From 38df8fac573f0883493a103dcad8eb7c413ff532 Mon Sep 17 00:00:00 2001 From: Salvatore Filippone Date: Mon, 17 Jul 2017 15:39:18 +0100 Subject: [PATCH] New docs, merged comments from Daniela, updated PDF. --- docs/html/img10.png | Bin 412 -> 408 bytes docs/html/img100.png | Bin 176 -> 0 bytes docs/html/img101.png | Bin 610 -> 0 bytes docs/html/img102.png | Bin 252 -> 0 bytes docs/html/img103.png | Bin 193 -> 0 bytes docs/html/img104.png | Bin 211 -> 0 bytes docs/html/img105.png | Bin 583 -> 0 bytes docs/html/img106.png | Bin 652 -> 0 bytes docs/html/img107.png | Bin 337 -> 0 bytes docs/html/img108.png | Bin 660 -> 0 bytes docs/html/img12.png | Bin 243 -> 242 bytes docs/html/img13.png | Bin 780 -> 766 bytes docs/html/img15.png | Bin 209 -> 211 bytes docs/html/img16.png | Bin 399 -> 373 bytes docs/html/img25.png | Bin 201 -> 206 bytes docs/html/img26.png | Bin 318 -> 270 bytes docs/html/img27.png | Bin 223 -> 255 bytes docs/html/img28.png | Bin 675 -> 176 bytes docs/html/img29.png | Bin 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  • Adding new smoothers and solvers to MLD2P4 + HREF="node24.html">Error Handling
  • Error Handling + HREF="node25.html">License
  • License + HREF="node26.html">Bibliography
  • Bibliography -
  • About this document ... + HREF="node27.html">About this document ...
    diff --git a/docs/html/node1.html b/docs/html/node1.html index cf182508..a09374a8 100644 --- a/docs/html/node1.html +++ b/docs/html/node1.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Contents - Up: Up: userhtml - Previous: Previous: userhtml -   Contents

    @@ -90,26 +90,26 @@ the user interface of MLD2P4.
    - next - up - previous - contents
    - Next: Next: Contents - Up: Up: userhtml - Previous: Previous: userhtml -   Contents diff --git a/docs/html/node10.html b/docs/html/node10.html index a64fde3f..5a1c9cf7 100644 --- a/docs/html/node10.html +++ b/docs/html/node10.html @@ -25,26 +25,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Multigrid Background - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Bug reporting -   Contents

    diff --git a/docs/html/node11.html b/docs/html/node11.html index 0b2a7787..1c0c92ad 100644 --- a/docs/html/node11.html +++ b/docs/html/node11.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: AMG preconditioners - Up: Up: userhtml - Previous: Previous: Example and test programs -   Contents

    @@ -55,46 +55,46 @@ original version by: Nikos Drakos, CBLU, University of Leeds
    Multigrid Background Multigrid preconditioners, coupled with Krylov iterative solvers, are widely used in the parallel solution of large and sparse linear systems, because of their optimality in the solution of linear systems arising from the discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids. Optimality, also known as algorithmic scalability, is the property of having a computational cost per iteration that depends linearly 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 consisting of smoother iterations and a coarse-space (or coarse-level) correction. The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones, or more complex subspace-correction methods, such as the Schwarz ones. The coarse-space correction consists of solving, in an appropriately chosen coarse space, the residual equation associated with the approximate solