 0 Td [(set)]TJ/F15 10.9091 Tf 17.182 0 Td [(.)-505(This)-354(routine)-354(m)28(ust)-353(b)-28(e)-354(called)-354(onl)1(y)-354(if)]TJ -209.698 -13.549 Td [(the)-296(user)-296(w)28(an)27(ts)-296(to)-296(mo)-28(dify)-296(the)-296(default)-296(v)56(alues)-296(of)-297(th)1(e)-297(parameters)-296(asso)-28(ciated)-296(with)-296(the)]TJ 0 -13.549 Td [(selected)-263(pr)1(e)-1(cond)1(itioner)-263(t)28(yp)-28(e,)-276(to)-263(obtain)-262(a)-263(v)56(arian)28(t)-263(of)-262(that)-262(preconditioner.)-421(Examples)]TJ 0 -13.549 Td [(of)-306(use)-306(of)]TJ/F45 10.9091 Tf 42.808 0 Td [(set)]TJ/F15 10.9091 Tf 20.522 0 Td [(are)-306(giv)28(en)-307(in)-306(Section)]TJ -0 0 1 rg 0 0 1 RG - [-306(5.1)]TJ -0 g 0 G - [(;)-315(a)-306(complete)-306(list)-307(of)-306(all)-306(the)-306(preconditioner)-306(pa-)]TJ -63.33 -13.549 Td [(rameters)-283(and)-283(their)-283(allo)27(w)28(ed)-283(and)-283(default)-283(v)55(alues)-283(is)-283(pro)28(vided)-283(in)-283(Section)]TJ -0 0 1 rg 0 0 1 RG - [-283(6)]TJ -0 g 0 G - [(,)-294(T)84(ables)]TJ -0 0 1 rg 0 0 1 RG - [-284(2)]TJ -0 g 0 G - [(-)]TJ -0 0 1 rg 0 0 1 RG - [(8)]TJ -0 g 0 G - [(.)]TJ -0 g 0 G - -13.94 -23.442 Td [(4.)]TJ -0 g 0 G -/F18 10.9091 Tf 13.94 0 Td [(Build)-508(the)-508(pr)51(e)51(c)51(onditioner)-508(for)-508(a)-508(given)-508(matrix)]TJ/F15 10.9091 Tf 214.941 0 Td [(.)-935(If)-496(the)-497(selected)-497(preconditioner)-497(is)]TJ -214.941 -13.55 Td [(m)28(ultilev)28(el,)-334(then)-333(t)28(w)28(o)-334(steps)-333(m)28(ust)-334(b)-27(e)-334(p)-28(erformed,)-333(as)-333(sp)-28(eci\014ed)-333(next.)]TJ -0 g 0 G - 4.606 -23.442 Td [(4.1)]TJ -0 g 0 G -/F18 10.9091 Tf 19.394 0 Td [(Build)-328(the)-328(aggr)51(e)51(gation)-328(hier)51(ar)52(chy)-329(f)1(or)-329(a)-328(given)-328(matrix.)]TJ/F15 10.9091 Tf 244.224 0 Td [(This)-301(is)-301(p)-28(erformed)-301(b)28(y)-301(the)]TJ -244.224 -13.549 Td [(routine)]TJ/F45 10.9091 Tf 37.606 0 Td [(hierarchy_build)]TJ/F15 10.9091 Tf 85.908 0 Td [(.)]TJ -0 g 0 G - -142.908 -17.997 Td [(4.2)]TJ -0 g 0 G -/F18 10.9091 Tf 19.394 0 Td [(Build)-295(the)-295(pr)51(e)52(c)51(onditioner)-295(for)-295(a)-295(given)-295(matrix.)]TJ/F15 10.9091 Tf 208.934 0 Td [(This)-265(is)-265(p)-28(erformed)-265(b)28(y)-265(the)-265(routine)]TJ/F45 10.9091 Tf -208.934 -13.55 Td [(smoothers_build)]TJ/F15 10.9091 Tf 85.908 0 Td [(.)]TJ -109.908 -23.442 Td [(If)-292(the)-292(selected)-292(preconditi)1(oner)-292(is)-292(one-lev)28(el,)-301(it)-291(is)-292(built)-292(in)-292(a)-292(single)-291(ste)-1(p)1(,)-301(p)-27(erformed)-292(b)28(y)]TJ 0 -13.549 Td [(the)-333(routine)]TJ/F45 10.9091 Tf 56.394 0 Td [(bld)]TJ/F15 10.9091 Tf 17.181 0 Td [(.)]TJ -0 g 0 G - -87.515 -23.442 Td [(5.)]TJ -0 g 0 G -/F18 10.9091 Tf 13.94 0 Td [(Apply)-387(the)-388(pr)51(e)52(c)51(onditioner)-388(at)-387(e)51(ach)-387(iter)51(ation)-388(of)-387(a)-387(Krylov)-388(solver.)]TJ/F15 10.9091 Tf 300.051 0 Td [(This)-366(is)-365(p)-28(erformed)]TJ -300.051 -13.549 Td [(b)28(y)-487(the)-487(metho)-27(d)]TJ/F45 10.9091 Tf 78.657 0 Td [(apply)]TJ/F15 10.9091 Tf 28.636 0 Td [(.)-905(When)-486(using)-487(the)-487(PSBLAS)-486(Krylo)28(v)-487(solv)28(ers,)-525(this)-487(step)-487(is)]TJ -107.293 -13.549 Td [(completely)-324(transparen)28(t)-323(to)-324(the)-324(user,)-325(since)]TJ/F45 10.9091 Tf 197.842 0 Td [(apply)]TJ/F15 10.9091 Tf 32.167 0 Td [(is)-324(called)-323(b)27(y)-323(the)-324(PSBLAS)-323(routine)]TJ -230.009 -13.55 Td [(implemen)28(ting)-333(the)-334(Krylo)28(v)-333(solv)28(e)-1(r)-333(\050)]TJ/F45 10.9091 Tf 158.303 0 Td [(psb_krylov)]TJ/F15 10.9091 Tf 57.272 0 Td [(\051.)]TJ -0 g 0 G - -229.515 -23.442 Td [(6.)]TJ -0 g 0 G -/F18 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[(Finally)83(,)-451(Figure)]TJ -0 0 1 rg 0 0 1 RG - [-427(5)]TJ -0 g 0 G - [-428(sho)28(ws)-428(the)-428(setup)-427(of)-428(a)-427(one-le)-1(v)28(el)-427(additiv)28(e)-428(Sc)28(h)27(w)28(arz)-427(preconditioner,)]TJ -16.936 -13.549 Td [(i.e.,)-373(RAS)-364(with)-365(o)28(v)27(erlap)-364(2.)-539(Note)-365(also)-365(that)-364(a)-365(Krylo)28(v)-365(metho)-28(d)-365(di\013eren)28(t)-365(from)-364(C)-1(G)-364(m)27(ust)-364(b)-28(e)]TJ 0 -13.549 Td [(used)-294(to)-294(solv)28(e)-294(the)-294(p)1(re)-1(cond)1(itioned)-294(system,)-302(since)-294(the)-294(preconditione)-293(in)-294(nonsymmetric.)-432(Th)1(e)]TJ 0 -13.549 Td [(corresp)-28(onding)-333(example)-333(program)-334(is)-333(a)28(v)55(ailable)-333(in)-333(the)-333(\014le)]TJ/F45 10.9091 Tf 261.455 0 Td [(mld_dexample_1lev.f90)]TJ/F15 10.9091 Tf 120.271 0 Td [(.)]TJ -364.79 -13.549 Td [(F)83(or)-452(all)-452(the)-452(previous)-452(preconditi)1(oners)-1(,)-481(example)-452(programs)-452(where)-452(the)-452(sparse)-452(matrix)]TJ -16.936 -13.55 Td [(and)-370(the)-371(righ)28(t-hand)-371(side)-370(are)-371(generated)-370(b)27(y)-370(discretizing)-371(a)-370(PDE)-371(with)-370(Diric)28(hlet)-371(b)-28(oundary)]TJ 0 -13.549 Td [(conditions)-333(are)-334(also)-333(a)28(v)55(ailab)1(le)-334(in)-333(the)-333(directory)]TJ/F45 10.9091 Tf 215.061 0 Td [(examples/pdegen)]TJ/F15 10.9091 Tf 85.908 0 Td [(.)]TJ -0 g 0 G -0 g 0 G -ET - -endstream -endobj -455 0 obj -<< -/Length 3337 ->> -stream -0 g 0 G -0 0 1 rg 0 0 1 RG -BT -/F42 10.9091 Tf 93.6 740.002 Td [(5)]TJ -0 g 0 G - [-378(Getting)-378(St)67(ar)67(ted)]TJ/F15 10.9091 Tf 401.542 0 Td [(19)]TJ -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -/F45 9.9626 Tf -370.457 -30.995 Td [(use)-525(psb_base_mod)]TJ 0 -11.955 Td [(use)-525(mld_prec_mod)]TJ 0 -11.955 Td [(use)-525(psb_krylov_mod)]TJ -10.461 -11.956 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(sparse)-525(matrix)]TJ 10.461 -11.955 Td [(type\050psb_dspmat_type\051)-525(::)-525(A)]TJ -10.461 -11.955 Td [(!)-525(sparse)-525(matrix)-525(descriptor)]TJ 10.461 -11.955 Td [(type\050psb_desc_type\051)-1575(::)-525(desc_A)]TJ -10.461 -11.956 Td [(!)-525(preconditioner)]TJ 10.461 -11.955 Td [(type\050mld_dprec_type\051)-1050(::)-525(P)]TJ -10.461 -11.955 Td [(!)-525(right-hand)-525(side)-525(and)-525(solution)-525(vectors)]TJ 10.461 -11.955 Td [(type\050psb_d_vect_type\051)-525(::)-525(b,)-525(x)]TJ -10.461 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.956 Td [(!)-525(initialize)-525(the)-525(parallel)-525(environment)]TJ 10.461 -11.955 Td [(call)-525(psb_init\050ictxt\051)]TJ 0 -11.955 Td [(call)-525(psb_info\050ictxt,iam,np\051)]TJ -10.461 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(read)-525(and)-525(assemble)-525(the)-525(spd)-525(matrix)-525(A)-525(and)-525(the)-525(right-hand)-525(side)-525(b)]TJ 0 -11.956 Td [(!)-525(using)-525(PSBLAS)-525(routines)-525(for)-525(sparse)-525(matrix)-525(/)-525(vector)-525(management)]TJ 0 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(initialize)-525(the)-525(default)-525(multilevel)-525(preconditioner,)-525(i.e.)-525(V-cycle)]TJ 0 -11.955 Td [(!)-525(with)-525(basic)-525(smoothed)-525(aggregation,)-525(1)-525(hybrid)-525(forward/backward)]TJ 0 -11.956 Td [(!)-525(GS)-525(sweep)-525(as)-525(pre/post-smoother)-525(and)-525(UMFPACK)-525(as)-525(coarsest-level)]TJ 0 -11.955 Td [(!)-525(solver)]TJ 10.461 -11.955 Td [(call)-525(P%init\050'ML',info\051)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(build)-525(the)-525(preconditioner)]TJ 10.461 -11.955 Td [(call)-525(P%hierarchy_build\050A,desc_A,info\051)]TJ 0 -11.956 Td [(call)-525(P%smoothers_build\050A,desc_A,info\051)]TJ -10.461 -23.91 Td [(!)]TJ 0 -11.955 Td [(!)-525(set)-525(the)-525(solver)-525(parameters)-525(and)-525(the)-525(initial)-525(guess)]TJ 10.461 -11.955 Td [(...)-525(...)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.956 Td [(!)-525(solve)-525(Ax=b)-525(with)-525(preconditioned)-525(CG)]TJ 10.461 -11.955 Td [(call)-525(psb_krylov\050'CG',A,P,b,x,tol,desc_A,info\051)]TJ 0 -11.955 Td [(...)-525(...)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(deallocate)-525(the)-525(preconditioner)]TJ 10.461 -11.955 Td [(call)-525(P%free\050info\051)]TJ -10.461 -11.956 Td [(!)]TJ 0 -11.955 Td [(!)-525(deallocate)-525(other)-525(data)-525(structures)]TJ 10.461 -11.955 Td [(...)-525(...)]TJ -10.461 -11.955 Td [(!)]TJ 0 -11.955 Td [(!)-525(exit)-525(the)-525(parallel)-525(environment)]TJ 10.461 -11.956 Td [(call)-525(psb_exit\050ictxt\051)]TJ 0 -11.955 Td [(stop)]TJ -0 g 0 G -/F15 10.9091 Tf -29.405 -21.354 Td [(Figure)-333(2:)-445(setup)-333(and)-333(application)-333(of)-334(the)-333(default)-333(m)27(u)1(ltilev)27(el)-333(preconditioner)-333(\050example)-334(1\051.)]TJ -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -ET - -endstream -endobj -460 0 obj -<< -/Length 3194 ->> -stream -0 g 0 G -BT -/F15 10.9091 Tf 86.4 740.002 Td [(20)]TJ/F42 10.9091 Tf 203.265 0 Td [(MLD2P4)-378(User)67('s)-378(and)-378(Ref)1(erence)-378(Guide)]TJ -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -/F45 9.9626 Tf -182.641 -47.344 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)-525(build)-525(a)-525(V-cycle)-525(preconditioner)-525(with)-525(1)-525(block-Jacobi)-525(sweep)-525(\050with)]TJ 0 -11.955 Td [(!)-525(ILU\0500\051)-525(on)-525(the)-525(blocks\051)-525(as)-525(pre-)-525(and)-525(post-smoother,)-525(and)-525(8)-1050(block-Jacobi)]TJ 0 -11.955 Td [(!)-525(sweeps)-525(\050with)-525(ILU\0500\051)-525(on)-525(the)-525(blocks\051)-525(as)-525(coarsest-level)-525(solver)]TJ 10.46 -11.956 Td [(call)-525(P%init\050'ML',info\051)]TJ 0 -11.955 Td [(call_P%set\050'SMOOTHER_TYPE','BJAC',info\051)]TJ 0 -11.955 Td [(call)-525(P%set\050'COARSE_SOLVE','BJAC',info\051)]TJ 0 -11.955 Td [(call)-525(P%set\050'COARSE_SWEEPS',8,info\051)]TJ 0 -11.955 Td [(call)-525(P%hierarchy_build\050A,desc_A,info\051)]TJ 0 -11.956 Td [(call)-525(P%smoothers_build\050A,desc_A,info\051)]TJ -10.46 -11.955 Td [(...)-525(...)]TJ -0 g 0 G -/F15 10.9091 Tf 77.116 -18.534 Td [(Figure)-333(3:)-445(setup)-333(of)-333(a)-334(m)28(ultilev)28(el)-333(preconditioner)]TJ -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -/F45 9.9626 Tf -77.116 -65.676 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)-525(build)-525(a)-525(W-cycle)-525(preconditioner)-525(with)-525(2)-525(hybrid)-525(Gauss-Seidel)-525(sweeps)]TJ 0 -11.955 Td [(!)-525(as)-525(pre-)-525(and)-525(post-smoother,)-525(a)-525(distributed)-525(coarsest)]TJ 0 -11.955 Td [(!)-525(matrix,)-525(and)-525(MUMPS)-525(as)-525(coarsest-level)-525(solver)]TJ 10.46 -11.956 Td [(call)-525(P%init\050'ML',info\051)]TJ 0 -11.955 Td [(call)-525(P%set\050'ML_CYCLE','WCYCLE',info\051)]TJ 0 -11.955 Td [(call)-525(P%set\050'SMOOTHER_TYPE','FBGS',info\051)]TJ 0 -11.955 Td [(call)-525(P%set\050'SMOOTHER_SWEEPS',2,info\051)]TJ 0 -11.955 Td [(call)-525(P%set\050'COARSE_SOLVE','MUMPS',info\051)]TJ 0 -11.955 Td [(call)-525(P%set\050'COARSE_MAT','DIST',info\051)]TJ 0 -11.956 Td [(call)-525(P%hierarchy_build\050A,desc_A,info\051)]TJ 0 -11.955 Td [(call)-525(P%smoothers_build\050A,desc_A,info\051)]TJ -10.46 -11.955 Td [(...)-525(...)]TJ -0 g 0 G -/F15 10.9091 Tf 77.116 -18.535 Td [(Figure)-333(4:)-445(setup)-333(of)-333(a)-334(m)28(ultilev)28(el)-333(preconditioner)]TJ -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -/F45 9.9626 Tf -77.116 -65.675 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)-525(set)-525(RAS)-525(with)-525(overlap)-525(2)-525(and)-525(ILU\0500\051)-525(on)-525(the)-525(local)-525(blocks)]TJ 10.46 -11.956 Td [(call)-525(P%init\050'AS',info\051)]TJ 0 -11.955 Td [(call)-525(P%set\050'SUB_OVR',2,info\051)]TJ 0 -11.955 Td [(call)-525(P%bld\050A,desc_A,info\051)]TJ -10.46 -11.955 Td [(...)-525(...)]TJ 0 -11.955 Td [(!)-525(solve)-525(Ax=b)-525(with)-525(preconditioned)-525(BiCGSTAB)]TJ 10.46 -11.955 Td [(call)-525(psb_krylov\050'BICGSTAB',A,P,b,x,tol,desc_A,info\051)]TJ -0 g 0 G -/F15 10.9091 Tf 46.641 -20.749 Td [(Figure)-333(5:)-445(setup)-333(of)-333(a)-334(one-lev)28(el)-333(Sc)28(h)27(w)28(arz)-333(preconditioner.)]TJ -0 g 0 G -0 g 0 G -0 g 0 G -0 g 0 G -ET - -endstream -endobj -466 0 obj -<< -/Length 6306 ->> -stream -0 g 0 G -0 0 1 rg 0 0 1 RG -BT -/F42 10.9091 Tf 93.6 740.002 Td [(6)]TJ -0 g 0 G - [-378(User)-377(Interf)88(a)23(ce)]TJ/F15 10.9091 Tf 401.542 0 Td [(21)]TJ -0 g 0 G -/F17 14.3462 Tf -401.542 -35.866 Td [(6)-1125(User)-375(In)31(terface)]TJ/F15 10.9091 Tf 0 -24.352 Td [(The)-325(basic)-325(user)-324(in)28(te)-1(r)1(face)-325(of)-325(MLD2P4)-325(consists)-324(of)-325(eigh)28(t)-325(metho)-28(ds.)-442(Th)1(e)-325(six)-325(metho)-28(ds)]TJ/F45 10.9091 Tf 386.512 0 Td [(init)]TJ/F15 10.9091 Tf 22.909 0 Td [(,)]TJ/F45 10.9091 Tf -409.421 -13.549 Td [(set)]TJ/F15 10.9091 Tf 17.182 0 Td [(,)]TJ/F45 10.9091 Tf 6.791 0 Td [(build)]TJ/F15 10.9091 Tf 28.636 0 Td [(,)]TJ/F45 10.9091 Tf 6.791 0 Td [(hierarchy_build)]TJ/F15 10.9091 Tf 85.908 0 Td [(,)]TJ/F45 10.9091 Tf 6.791 0 Td [(smoothers_build)]TJ/F15 10.9091 Tf 89.644 0 Td [(and)]TJ/F45 10.9091 Tf 21.311 0 Td [(apply)]TJ/F15 10.9091 Tf 32.372 0 Td [(encapsulate)-342(all)-343(the)-342(func-)]TJ -295.426 -13.549 Td [(tionalities)-392(for)-393(th)1(e)-393(setup)-392(and)-392(the)-393(application)-392(of)-392(an)28(y)-393(m)28(ultilev)28(el)-392(and)-393(one-lev)28(el)-392(precondi-)]TJ 0 -13.549 Td [(tioner)-398(implemen)28(ted)-398(in)-398(the)-399(pac)28(k)56(age.)-639(The)-398(metho)-28(d)]TJ/F45 10.9091 Tf 240.937 0 Td [(free)]TJ/F15 10.9091 Tf 27.252 0 Td [(deallo)-28(cates)-398(the)-398(preconditioner)]TJ -268.189 -13.55 Td [(data)-274(s)-1(t)1(ructure,)-287(while)]TJ/F45 10.9091 Tf 101.355 0 Td [(descr)]TJ/F15 10.9091 Tf 31.63 0 Td [(prin)28(ts)-275(a)-274(description)-275(of)-274(the)-274(prec)-1(on)1(ditioner)-275(setup)-274(b)28(y)-275(the)-274(use)-1(r)1(.)]TJ -132.985 -13.549 Td [(F)83(or)-333(bac)28(kw)28(ard)-334(compatibilit)28(y)84(,)-334(metho)-28(d)1(s)-334(are)-333(also)-334(accessible)-333(as)-333(s)-1(tan)1(d-alone)-334(subroutines.)]TJ 16.937 -13.549 Td [(F)83(or)-419(e)-1(ac)28(h)-420(metho)-27(d,)-442(the)-420(same)-420(user)-420(in)28(terface)-420(is)-420(o)28(v)28(erloaded)-420(with)-420(resp)-28(ect)-420(to)-420(the)-420(real/)]TJ -16.937 -13.549 Td [(complex)-367(case)-367(and)-367(the)-367(single/double)-366(precis)-1(i)1(on;)-384(argumen)28(ts)-367(with)-367(appropriate)-367(data)-367(t)28(yp)-28(es)]TJ 0 -13.549 Td [(m)28(ust)-334(b)-27(e)-334(passed)-333(to)-333(the)-334(metho)-28(d)1(,)-334(i.e.,)]TJ -0 g 0 G -/F25 10.9091 Tf 16.364 -22.516 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-463(sparse)-463(matrix)-463(data)-463(structure,)-496(con)28(taining)-463(the)-463(matrix)-463(to)-463(b)-28(e)-463(preconditioned,)]TJ 0 -13.549 Td [(m)28(ust)-424(b)-28(e)-424(of)-423(t)27(yp)-27(e)]TJ/F45 10.9091 Tf 82.798 0 Td [(psb_)]TJ/F18 10.9091 Tf 22.909 0 Td [(x)]TJ/F45 10.9091 Tf 6.374 0 Td [(spmat_type)]TJ/F15 10.9091 Tf 61.896 0 Td [(with)]TJ/F18 10.9091 Tf 25.835 0 Td [(x)]TJ/F15 10.9091 Tf 10.998 0 Td [(=)]TJ/F45 10.9091 Tf 13.109 0 Td [(s)]TJ/F15 10.9091 Tf 10.351 0 Td [(for)-424(real)-424(single)-423(precision,)]TJ/F18 10.9091 Tf 121.075 0 Td [(x)]TJ/F15 10.9091 Tf 10.998 0 Td [(=)]TJ/F45 10.9091 Tf 13.108 0 Td [(d)]TJ/F15 10.9091 Tf -379.451 -13.549 Td [(for)-371(real)-371(double)-371(precision,)]TJ/F18 10.9091 Tf 123.413 0 Td [(x)]TJ/F15 10.9091 Tf 10.422 0 Td [(=)]TJ/F45 10.9091 Tf 12.532 0 Td [(c)]TJ/F15 10.9091 Tf 9.774 0 Td [(for)-371(complex)-371(single)-371(precision,)]TJ/F18 10.9091 Tf 140.11 0 Td [(x)]TJ/F15 10.9091 Tf 10.422 0 Td [(=)]TJ/F45 10.9091 Tf 12.532 0 Td [(z)]TJ/F15 10.9091 Tf 9.774 0 Td [(for)-371(complex)]TJ -328.979 -13.55 Td [(double)-333(precision;)]TJ -0 g 0 G -/F25 10.9091 Tf -10.909 -22.515 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-337(preconditioner)-337(data)-337(structure)-337(m)28(ust)-337(b)-28(e)-337(of)-337(t)28(yp)-28(e)]TJ/F45 10.9091 Tf 241.776 0 Td [(mld_)]TJ/F18 10.9091 Tf 22.908 0 Td [(x)]TJ/F45 10.9091 Tf 6.375 0 Td [(prec_type)]TJ/F15 10.9091 Tf 51.544 0 Td [(,)-338(with)]TJ/F18 10.9091 Tf 31.606 0 Td [(x)]TJ/F15 10.9091 Tf 10.05 0 Td [(=)]TJ/F45 10.9091 Tf 12.162 0 Td [(s)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)]TJ/F45 10.9091 Tf -382.148 -13.549 Td [(d)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)]TJ/F45 10.9091 Tf 6.667 0 Td [(c)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)]TJ/F45 10.9091 Tf 6.667 0 Td [(z)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)-333(according)-334(to)-333(the)-333(sparse)-334(matrix)-333(data)-333(structure;)]TJ -0 g 0 G -/F25 10.9091 Tf -41.424 -22.516 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-321(arra)28(ys)-321(con)27(tain)1(ing)-321(the)-322(v)28(ectors)]TJ/F22 10.9091 Tf 159.847 0 Td [(v)]TJ/F15 10.9091 Tf 9.182 0 Td [(and)]TJ/F22 10.9091 Tf 21.078 0 Td [(w)]TJ/F15 10.9091 Tf 11.606 0 Td [(in)28(v)28(olv)27(ed)-321(in)-321(the)-321(preconditioner)-321(appli)1(c)-1(a-)]TJ -201.713 -13.549 Td [(tion)]TJ/F22 10.9091 Tf 21.975 0 Td [(w)]TJ/F15 10.9091 Tf 11.134 0 Td [(=)]TJ/F22 10.9091 Tf 11.515 0 Td [(B)]TJ/F26 7.9701 Tf 8.822 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -3.959 Td [(v)]TJ/F15 10.9091 Tf 8.866 0 Td [(m)28(ust)-292(b)-28(e)-292(of)-293(t)28(yp)-27(e)]TJ/F45 10.9091 Tf 77.053 0 Td [(psb_)]TJ/F18 10.9091 Tf 22.909 0 Td [(x)]TJ/F45 10.9091 Tf 6.374 0 Td [(vect_type)]TJ/F15 10.9091 Tf 54.732 0 Td [(with)]TJ/F18 10.9091 Tf 24.399 0 Td [(x)]TJ/F15 10.9091 Tf 9.562 0 Td [(=)]TJ/F45 10.9091 Tf 11.672 0 Td [(s)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)]TJ/F45 10.9091 Tf 6.308 0 Td [(d)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)]TJ/F45 10.9091 Tf 6.307 0 Td [(c)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)]TJ/F45 10.9091 Tf 6.308 0 Td [(z)]TJ/F15 10.9091 Tf 5.727 0 Td [(,)-300(in)-292(a)-293(manner)]TJ -322.163 -13.549 Td 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[(Dist)-443(requires)-443(it)-443(to)-442(b)-28(e)-443(distributed.)-773(In)-443(these)-443(cases,)-470(setting)]TJ -131.07 -13.549 Td [(the)-445(coarsest-le)-1(v)28(el)-445(solv)28(er)-446(implies)-445(that)-445(the)-446(la)28(y)28(out)-445(is)-446(rede\014ned)-445(according)-445(to)-446(the)-445(solv)28(er,)]TJ -0 g 0 G -0 g 0 G -ET - -endstream -endobj -515 0 obj -<< -/Length 3880 ->> -stream -0 g 0 G -0 0 1 rg 0 0 1 RG -BT -/F42 10.9091 Tf 93.6 740.002 Td [(6)]TJ -0 g 0 G - [-378(User)-377(Interf)88(a)23(ce)]TJ/F15 10.9091 Tf 401.542 0 Td [(25)]TJ -0 g 0 G - -401.542 -35.866 Td [(o)28(vv)28(erriding)-487(an)28(y)-486(previous)-486(settings.)-904(M)1(UMPS,)-487(p)-27(oin)27(t-Jacobi,)-524(h)28(ybrid)-486(Gauss-Seidel)-487(and)]TJ 0 -13.549 Td [(blo)-28(c)28(k-Jacobi)-416(can)-417(b)-27(e)-417(app)1(lied)-417(to)-416(replicated)-416(and)-416(distributed)-416(matrice)-1(s,)-437(th)28(us)-416(their)-416(c)28(hoice)]TJ 0 -13.549 Td 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0 1 406.628 788.225 cm -[]0 d 0 J 0.398 w 0 0 m 0 13.549 l S -Q -BT -/F42 10.9091 Tf 412.605 792.29 Td [(d)22(a)67(t)66(a)-377(type)]TJ -ET -q -1 0 0 1 510.218 788.225 cm -[]0 d 0 J 0.398 w 0 0 m 0 13.549 l S -Q -BT -/F45 10.9091 Tf 516.196 792.29 Td [(val)]TJ -ET -q -1 0 0 1 590.205 788.225 cm -[]0 d 0 J 0.398 w 0 0 m 0 13.549 l S -Q -BT -/F42 10.9091 Tf 596.183 792.29 Td [(def)89(a)22(ul)67(t)]TJ -ET -q -1 0 0 1 670.191 788.225 cm -[]0 d 0 J 0.398 w 0 0 m 0 13.549 l S -Q -BT -/F42 10.9091 Tf 676.169 792.29 Td [(comments)]TJ -ET -q -1 0 0 1 886.241 788.225 cm -[]0 d 0 J 0.398 w 0 0 m 0 13.549 l S -Q -q -1 0 0 1 292.625 788.026 cm -[]0 d 0 J 0.398 w 0 0 m 593.616 0 l S -Q -q -1 0 0 1 292.625 692.982 cm -[]0 d 0 J 0.398 w 0 0 m 0 94.844 l S -Q -BT -/F45 10.9091 Tf 298.603 778.342 Td [('ML_CYCLE')]TJ -ET -q -1 0 0 1 406.628 692.982 cm -[]0 d 0 J 0.398 w 0 0 m 0 94.844 l S -Q -BT -/F45 10.9091 Tf 412.605 778.342 Td [(character\050len=*\051)]TJ -ET -q -1 0 0 1 510.218 692.982 cm -[]0 d 0 J 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886.241 665.485 cm -[]0 d 0 J 0.398 w 0 0 m 0 27.098 l S -Q -q -1 0 0 1 292.625 665.286 cm -[]0 d 0 J 0.398 w 0 0 m 593.616 0 l S -Q -0 g 0 G -BT -/F15 10.9091 Tf 373.473 634.597 Td [(T)83(able)-333(2:)-444(P)27(arameters)-333(de\014ning)-333(the)-334(m)28(ultilev)28(el)-333(cycle)-334(and)-333(the)-333(n)28(um)27(b)-27(er)-334(of)-333(cycles)-334(t)1(o)-334(b)-28(e)-333(applied.)]TJ -0 g 0 G -0 g 0 G -ET -1 0 0 1 292.625 715.095 cm -Q -0 g 0 G -0 g 0 G -0 g 0 G - -endstream -endobj -525 0 obj -<< -/Length 10624 ->> -stream -0 g 0 G -0 0 1 rg 0 0 1 RG -BT -/F42 10.9091 Tf 93.6 740.002 Td [(6)]TJ -0 g 0 G - [-378(User)-377(Interf)88(a)23(ce)]TJ/F15 10.9091 Tf 401.542 0 Td [(27)]TJ -0 g 0 G -0 g 0 G -ET -1 0 0 1 299.826 121.521 cm -q -0 1 -1 0 0 0 cm -0 g 0 G -0 g 0 G -0 g 0 G -q -1 0 0 1 0 216.592 cm -[]0 d 0 J 0.398 w 0 0 m 600.087 0 l S -Q -q -1 0 0 1 0 202.843 cm -[]0 d 0 J 0.398 w 0 0 m 0 13.549 l S -Q -1 0 0 1 -299.826 -121.521 cm -BT -/F45 10.9091 Tf 305.803 328.429 Td [(what)]TJ 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d 0 J 0.398 w 0 0 m 0 54.197 l S -Q -BT -/F15 10.9091 Tf 527.032 314.481 Td [(An)28(y)-334(n)28(um)28(b)-28(er)]TJ/F22 10.9091 Tf 0 -13.549 Td [(>)]TJ/F15 10.9091 Tf 11.516 0 Td [(0)]TJ -ET -q -1 0 0 1 598.207 269.769 cm -[]0 d 0 J 0.398 w 0 0 m 0 54.197 l S -Q -BT -/F25 10.9091 Tf 604.185 314.481 Td [(b)]TJ/F15 10.9091 Tf 4.848 0 Td [(40)]TJ/F21 5.9776 Tf 13.939 3.67 Td [(3)]TJ/F25 10.9091 Tf -2.407 4.188 Td [(p)]TJ -ET -q -1 0 0 1 629.656 322.557 cm -[]0 d 0 J 0.436 w 0 0 m 6.548 0 l S -Q -BT -/F22 10.9091 Tf 629.656 314.481 Td [(n)]TJ/F25 10.9091 Tf 6.548 0 Td [(c)]TJ/F15 10.9091 Tf 4.848 0 Td [(,)-363(where)]TJ/F22 10.9091 Tf 38.789 0 Td [(n)]TJ/F15 10.9091 Tf -75.656 -13.549 Td [(is)-540(the)-540(dimension)]TJ 0 -13.549 Td [(of)-511(the)-511(matrix)-510(at)]TJ 0 -13.549 Td [(the)-333(\014nest)-334(lev)28(el)]TJ -ET -q -1 0 0 1 692.367 269.769 cm -[]0 d 0 J 0.398 w 0 0 m 0 54.197 l S -Q -BT -/F15 10.9091 Tf 698.344 314.481 Td [(Coarse)-475(size)-475(threshold.)-868(The)-475(aggregation)]TJ 0 -13.549 Td [(stops)-531(if)-531(the)-530(global)-531(n)28(um)28(b)-28(er)-531(of)-531(v)56(ariables)]TJ 0 -13.549 Td [(of)-354(the)-355(computed)-354(coarsest)-355(matrix)-354(is)-354(lo)27(w)28(er)]TJ 0 -13.549 Td [(than)-249(or)-248(equal)-249(to)-248(this)-249(threshold)-248(\050see)-249(Note\051.)]TJ -ET -q -1 0 0 1 899.912 269.769 cm -[]0 d 0 J 0.398 w 0 0 m 0 54.197 l S -Q -q -1 0 0 1 299.826 269.57 cm -[]0 d 0 J 0.398 w 0 0 m 600.087 0 l S -Q -q -1 0 0 1 299.826 201.625 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 305.803 259.886 Td [('MIN_CR_RATIO')]TJ -ET -q -1 0 0 1 422.332 201.625 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 428.31 259.886 Td [(real)]TJ -ET -q -1 0 0 1 521.055 201.625 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F15 10.9091 Tf 527.032 259.886 Td [(An)28(y)-334(n)28(um)28(b)-28(er)]TJ/F22 10.9091 Tf 0 -13.549 Td [(>)]TJ/F15 10.9091 Tf 11.516 0 Td [(1)]TJ -ET -q -1 0 0 1 598.207 201.625 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F15 10.9091 Tf 604.185 259.886 Td [(1.5)]TJ -ET -q -1 0 0 1 692.367 201.625 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F15 10.9091 Tf 698.344 259.886 Td [(Minim)28(um)-325(coarsening)-324(ratio.)-442(The)-324(aggrega-)]TJ 0 -13.549 Td [(tion)-480(stops)-480(if)-480(the)-480(ratio)-480(b)-27(et)27(w)28(een)-480(the)-480(ma-)]TJ 0 -13.549 Td [(trix)-364(dimensions)-364(at)-364(t)27(w)28(o)-364(consecutiv)28(e)-364(lev)27(els)]TJ 0 -13.549 Td [(is)-459(lo)28(w)28(er)-459(th)1(an)-459(or)-458(e)-1(q)1(ual)-459(to)-458(this)-459(threshold)]TJ 0 -13.55 Td [(\050see)-333(Note)-1(\051.)]TJ -ET -q -1 0 0 1 899.912 201.625 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -q -1 0 0 1 299.826 201.425 cm -[]0 d 0 J 0.398 w 0 0 m 600.087 0 l S -Q -q -1 0 0 1 299.826 160.579 cm -[]0 d 0 J 0.398 w 0 0 m 0 40.648 l S -Q -BT -/F45 10.9091 Tf 305.803 191.742 Td [('MAX_LEVS')]TJ -ET -q -1 0 0 1 422.332 160.579 cm 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600.087 0 l S -Q -q -1 0 0 1 299.826 92.434 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 305.803 150.696 Td [('PAR_AGGR_ALG')]TJ -ET -q -1 0 0 1 422.332 92.434 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 428.31 150.696 Td [(character\050len=*\051)]TJ -ET -q -1 0 0 1 521.055 92.434 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 527.032 150.696 Td [('DEC')]TJ/F15 10.9091 Tf 28.637 0 Td [(,)]TJ/F45 10.9091 Tf -28.637 -13.55 Td [('SYMDEC')]TJ -ET -q -1 0 0 1 598.207 92.434 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 604.185 150.696 Td [('DEC')]TJ -ET -q -1 0 0 1 692.367 92.434 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F15 10.9091 Tf 698.344 150.696 Td [(P)28(arallel)-333(aggregation)-334(algorithm.)]TJ 0 -13.55 Td [(Curren)28(tly)83(,)-604(only)-550(the)-550(decoupled)-550(aggrega-)]TJ 0 -13.549 Td [(tion)-518(\050)]TJ/F45 10.9091 Tf 28.679 0 Td [(DEC)]TJ/F15 10.9091 Tf 17.182 0 Td [(\051)-518(is)-517(a)27(v)56(ailable;)-610(the)]TJ/F45 10.9091 Tf 94.568 0 Td [(SYMDEC)]TJ/F15 10.9091 Tf 40.01 0 Td [(op-)]TJ -180.439 -13.549 Td [(tion)-306(applies)-307(decoupled)-306(aggregation)-306(to)-307(the)]TJ 0 -13.549 Td [(sparsit)28(y)-334(p)1(atte)-1(r)1(n)-334(of)]TJ/F22 10.9091 Tf 92 0 Td [(A)]TJ/F15 10.9091 Tf 10.606 0 Td [(+)]TJ/F22 10.9091 Tf 10.909 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(T)]TJ/F15 10.9091 Tf 6.605 -3.959 Td [(.)]TJ -ET -q -1 0 0 1 899.912 92.434 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -q -1 0 0 1 299.826 92.235 cm -[]0 d 0 J 0.398 w 0 0 m 600.087 0 l S -Q -q -1 0 0 1 299.826 24.29 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 305.803 82.551 Td [('AGGR_TYPE')]TJ -ET -q -1 0 0 1 422.332 24.29 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 428.31 82.551 Td [(character\050len=*\051)]TJ -ET -q -1 0 0 1 521.055 24.29 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 527.032 82.551 Td [('SOC1')]TJ -ET -q -1 0 0 1 598.207 24.29 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F45 10.9091 Tf 604.185 82.551 Td [('SOC1')]TJ/F15 10.9091 Tf 34.363 0 Td [(,)]TJ/F45 10.9091 Tf 6.667 0 Td [('SOC2')]TJ -ET -q -1 0 0 1 692.367 24.29 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -BT -/F15 10.9091 Tf 698.344 82.551 Td [(T)28(yp)-28(e)-254(of)-255(aggregation)-254(algorithm:)-405(curren)28(tly)83(,)]TJ 0 -13.549 Td [(w)28(e)-355(implemen)28(t)-355(to)-355(measures)-355(of)-355(strength)-355(of)]TJ 0 -13.549 Td [(connection,)-595(the)-543(one)-542(b)28(y)-543(V)83(an)28(\024)472(ek,)-595(Mandel)]TJ 0 -13.549 Td [(and)-454(Brezina)-454([)]TJ -1 0 0 rg 1 0 0 RG - [(26)]TJ -0 g 0 G - [(],)-484(and)-454(the)-454(one)-454(b)28(y)-454(Grat-)]TJ 0 -13.55 Td [(ton)-333(et)-334(al)-333([)]TJ -1 0 0 rg 1 0 0 RG - [(16)]TJ -0 g 0 G - [(].)]TJ -ET -q -1 0 0 1 899.912 24.29 cm -[]0 d 0 J 0.398 w 0 0 m 0 67.746 l S -Q -q -1 0 0 1 299.826 24.09 cm -[]0 d 0 J 0.398 w 0 0 m 600.087 0 l S -Q -q -1 0 0 1 299.826 -16.756 cm -[]0 d 0 J 0.398 w 0 0 m 0 40.648 l S -Q -BT -/F45 10.9091 Tf 305.803 14.407 Td [('AGGR_PROL')]TJ -ET -q -1 0 0 1 422.332 -16.756 cm -[]0 d 0 J 0.398 w 0 0 m 0 40.648 l S -Q -BT -/F45 10.9091 Tf 428.31 14.407 Td [(character\050len=*\051)]TJ -ET -q -1 0 0 1 521.055 -16.756 cm -[]0 d 0 J 0.398 w 0 0 m 0 40.648 l S -Q -BT -/F45 10.9091 Tf 527.032 14.407 Td [('SMOOTHED')]TJ/F15 10.9091 Tf 57.273 0 Td [(,)]TJ/F45 10.9091 Tf -57.273 -13.549 Td [('UNSMOOTHED')]TJ -ET -q -1 0 0 1 598.207 -16.756 cm -[]0 d 0 J 0.398 w 0 0 m 0 40.648 l S -Q -BT -/F45 10.9091 Tf 604.185 14.407 Td [('SMOOTHED')]TJ -ET -q -1 0 0 1 692.367 -16.756 cm -[]0 d 0 J 0.398 w 0 0 m 0 40.648 l S -Q -BT -/F15 10.9091 Tf 698.344 14.407 Td [(Prolongator)-436(used)-435(b)27(y)-435(the)-436(aggregation)-436(al-)]TJ 0 -13.549 Td [(gorithm:)-609(smo)-28(othed)-416(or)-416(unsmo)-28(oth)1(e)-1(d)-415(\050i.e.,)]TJ 0 -13.55 Td [(ten)28(tativ)28(e)-334(prolongator\051.)]TJ -ET -q -1 0 0 1 899.912 -16.756 cm 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m 0 13.549 l S -Q -q -1 0 0 1 299.826 -57.802 cm -[]0 d 0 J 0.398 w 0 0 m 0 13.549 l S -Q -BT -/F15 10.9091 Tf 305.803 -53.738 Td [(smaller)-333(than)-334(the)-333(sp)-28(eci\014ed)-333(maxim)28(um)-334(n)28(um)28(b)-28(er)-333(of)-333(lev)27(els.)]TJ -ET -q -1 0 0 1 899.912 -57.802 cm -[]0 d 0 J 0.398 w 0 0 m 0 13.549 l S -Q -q -1 0 0 1 299.826 -58.002 cm -[]0 d 0 J 0.398 w 0 0 m 600.087 0 l S -Q -0 g 0 G -BT -/F15 10.9091 Tf 459.779 -88.691 Td [(T)83(able)-333(3:)-444(P)27(arameters)-333(de\014ning)-333(the)-334(aggregation)-333(algorithm.)]TJ -0 g 0 G -0 g 0 G -ET -1 0 0 1 299.826 121.521 cm -Q -0 g 0 G -0 g 0 G -0 g 0 G - -endstream -endobj -531 0 obj -<< -/Length 6126 ->> -stream -0 g 0 G -BT -/F15 10.9091 Tf 86.4 740.002 Td [(28)]TJ/F42 10.9091 Tf 203.265 0 Td [(MLD2P4)-378(User)67('s)-378(and)-378(Ref)1(erence)-378(Guide)]TJ -0 g 0 G -0 g 0 G -ET -1 0 0 1 292.625 715.095 cm -q -0 -1 1 0 0 0 cm -0 g 0 G -0 g 0 G -0 g 0 G -q -1 0 0 1 0 108.207 cm -[]0 d 0 J 0.398 w 0 0 m 598.736 0 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E G DÑ F J O I M HÔ^ K Q Nñ” P T S W Rù U€W [ X8 Y€ \þ ] }€€W€€€qH€€€€€€ €!€"€#€$€%€&€'€(€)€*€+€,v+€-Ãc€.}€/IN€0jØ€1¥ó€2ÅB€3îò€4UT€5î€6¯$€7ÍŸ€8ôƒ€9M€:h«€;ž€<ö¿€=€X€Y€Z€[ SyÀÀÀÀÀÀ À!À"À#À$À%À&À'À(À)À*À+À,À-À.À/À0À1À2À3À4À5À6À7À8À9À:À; P© s• -endstream -endobj -startxref -619413 -%%EOF diff --git a/docs/src/Makefile b/docs/src/Makefile index 81417e58..e549a598 100644 --- a/docs/src/Makefile +++ b/docs/src/Makefile @@ -140,7 +140,7 @@ PDF = $(join $(BASEFILE),.pdf) PS = $(join $(BASEFILE),.ps) GXS = $(join $(BASEFILE),.gxs) GLX = $(join $(BASEFILE),.glx) -TARGETPDF= ../mld2p4-2.2-guide.pdf +TARGETPDF= ../amg4psblas_1.0-guide.pdf BASEHTML = $(patsubst %.tex,%,$(HTMLFILE)) HTML = $(join $(BASEHTML),.html) HTMLDIR = ../html diff --git a/docs/src/abstract.tex b/docs/src/abstract.tex index ad3e67b4..6146c18d 100644 --- a/docs/src/abstract.tex +++ b/docs/src/abstract.tex @@ -1,25 +1,25 @@ \section*{Abstract} -\addcontentsline{toc}{section}{Abstract} +\addcontentsline{toc}{section}{Abstract} -\textsc{MLD2P4 (MultiLevel Domain Decomposition Parallel Preconditioners Package -based on PSBLAS}) is a package of parallel algebraic multilevel preconditioners. -The first release of MLD2P4 made available multilevel additive and hybrid Schwarz -preconditioners, as well as one-level additive Schwarz preconditioners. The package -has been extended to include further multilevel cycles and smoothers widely used in -multigrid methods. In the multilevel case, a purely algebraic approach is applied to -generate coarse-level corrections, so that no geometric background is needed -concerning the matrix to be preconditioned. The matrix is assumed to be square, -real or complex. +\textsc{AMG4PSBLAS (Algebraic MultiGrid Preconditioners Package +based on PSBLAS}) is a package of parallel algebraic multilevel preconditioners included in the PSCToolkit (Parallel Sparse Computation Toolkit) software framework. +It is a progress of a software development project started in 2007, named MLD2P4, which implemented a multilevel version of some domain decomposition preconditioners of additive-Schwarz type and was based on a parallel decoupled version of the well known smoothed +aggregation method to generate the multilevel hierarchy of coarser matrices. In the last years, within the context of the EU-H2020 EoCoE project (Energy Oriented Center of Excellence), the package was extended including new algorithms and functionalities for setup and application of new AMG preconditioners with the final aims of improving efficiency and scalability when tens of thousands cores are +used and of boosting reliability in dealing with general symmetric positive definite linear systems. Due to the significant number of changes and the increase in scope, we decided to rename the package as AMG4PSBLAS. -MLD2P4 has been designed to provide scalable and easy-to-use preconditioners +AMG4PSBLAS has been designed to provide scalable and easy-to-use preconditioners in the context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms) computational framework and can be used in conjuction with the Krylov solvers -available in this framework. MLD2P4 enables the user to easily specify different -features of an algebraic multilevel preconditioner, thus allowing to search -for the ``best'' preconditioner for the problem at hand. +available in this framework. +Our package is based on a completely algebraic approach and users level interfaces +assume that the system matrix and preconditioners are represented as PSBLAS +distributed sparse matrices. +AMG4PSBLAS enables the user to easily specify different +features of an algebraic multilevel preconditioner, thus allowing to experiment +with different preconditioners for the problem and parallel computers at hand. The package employs object-oriented design techniques in -Fortran~2003, with interfaces to additional third party libraries +Fortran~2003, with interfaces to additional third party libraries such as MUMPS, UMFPACK, SuperLU, and SuperLU\_Dist, which can be exploited in building multilevel preconditioners. The parallel implementation is based on a Single Program Multiple Data (SPMD) @@ -27,4 +27,4 @@ paradigm; the inter-process communication is based on MPI and is managed mainly through PSBLAS. This guide provides a brief description of the functionalities and -the user interface of MLD2P4. +the user interface of AMG4PSBLAS. \ No newline at end of file diff --git a/docs/src/bibliography.tex b/docs/src/bibliography.tex index d6716846..c8b21bbe 100644 --- a/docs/src/bibliography.tex +++ b/docs/src/bibliography.tex @@ -1,6 +1,6 @@ %\section{Bibliography\label{sec:bib}} \begin{thebibliography}{99} -\addcontentsline{toc}{section}{\refname} +\addcontentsline{toc}{section}{\refname} \markboth{\textsc{MLD2P4 User's and Reference Guide}} {\textsc{References}} @@ -18,7 +18,7 @@ M.~Brezina, P.~Van\v{e}k, Computing, 63, 1999, 233--263. % \bibitem{Briggs2000} -W.~L.~Briggs, V.~E.~Henson, S.~F.~McCormick, +W.~L.~Briggs, V.~E.~Henson, S.~F.~McCormick, {\em A Multigrid Tutorial, Second Edition}, SIAM, 2000. % @@ -30,11 +30,11 @@ Proceedings of PARA~04 Workshop on State of the Art in Scientific Computing, Lecture Notes in Computer Science, Springer, 2005, 593--602. % -\bibitem{aaecc_07} +\bibitem{aaecc_07} A.~Buttari, P.~D'Ambra, D.~di~Serafino, S.~Filippone, {\em 2LEV-D2P4: a package of high-performance preconditioners for scientific and engineering applications}, -Applicable Algebra in Engineering, Communications and Computing, +Applicable Algebra in Engineering, Communications and Computing, 18 (3) 2007, 223--239. %Published online: 13 February 2007, {\tt http://dx.doi.org/10.1007/s00200-007-0035-z} % @@ -57,19 +57,29 @@ SIAM Journal on Scientific Computing, 21 (2), 1999, 792--797. \bibitem{apnum_07} P.~D'Ambra, S.~Filippone, D.~di~Serafino, {\em On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners}, -Applied Numerical Mathematics, Elsevier Science, +Applied Numerical Mathematics, Elsevier Science, 57 (11-12), 2007, 1181-1196. %published online 3 February 2007, {\tt % http://dx.doi.org/10.1016/j.apnum.2007.01.006} % -\bibitem{MLD2P4_TOMS} +\bibitem{MLD2P4_TOMS} P.~D'Ambra, D.~di~Serafino, S.~Filippone, \emph{MLD2P4: a Package of Parallel Multilevel Algebraic Domain Decomposition Preconditioners in Fortran 95}, ACM Trans. Math. Softw., 37(3), 2010, art. 30. % +\bibitem{BDDF2007} +A.~Buttari, P.~D'Ambra, D.~di Serafino, S.~Filippone, +\emph{2{LEV}-{D}2{P}4: a Package of High-Performance Preconditioners +for Scientific and Engineering Applications}, Appl. Algebra Engrg. Comm. Comput., +18(3), 2007, 223--239. +% +\bibitem{DDF2020} +P.~D'Ambra, F~Durastante, S.~Filippone, +\emph{AMG preconditioners for Linear Solvers towards Extreme Scale}, 2020, arXiv:2006.16147v2. +% \bibitem{UMFPACK} -T.~A.~Davis, +T.~A.~Davis, {\em Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal Method with a Column Pre-ordering Strategy}, ACM Transactions on Mathematical Software, 30, 2004, 196--199. @@ -96,13 +106,13 @@ ACM Transactions on Mathematical Software, 14 (1) 1988, 1--17. %Lapack Working Note 94, Tech.\ Rep.\ UT-CS-95-281, University of %Tennessee, March 1995 (updated May 1997). % -%\bibitem{EFSTATHIOU} +%\bibitem{EFSTATHIOU} %E.~Efstathiou, J.~G.~Gander, %{\em Why Restricted Additive Schwarz Converges Faster than Additive Schwarz}, %BIT Numerical Mathematics, 43 (5), 2003, 945--959. % \bibitem{PSBLASGUIDE} -S.~Filippone, A.~Buttari, +S.~Filippone, A.~Buttari, {\em PSBLAS 3.5.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library}, 2012, available from \texttt{https://github.com/sfilippone/psblas3/tree/master/docs}. % @@ -112,19 +122,19 @@ S.~Filippone, A.~Buttari, ACM Transactions on on Mathematical Software, 38 (4), 2012, art.~23. % \bibitem{psblas_00} -S.~Filippone, M.~Colajanni, +S.~Filippone, M.~Colajanni, {\em PSBLAS: A Library for Parallel Linear Algebra Computation on Sparse Matrices}, ACM Transactions on Mathematical Software, 26 (4), 2000, 527--550. % \bibitem{GrHeJi:16} S. Gratton, P. Henon, P. Jiranek and X. Vasseur, - {\em Reducing complexity of algebraic multigrid by aggregation}, + {\em Reducing complexity of algebraic multigrid by aggregation}, Numerical Lin. Algebra with Applications, 2016, 23:501-518 % \bibitem{MPI2} -W.~Gropp, S.~Huss-Lederman, A.~Lumsdaine, E.~Lusk, B.~Nitzberg, W.~Saphir, M.~Snir, +W.~Gropp, S.~Huss-Lederman, A.~Lumsdaine, E.~Lusk, B.~Nitzberg, W.~Saphir, M.~Snir, {\em MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions}, MIT Press, 1998. % @@ -142,7 +152,7 @@ ACM Transactions on Mathematical Software, 29 (2), 2003, 110--140. \bibitem{Notay2008} Y.~Notay, P.~S.~Vassilevski, {\em Recursive Krylov-based multigrid cycles}, -Numerical Linear Algebra with Applications, 15 (5), 2008, 473--487. +Numerical Linear Algebra with Applications, 15 (5), 2008, 473--487. % \bibitem{Saad_book} Y.~Saad, @@ -173,5 +183,15 @@ R.~S.~Tuminaro, C.~Tong, P.~Van\v{e}k, J.~Mandel, M.~Brezina, {\em Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems}, Computing, 56 (3) 1996, 179--196. - +% +\bibitem{DV2013} +P.~D'Ambra and P.\,S.~Vassilevski, +{\em Adaptive AMG with coarsening based on compatible weighted matching}, +Computing and Visualization in Science, 16, (2013) 59--76. +% +\bibitem{DFV2018} +P.~D'Ambra, S.~Filippone and P.\,S.~Vassilevski, +{\em BootCMatch: a software package for bootstrap AMG based on graph weighted matching}, +ACM Transactions on Mathematical Software, 44, (2018) 39:1--39:25. +% \end{thebibliography} diff --git a/docs/src/building.tex b/docs/src/building.tex index 836e5470..baf47463 100644 --- a/docs/src/building.tex +++ b/docs/src/building.tex @@ -1,33 +1,33 @@ -\section{Configuring and Building MLD2P4\label{sec:building}} -\markboth{\textsc{MLD2P4 User's and Reference Guide}} - {\textsc{\ref{sec:building} Configuring and Building MLD2P4}} -In order to build MLD2P4 it is necessary to set up a Makefile with appropriate +\section{Configuring and Building AMG4PSBLAS\label{sec:building}} +\markboth{\textsc{AMG4PSBLAS User's and Reference Guide}} + {\textsc{\ref{sec:building} Configuring and Building AMG4PSBLAS}} +In order to build AMG4PSBLAS it is necessary to set up a Makefile with appropriate system-dependent variables; this is done by means of the \verb|configure| script. The distribution also includes the autoconf and automake sources employed to generate the script, but usually this is not needed -to build the software. +to build the software. -MLD2P4 is implemented almost entirely in Fortran~2003, with some +AMG4PSBLAS is implemented almost entirely in Fortran~2003, with some interfaces to external libraries in C; the Fortran compiler must support the Fortran~2003 standard plus the extension \verb|MOLD=| -feature, which enhances the usability of \verb|ALLOCATE|. +feature, which enhances the usability of \verb|ALLOCATE|. Many compilers do this; in particular, this is -supported by the GNU Fortran compiler, for which we -recommend to use at least version 4.8. +supported by the GNU Fortran compiler, for which we +recommend to use at least version 4.8. The software defines data types and interfaces for -real and complex data, in both single and double precision. +real and complex data, in both single and double precision. -Building MLD2P4 requires some base libraries (see Section~\ref{sec:prerequisites}); -interfaces to optional third-party libraries, which extend the functionalities of MLD2P4 +Building AMG4PSBLAS requires some base libraries (see Section~\ref{sec:prerequisites}); +interfaces to optional third-party libraries, which extend the functionalities of AMG4PSBLAS (see Section~\ref{sec:third-party}), are also available. Many Linux distributions (e.g., Ubuntu, Fedora, CentOS) provide precompiled packages for the prerequisite and optional software. In many cases these packages are split between a runtime part and a -``developer'' part; in order to build MLD2P4 you need both. A description of the base and -optional software used by MLD2P4 is given in the next sections. +``developer'' part; in order to build AMG4PSBLAS you need both. A description of the base and +optional software used by AMG4PSBLAS is given in the next sections. \subsection{Prerequisites\label{sec:prerequisites}} -The following base libraries are needed: +The following base libraries are needed: \begin{description} \item[BLAS] \cite{blas3,blas2,blas1} Many vendors provide optimized versions of BLAS; if no vendor version is @@ -36,38 +36,38 @@ The following base libraries are needed: may be employed. The reference BLAS from Netlib (\url{www.netlib.org/blas}) are meant to define the standard behaviour of the BLAS interface, so they are not optimized for any - particular plaftorm, and should only be used as a last + particular platform, and should only be used as a last resort. Note that BLAS computations form a relatively small part of - the MLD2P4/PSBLAS computations; they are however critical when using + the AMG4PSBLAS/PSBLAS computations; they are however critical when using preconditioners based on MUMPS, UMFPACK or SuperLU third party libraries. Note that UMFPACK requires a full LAPACK library; our experience is that configuring ATLAS for building full LAPACK does not work in the correct way. Our advice is first to download the LAPACK tarfile from \url{www.netlib.org/lapack} and install it independently of ATLAS. In this case, you need to modify the OPTS and NOOPT definitions for including -fPIC compilation option -in the make.inc file of the LAPACK library. +in the make.inc file of the LAPACK library. \item[MPI] \cite{MPI2,MPI1} A version of MPI is available on most high-performance computing systems. \item[PSBLAS] \cite{PSBLASGUIDE,psblas_00} Parallel Sparse BLAS (PSBLAS) is - available from \url{github.com/sfilippone/psblas3}; version + available from \url{github.com/sfilippone/psblas3} ?? {\bf Mettere tutto sotto psctoolkit}; version 3.5.0 (or later) is required. Indeed, all the prerequisites listed so far are also prerequisites of PSBLAS. \end{description} Please note that the four previous libraries must have Fortran -interfaces compatible with MLD2P4; +interfaces compatible with AMG4PSBLAS; usually this means that they should all be built with the same -compiler as MLD2P4. +compiler as AMG4PSBLAS. \subsection{Optional third party libraries\label{sec:third-party}} We provide interfaces to the following third-party software libraries; note that these are optional, but if you enable them some defaults -for multilevel preconditioners may change to reflect their presence. +for multilevel preconditioners may change to reflect their presence. \begin{description} \item[UMFPACK] \cite{UMFPACK} - A sparse LU factorization package included in the SuiteSparse library, available from - \url{faculty.cse.tamu.edu/davis/suitesparse.html}; + A sparse LU factorization package included in the SuiteSparse library, available from + \url{faculty.cse.tamu.edu/davis/suitesparse.html}; it provides sequential factorization and triangular system solution for double precision real and complex data. We tested version 4.5.4 of SuiteSparse. Note that for configuring SuiteSparse you should provide the right path to the BLAS @@ -97,10 +97,13 @@ for multilevel preconditioners may change to reflect their presence. \subsection{Configuration options} -In order to build MLD2P4, the first step is to use the \verb|configure| script +In order to build AMG4PSBLAS, the first step is to use the \verb|configure| script in the main directory to generate the necessary makefile. %\textbf{Sono necessarie le parentesi intorno a s?} + +\bf{DA RISCRIVERE} + As a minimal example consider the following: \begin{verbatim} ./configure --with-psblas=PSB-INSTALL-DIR @@ -290,7 +293,7 @@ generated the file \verb|Make.inc| which will then be used by all Makefiles in the directory tree; this file will be copied in the install directory under the name \verb|Make.inc.MLD2P4|. -To use the MUMPS solver package, +To use the MUMPS solver package, the user has to add the appropriate options to the configure script; by default we are looking for the libraries \verb|-ldmumps -lsmumps| \verb| -lzmumps -lcmumps -mumps_common -lpord|. @@ -298,30 +301,22 @@ MUMPS often uses additional packages such as ScaLAPACK, ParMETIS, SCOTCH, as well as enabling OpenMP; in such cases it is necessary to add linker options with the \verb|--with-extra-libs| configure option. -To build the library the user will now enter +To build the library the user will now enter \begin{verbatim} make \end{verbatim} -followed (optionally) by +followed (optionally) by \begin{verbatim} make install \end{verbatim} \subsection{Bug reporting} If you find any bugs in our codes, please report them through our issues page on \\[2mm] -\url{https://github.com/sfilippone/mld2p4-2/issues}\\ -% please let us know at -%\begin{rawhtml} -% -%\end{rawhtml} -%\texttt{bugreport@mld2p4.it} -%\begin{rawhtml} -% -%\end{rawhtml} -%; +\url{https://github.com/psctoolkit/amg4psblas/issues}\\ + To enable us to track the bug, please provide a log from the failing application, the test conditions, and ideally a self-contained test -program reproducing the issue. +program reproducing the issue. \subsection{Example and test programs\label{sec:ex_and_test}} The package contains the \verb|examples| and \verb|tests| directories; @@ -331,7 +326,7 @@ both of them are further divided into \verb|fileread| and \item[\tt examples] contains a set of simple example programs with a predefined choice of preconditioners, selectable via integer values. These are intended to get an acquaintance with the - multilevel preconditioners available in MLD2P4. + multilevel preconditioners available in AMG4PSBLAS. \item[\tt tests] contains a set of more sophisticated examples that will allow the user, via the input files in the \verb|runs| subdirectories, to experiment with the full range of preconditioners @@ -341,4 +336,4 @@ The \verb|fileread| directories contain sample programs that read sparse matrices from files, according to the Matrix Market or the Harwell-Boeing storage format; the \verb|pdegen| programs generate matrices in full parallel mode from the discretization of a sample partial -differential equation. +differential equation. diff --git a/docs/src/distribution.tex b/docs/src/distribution.tex index 5682acfe..8aec100c 100644 --- a/docs/src/distribution.tex +++ b/docs/src/distribution.tex @@ -1,11 +1,11 @@ \section{Code Distribution\label{sec:distribution}} -\markboth{\textsc{MLD2P4 User's and Reference Guide}} +\markboth{\textsc{AMG4PSBLAS User's and Reference Guide}} {\textsc{\ref{sec:distribution} Code Distribution}} \noindent -MLD2P4 is available from the web site +AMG4PSBLAS is available from the web site \begin{quotation} -\texttt{https://github.com/sfilippone/mld2p4-2} +\texttt{https://github.com/psctoolkit/amg4psblas} \end{quotation} where contact points for further information can be also found. @@ -13,25 +13,16 @@ The software is available under a modified BSD license, as specified in Appendix~\ref{sec:license}; please note that some of the optional third party libraries may be licensed under a different and more stringent license, most notably the GPL, and this should be taken into -account when treating derived works. +account when treating derived works. The library defines a version string with the -constant -\[ \verb|mld_version_string_|\] -whose current value is \verb|2.1.0|. +constant +\[ \verb|amg_version_string_|\] +whose current value is \verb|1.0|. \subsection*{Contributors} -Contributors to version 2: \begin{itemize} -\item Salvatore Filippone, Cranfield University, UK; -\item Pasqua D'Ambra, IAC-CNR, Naples, IT; -\item Daniela di Serafino, University of Campania ``L. Vanvitelli'', Caserta, IT; -\item Ambra Abdullahi Hassan, University of Rome ``Tor Vergata'', IT. -\end{itemize} -Contributors to version 1: -\begin{itemize} -\item Salvatore Filippone; -\item Pasqua D'Ambra; -\item Daniela di Serafino; -\item Alfredo Buttari, CNRS-IRIT, Toulouse, F. -\end{itemize} +\item Pasqua D'Ambra, IAC-CNR, IT; +\item Fabio Durastante, University of Pisa and IAC-CNR, IT; +\item Salvatore Filippone, University of Rome Tor-Vergata and IAC-CNR, IT; +\end{itemize} \ No newline at end of file diff --git a/docs/src/errors.tex b/docs/src/errors.tex index 67ee191d..5c67ede4 100644 --- a/docs/src/errors.tex +++ b/docs/src/errors.tex @@ -1,8 +1,8 @@ \section{Error Handling\label{sec:errors}} -\markboth{\textsc{MLD2P4 User's and Reference Guide}} +\markboth{\textsc{AMG4PSBLAS User's and Reference Guide}} {\textsc{\ref{sec:errors} Error handling}} -The error handling in MLD2P4 is based on the PSBLAS error +The error handling in AMG4PSBLAS is based on the PSBLAS error handling. Error conditions are signaled via an integer argument \verb|info|; whenever an error condition is detected, an error trace stack is built by the library up to the top-level, user-callable @@ -12,9 +12,9 @@ program or by returning the error condition to the user code, which will then take action, and whether an error message should be printed. 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\par $> 1$ & 1.5 & Minimum coarsening ratio. The aggregation stops if the ratio between the matrix dimensions at two consecutive levels is lower than or equal to this - threshold (see Note).\\ \hline -%\verb|mld_max_levs_| \par + threshold (see Note).\\ \hline + \verb|'MAX_LEVS'| & \verb|integer| & Any integer \par number $> 1$ & 20 & Maximum number of levels. The aggregation stops - if the number of levels reaches this value (see Note). \\ \hline -%\verb|mld_par_aggr_alg_| \par + if the number of levels reaches this value (see Note). \\ \hline + \verb|'PAR_AGGR_ALG'| & \verb|character(len=*)| \hspace*{-3mm} & \texttt{'DEC'}, \texttt{'SYMDEC'} & \texttt{'DEC'} @@ -328,24 +297,24 @@ be applied. \verb|SYMDEC| option applies decoupled aggregation to the sparsity pattern of $A+A^T$.\\ \hline -%\verb|mld_aggr_type_| \par + \verb|'AGGR_TYPE'| & \verb|character(len=*)| \hspace*{-3mm} & \textbf{\texttt{'SOC1'}} & \textbf{\texttt{'SOC1'}}, - \textbf{\texttt{'SOC2'}} + \textbf{\texttt{'SOC2'}} & Type of aggregation algorithm: currently, - we implement two measures of strength of + we implement to measures of strength of connection, the one by Van\v{e}k, Mandel and Brezina~\cite{VANEK_MANDEL_BREZINA}, and the one by Gratton et al~\cite{GrHeJi:16}. \\ \hline -%\verb|mld_aggr_prol_| \par + \verb|'AGGR_PROL'| & \verb|character(len=*)| \hspace*{-3mm} & \texttt{'SMOOTHED'}, \texttt{'UNSMOOTHED'} & \texttt{'SMOOTHED'} & Prolongator used by the aggregation algorithm: smoothed or unsmoothed (i.e., tentative prolongator). \\ \hline \multicolumn{5}{|l|}{{\bfseries Note.} The aggregation algorithm stops when -at least one of the following criteria is met: +at least one of the following criteria is met: the coarse size threshold, the} \\ \multicolumn{5}{|l|}{minimum coarsening ratio, or the maximum number of levels is reached. Therefore, the actual number of levels may be} \\ @@ -355,7 +324,7 @@ of levels. } \\ \end{tabular} \end{center} \caption{Parameters defining the aggregation algorithm. -\label{tab:p_aggregation}} +\label{tab:p_aggregation}} \esideways \bsideways @@ -365,67 +334,31 @@ of levels. } \\ \hline \verb|what| & \textsc{data type} & \verb|val| & \textsc{default} & \textsc{comments} \\ \hline -%\verb|mld_aggr_ord_| \par + \verb|'AGGR_ORD'| & \verb|character(len=*)| & \texttt{'NATURAL'} \par \texttt{'DEGREE'} & \texttt{'NATURAL'} & Initial ordering of indices for the aggregation algorithm: either natural ordering or sorted by descending degrees of the nodes in the - matrix graph. \\ \hline + matrix graph. \\ \hline %Since aggregation is %heuristic, results will be different. -%\verb|mld_aggr_thresh_| \par + \verb|'AGGR_THRESH'| & \verb|real(|\emph{kind\_parameter}\verb|)| & Any~real \par number~$\in [0, 1]$ & 0.01 & The threshold $\theta$ in the aggregation algorithm, see (\ref{eq:strongly_coup}) in Section~\ref{sec:aggregation}. See also the note at the bottom of this table. \\ \hline -%%\verb|mld_aggr_scale_| \par -% \verb|AGGR_SCALE| & \verb|real(|\emph{kind\_parameter}\verb|)| -% & Any~real \par number~$\in [0, 1]$ -% & 1.0 -% & Scale factor applied to the threshold in going -% from level $ilev$ to level $ilev+1$. \\ \hline -%\verb|mld_aggr_omega_alg_| \par -%\verb|'AGGR_OMEGA_ALG'|& \verb|character(len=*)| -% & \texttt{'EIG\_EST'} \par \texttt{'USER\_CHOICE'} -% & \texttt{'EIG\_EST'} -% & How the damping parameter $\omega$ in the -% smoothed aggregation is obtained: -% either via an estimate of the spectral radius of -% $D^{-1}A$, where $A$ is the matrix at the current -% level and $D$ is the diagonal matrix with -% the same diagonal entires as $A$, or explicily -% specified by the user. \\ \hline -%\verb|mld_aggr_eig_| \par -%\verb|'AGGR_EIG'| & \verb|character(len=*)| -% & \texttt{'A\_NORMI'} -% & \texttt{'A\_NORMI'} -% & How to estimate the spectral radius of $D^{-1}A$. -% Currently only the infinity norm estimate -% is available. \\ \hline -%\verb|mld_aggr_omega_val_| \par -%\verb|'AGGR_OMEGA_VAL'| & \verb|real(|\emph{kind\_parameter}\verb|)| -% & Any real \par number $>0$ -% & $4/(3\rho(D^{-1}A))$ -% & Damping parameter $\omega$ in the smoothed aggregation algorithm. -% It must be set by the user if -% \verb|USER_CHOICE| was specified for -% \verb|mld_aggr_omega_alg_|, -% otherwise it is computed by the library, using the -% selected estimate of the spectral radius $\rho(D^{-1}A)$ of -% $D^{-1}A$.\\ \hline -%\verb|mld_aggr_filter_| \par -\verb|'AGGR_FILTER'| +\verb|'AGGR_FILTER'| & \verb|character(len=*)| & \texttt{'FILTER'} \par \texttt{'NOFILTER'} & \texttt{'NOFILTER'} & Matrix used in computing the smoothed prolongator: filtered or unfiltered (see~(\ref{eq:filtered}) in Section~\ref{sec:aggregation}). \\ \hline \multicolumn{5}{|l|}{{\bfseries Note.} Different thresholds at different levels, such as -those used in \cite[Section~5.1]{VANEK_MANDEL_BREZINA}, can be easily set by +those used in \cite[Section~5.1]{VANEK_MANDEL_BREZINA}, can be easily set by invoking the rou-} \\ \multicolumn{5}{|l|}{tine \texttt{set} with the parameter \texttt{ilev}.} \\ @@ -433,31 +366,28 @@ the parameter \texttt{ilev}.} \\ \end{tabular} \end{center} \caption{Parameters defining the aggregation algorithm (continued). -\label{tab:p_aggregation_1}} +\label{tab:p_aggregation_1}} \esideways - + \bsideways \begin{center} \begin{tabular}{|p{3.9cm}|l|p{1.7cm}|p{1.7cm}|p{8.6cm}|} \hline \verb|what| & \textsc{data type} & \verb|val| & \textsc{default} & \textsc{comments} \\ \hline -%\multicolumn{5}{|c|}{\emph{coarse-space correction at the coarsest level}}\\ \hline -%\verb|mld_coarse_mat_| \par \verb|'COARSE_MAT'| & \verb|character(len=*)| & \texttt{'DIST'} \par \texttt{'REPL'} & \texttt{'REPL'} & Coarsest matrix layout: distributed among the processes or replicated on each of them. \\ \hline -%\verb|mld_coarse_solve_| \par \verb|'COARSE_SOLVE'| & \verb|character(len=*)| - & \texttt{'MUMPS'} \par \texttt{'UMF'} \par - \texttt{'SLU'} \par \texttt{'SLUDIST'} \par - \texttt{'JACOBI'} \par \texttt{'GS'} \par \texttt{'BJAC'} + & \texttt{'MUMPS'} \par \texttt{'UMF'} \par + \texttt{'SLU'} \par \texttt{'SLUDIST'} \par + \texttt{'JACOBI'} \par \texttt{'GS'} \par \texttt{'BJAC'} & See~Note. & Solver used at the coarsest level: sequential LU from MUMPS, UMFPACK, or SuperLU - (plus tri\-an\-gular solve); + (plus tri\-an\-gular solve); distributed LU from MUMPS or SuperLU\_Dist (plus triangular solve); point-Jacobi, hybrid Gauss-Seidel or block-Jacobi. \par @@ -467,35 +397,34 @@ the parameter \texttt{ilev}.} \\ distributed, and \texttt{MUMPS} can be used with either a replicated or a distributed matrix. When any of the previous solvers is specified, the matrix layout is set to a default - value which allows the use of the solver (see Remark 3, p.~24). + value which allows the use of the solver (see Remark 3, p.~24). Note also that UMFPACK and SuperLU\_Dist are available only in double precision. \\ \hline -%\verb|mld_coarse_subsolve_| \par \verb|'COARSE_SUBSOLVE'| & \verb|character(len=*)| & \texttt{'ILU'} \par \texttt{'ILUT'} \par \texttt{'MILU'} \par - \texttt{'MUMPS'} \par \texttt{'SLU'} \par \texttt{'UMF'} + \texttt{'MUMPS'} \par \texttt{'SLU'} \par \texttt{'UMF'} & See~Note. & Solver for the diagonal blocks of the coarse matrix, in case the block Jacobi solver is chosen as coarsest-level solver: ILU($p$), ILU($p,t$), - MILU($p$), LU from MUMPS, SuperLU or UMFPACK + MILU($p$), LU from MUMPS, SuperLU or UMFPACK (plus triangular solve). Note that UMFPACK and SuperLU\_Dist are available only in double precision. \\ \hline -\multicolumn{5}{|l|}{{\bfseries Note.} Defaults for \texttt{COARSE\_SOLVE} and +\multicolumn{5}{|l|}{{\bfseries Note.} Defaults for \texttt{COARSE\_SOLVE} and \texttt{COARSE\_SUBSOLVE} are chosen in the following order:} \\ -\multicolumn{5}{|l|}{single precision version -- \texttt{MUMPS} if installed, - then \texttt{SLU} if installed, +\multicolumn{5}{|l|}{single precision version -- \texttt{MUMPS} if installed, + then \texttt{SLU} if installed, \texttt{ILU} otherwise;}\\ -\multicolumn{5}{|l|}{double precision version -- \texttt{UMF} if installed, - then \texttt{MUMPS} if installed, then \texttt{SLU} if +\multicolumn{5}{|l|}{double precision version -- \texttt{UMF} if installed, + then \texttt{MUMPS} if installed, then \texttt{SLU} if installed, \texttt{ILU} otherwise.}\\ \hline \end{tabular} \end{center} \caption{Parameters defining the coarse-space correction at the coarsest -level.\label{tab:p_coarse}} +level.\label{tab:p_coarse}} \esideways \bsideways @@ -504,19 +433,15 @@ level.\label{tab:p_coarse}} \hline \verb|what| & \textsc{data type} & \verb|val| & \textsc{default} & \textsc{comments} \\ \hline -%\multicolumn{5}{|c|}{\emph{coarse-space correction at the coarsest level}}\\ \hline -%\verb|mld_coarse_sweeps_| \par -\verb|'COARSE_SWEEPS'| & \verb|integer| +\verb|'COARSE_SWEEPS'| & \verb|integer| & Any integer \par number $> 0$ & 10 & Number of sweeps when \verb|JACOBI|, \verb|GS| or \verb|BJAC| is chosen as coarsest-level solver. \\ \hline -%\verb|mld_coarse_fillin_| \par \verb|'COARSE_FILLIN'| & \verb|integer| & Any integer \par number $\ge 0$ & 0 & Fill-in level $p$ of the ILU factorizations. \\ \hline -%\verb|mld_coarse_iluthrs_| \par \verb|'COARSE_ILUTHRS'| & \verb|real(|\emph{kind\_parameter}\verb|)| & Any real \par number $\ge 0$ @@ -526,7 +451,7 @@ level.\label{tab:p_coarse}} \end{tabular} \end{center} \caption{Parameters defining the coarse-space correction at the coarsest -level (continued).\label{tab:p_coarse_1}} +level (continued).\label{tab:p_coarse_1}} \esideways \bsideways @@ -536,8 +461,7 @@ level (continued).\label{tab:p_coarse_1}} \hline \verb|what| & \textsc{data type} & \verb|val| & \textsc{default} & \textsc{comments} \\ \hline -%\multicolumn{5}{|c|}{\emph{basic one-level preconditioner (smoother)}} \\ \hline -%\verb|mld_smoother_type_| \par + \verb|'SMOOTHER_TYPE'| & \verb|character(len=*)| & \verb|'JACOBI'| \par \verb|'GS'| \par \verb|'BGS'| \par \verb|'BJAC'| \par \verb|'AS'| @@ -547,11 +471,10 @@ level (continued).\label{tab:p_coarse_1}} hybrid backward Gauss-Seidel, block-Jacobi, and Additive Schwarz. \par It is ignored by one-level preconditioners. \\ \hline -%\verb|mld_sub_solve_| \par \verb|'SUB_SOLVE'| & \verb|character(len=*)| & \texttt{'JACOBI'} \par - \texttt{'GS'} \par \texttt{'BGS'} \par \texttt{'ILU'} \par - \texttt{'ILUT'} \par \texttt{'MILU'} \par + \texttt{'GS'} \par \texttt{'BGS'} \par \texttt{'ILU'} \par + \texttt{'ILUT'} \par \texttt{'MILU'} \par \par \texttt{'MUMPS'} \par \texttt{'SLU'} \par \texttt{'UMF'} & \texttt{GS} and \texttt{BGS} for pre- and post-smoothers of multilevel preconditioners, respectively \par @@ -564,16 +487,14 @@ level (continued).\label{tab:p_coarse_1}} LU from MUMPS, SuperLU or UMFPACK (plus triangular solve). See Note for details on hybrid Gauss-Seidel. \\ \hline -%\verb|mld_moother_sweeps_| \par \verb|'SMOOTHER_SWEEPS'| & \verb|integer| & Any integer \par number~$\ge 0$ & 1 & Number of sweeps of the smoother or one-level preconditioner. In the multilevel case, no pre-smother or - post-smoother is used if this parameter is set to 0 + post-smoother is used if this parameter is set to 0 together with \verb|pos='PRE'| or \verb|pos='POST|, respectively. \\ \hline -%\verb|mld_sub_ovr_| \par \verb|'SUB_OVR'| & \verb|integer| & Any integer \par number~$\ge 0$ & 1 @@ -582,7 +503,7 @@ level (continued).\label{tab:p_coarse_1}} \end{tabular} \end{center} \caption{Parameters defining the smoother or the details of the one-level preconditioner. -\label{tab:p_smoother}} +\label{tab:p_smoother}} \esideways \bsideways @@ -592,16 +513,14 @@ level (continued).\label{tab:p_coarse_1}} \hline \verb|what| & \textsc{data type} & \verb|val| & \textsc{default} & \textsc{comments} \\ \hline -%\verb|mld_sub_restr_| \par \verb|'SUB_RESTR'| & \verb|character(len=*)| & \texttt{'HALO'} \par \texttt{'NONE'} & \texttt{'HALO'} & Type of restriction operator, for Additive Schwarz only: - \texttt{HALO} for taking into account the overlap, \texttt{NONE} + \texttt{HALO} for taking into account the overlap, \texttt{NONE} for neglecting it. \par Note that \texttt{HALO} must be chosen for the classical Addditive Schwarz smoother and its RAS variant.\\ \hline -%\verb|mld_sub_prol_| \par \verb|'SUB_PROL'| & \verb|character(len=*)| & \texttt{'SUM'} \par \texttt{'NONE'} & \texttt{'NONE'} @@ -610,12 +529,10 @@ level (continued).\label{tab:p_coarse_1}} for neglecting them. \par Note that \texttt{SUM} must be chosen for the classical Additive Schwarz smoother, and \texttt{NONE} for its RAS variant. \\ \hline -%\verb|mld_sub_fillin_| \par \verb|'SUB_FILLIN'| & \verb|integer| & Any integer \par number~$\ge 0$ & 0 & Fill-in level $p$ of the incomplete LU factorizations. \\ \hline -%\verb|mld_sub_iluthrs_| \par \verb|'SUB_ILUTHRS'| & \verb|real(|\emph{kind\_parameter}\verb|)| & Any real number~$\ge 0$ & 0 @@ -637,30 +554,18 @@ level (continued).\label{tab:p_coarse_1}} & 0 & Set an entry in the MUMPS real control array, as chosen via the \verb|idx| optional argument. \\ %\hline -%\verb|mld_sub_ren_| \par \verb|SUB_REN| & \verb|character(len=*)| -% & \texttt{'RENUM\_NONE'} \texttt{'RENUM\_GLOBAL'} %, \texttt{'RENUM_GPS'} -% & \texttt{'RENUM\_NONE'} -% & Row and column reordering of the local submatrices: no reordering, -% or reordering according to the global numbering of the rows and -% columns of the whole matrix. \\ -% \verb|mld_solver_eps_| \par \verb|SOLVER_EPS| & \verb|real| -% & Any~real number -% & 0 -% & Stopping tolerance for iterative local solver -% (currently only Gauss-Seidel); if $\le0$, then -% perform prespecified number of iterations. \\ \hline \hline \end{tabular} \end{center} \caption{Parameters defining the smoother or the details of the one-level preconditioner -(continued).\label{tab:p_smoother_1}} +(continued).\label{tab:p_smoother_1}} \esideways \clearpage \subsection{Method hierarchy\_build\label{sec:hier_bld}} - + \begin{center} \verb|call p%hierarchy_build(a,desc_a,info)|\\ \end{center} @@ -676,15 +581,13 @@ made by the user through the methods \verb|init| and \verb|set|. \verb|a| & \verb|type(psb_|\emph{x}\verb|spmat_type), intent(in)|. \\ & The sparse matrix structure containing the local part of the matrix to be preconditioned. Note that \emph{x} must be chosen according - to the real/complex, + to the real/complex, single/double precision version of MLD2P4 under use. See the PSBLAS User's Guide for details \cite{PSBLASGUIDE}.\\ \verb|desc_a| & \verb|type(psb_desc_type), intent(in)|. \\ & The communication descriptor of \verb|a|. See the PSBLAS User's Guide for details \cite{PSBLASGUIDE}.\\ -%\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\ -% & The preconditioner data structure. Note that \emph{x} must be chosen according -% to the real/complex, single/double precision version of MLD2P4 under use.\\ + \verb|info| & \verb|integer, intent(out)|.\\ & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ \end{tabular} @@ -693,7 +596,7 @@ single/double precision version of MLD2P4 under use. \clearpage \subsection{Method smoothers\_build\label{sec:smooth_bld}} - + \begin{center} \verb|call p%smoothers_build(a,desc_a,p,info[,amold,vmold,imold])|\\ @@ -704,7 +607,7 @@ This method builds the smoothers and the coarsest-level solvers for the multilevel preconditioner \verb|p|, according to the requirements made by the user through the methods \verb|init| and \verb|set|, and based on the aggregation hierarchy produced by a previous call to \verb|hierarchy_build| -(see Section~\ref{sec:hier_bld}). +(see Section~\ref{sec:hier_bld}). {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip @@ -717,9 +620,6 @@ hierarchy produced by a previous call to \verb|hierarchy_build| \verb|desc_a| & \verb|type(psb_desc_type), intent(in)|. \\ & The communication descriptor of \verb|a|. See the PSBLAS User's Guide for details \cite{PSBLASGUIDE}.\\ -%\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\ -% & The preconditioner data structure. Note that \emph{x} must be chosen according -% to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|info| & \verb|integer, intent(out)|.\\ & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ \verb|amold| & \verb|class(psb_|\emph{x}\verb|_base_sparse_mat), intent(in), optional|. \\ @@ -738,7 +638,7 @@ hierarchy produced by a previous call to \verb|hierarchy_build| \clearpage \subsection{Method build\label{sec:precbld}} - + \begin{center} \verb|call p%build(a,desc_a,info[,amold,vmold,imold])|\\ \end{center} @@ -750,8 +650,8 @@ made by the user through the methods \verb|init| and \verb|set| multilevel preconditioners). It is mostly provided for backward compatibility; indeed, it is internally implemented by invoking the two previous methods \verb|hierarchy_build| and -\verb|smoothers_build|, whose nomenclature would however be somewhat -unnatural when dealing with simple one-level preconditioners. +\verb|smoothers_build|, whose nomenclature would however be somewhat +unnatural when dealing with simple one-level preconditioners. {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip @@ -764,9 +664,6 @@ unnatural when dealing with simple one-level preconditioners. \verb|desc_a| & \verb|type(psb_desc_type), intent(in)|. \\ & The communication descriptor of \verb|a|. See the PSBLAS User's Guide for details \cite{PSBLASGUIDE}.\\ -%\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\ -% & The preconditioner data structure. Note that \emph{x} must be chosen according -% to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|info| & \verb|integer, intent(out)|.\\ & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ \verb|amold| & \verb|class(psb_|\emph{x}\verb|_base_sparse_mat), intent(in), optional|. \\ @@ -787,7 +684,7 @@ For compatibility with the previous versions of MLD2P4, this method can be also as follows: \begin{center} -\verb|call mld_precbld(p,what,val,info[,amold,vmold,imold])| +\verb|call amg_precbld(p,what,val,info[,amold,vmold,imold])| \end{center} \noindent @@ -813,12 +710,8 @@ and hence it is completely transparent to the user. {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip \begin{tabular}{p{1.2cm}p{12cm}} -%\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\ -% & The preconditioner data structure, containing the local part of $M$. -% Note that \emph{x} must be chosen according -% to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|x| & \emph{type}\verb|(|\emph{kind\_parameter}\verb|), dimension(:), intent(in)|.\\ - & The local part of the vector $x$. Note that \emph{type} and + & The local part of the vector $x$. Note that \emph{type} and \emph{kind\_parameter} must be chosen according to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|y| & \emph{type}\verb|(|\emph{kind\_parameter}\verb|), dimension(:), intent(out)|.\\ @@ -847,7 +740,7 @@ For compatibility with the previous versions of MLD2P4, this method can be also as follows: \begin{center} -\verb|call mld_precaply(p,what,val,info)| +\verb|call amg_precaply(p,what,val,info)| \end{center} \clearpage @@ -864,9 +757,6 @@ This method deallocates the preconditioner data structure \verb|p|. {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip \begin{tabular}{p{1.2cm}p{10.5cm}} -%\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\ -% & The preconditioner data structure. Note that \emph{x} must be chosen according - % to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|info| & \verb|integer, intent(out)|.\\ & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ \end{tabular} @@ -876,7 +766,7 @@ For compatibility with the previous versions of MLD2P4, this method can be also as follows: \begin{center} -\verb|call mld_precfree(p,info)| +\verb|call amg_precfree(p,info)| \end{center} @@ -896,9 +786,6 @@ or \verb|build|, have been called. {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip \begin{tabular}{p{1.2cm}p{12cm}} -%\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(in)|.\\ -% & The preconditioner data structure. Note that \emph{x} must be chosen according -% to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|info| & \verb|integer, intent(out)|.\\ & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ \verb|iout| & \verb|integer, intent(in), optional|.\\ @@ -911,16 +798,16 @@ For compatibility with the previous versions of MLD2P4, this method can be also as follows: \begin{center} -\verb|call mld_precdescr(p,info [,iout])| +\verb|call amg_precdescr(p,info [,iout])| \end{center} \subsection{Auxiliary Methods\label{sec:auxil}} Various functionalities are implemented as additional methods of the -preconditioner object. +preconditioner object. \subsubsection{Method: dump} - + \begin{center} \verb|call p%dump(info[,istart,iend,prefix,head,ac,rp,smoother,solver,global_num])|\\ \end{center} @@ -931,9 +818,6 @@ Dump on file. {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip \begin{tabular}{p{1.2cm}p{12cm}} -%\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\ -% & The preconditioner data structure. Note that \emph{x} must be chosen according -% to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|info| & \verb|integer, intent(out)|.\\ & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ \verb|amold| & \verb|class(psb_|\emph{x}\verb|_base_sparse_mat), intent(in), optional|. \\ @@ -945,20 +829,20 @@ Dump on file. \subsubsection{Method: clone} - + \begin{center} \verb|call p%clone(pout,info)|\\ \end{center} \noindent -Create a (deep) copy of the preconditioner object. +Create a (deep) copy of the preconditioner object. {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip \begin{tabular}{p{1.2cm}p{12cm}} -\verb|pout| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(out)|.\\ +\verb|pout| & \verb|type(amg_|\emph{x}\verb|prec_type), intent(out)|.\\ & The copy of the preconditioner data structure. Note - that \emph{x} must be chosen according + that \emph{x} must be chosen according to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|info| & \verb|integer, intent(out)|.\\ & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ @@ -967,26 +851,16 @@ Create a (deep) copy of the preconditioner object. \subsubsection{Method: sizeof} - + \begin{center} \verb|sz = p%sizeof()|\\ \end{center} \noindent -Return memory footprint in bytes. - -% {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip - -% \begin{tabular}{p{1.2cm}p{12cm}} -% %\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\ -% % & The preconditioner data structure. Note that \emph{x} must be chosen according -% % to the real/complex, single/double precision version of MLD2P4 under use.\\ -% \verb|info| & \verb|integer, intent(out)|.\\ -% & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ -% \end{tabular} +Return memory footprint in bytes. \subsubsection{Method: allocate\_wrk} - + \begin{center} \verb|call p%allocate_wrk(info[, vmold])|\\ \end{center} @@ -996,23 +870,20 @@ Allocate internal work vectors. Each application of the preconditioner uses a number of work vectors which are allocated internally as necessary; therefore allocation and deallocation of memory occurs multiple times during the execution of a Krylov method. In most cases -this strategy is perfectly acceptable, but +this strategy is perfectly acceptable, but on some platforms, most notably GPUs, memory allocation is a slow operation, and the default behaviour would lead to a -slowdown. This method allows to trade space for time by preallocating +slowdown. This method allows to trade space for time by preallocating the internal workspace outside of the invocation of a Krylov method. When using GPUs or other specialized devices, the \verb|vmold| argument is also necessary to ensure the internal work vectors are of the appropriate dynamic type to exploit the accelerator hardware; when allocation occurs internally this is taken care of based on the dynamic -type of the \verb|x| argument to the \verb|apply| method. +type of the \verb|x| argument to the \verb|apply| method. {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip \begin{tabular}{p{1.2cm}p{12cm}} -%\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\ -% & The preconditioner data structure. Note that \emph{x} must be chosen according -% to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|info| & \verb|integer, intent(out)|.\\ & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ \verb|vmold| & \verb|class(psb_|\emph{x}\verb|_base_vect_type), intent(in), optional|. \\ @@ -1023,20 +894,17 @@ type of the \verb|x| argument to the \verb|apply| method. \subsubsection{Method: free\_wrk} - + \begin{center} \verb|call p%free_wrk(info)|\\ \end{center} \noindent -Deallocate internal work vectors. +Deallocate internal work vectors. {\vskip1.5\baselineskip\noindent\large\bfseries Arguments} \smallskip \begin{tabular}{p{1.2cm}p{12cm}} -%\verb|p| & \verb|type(mld_|\emph{x}\verb|prec_type), intent(inout)|.\\ -% & The preconditioner data structure. Note that \emph{x} must be chosen according -% to the real/complex, single/double precision version of MLD2P4 under use.\\ \verb|info| & \verb|integer, intent(out)|.\\ & Error code. If no error, 0 is returned. See Section~\ref{sec:errors} for details.\\ \end{tabular} @@ -1045,7 +913,7 @@ Deallocate internal work vectors. -%%% Local Variables: +%%% Local Variables: %%% mode: latex %%% TeX-master: "userguide" -%%% End: +%%% End: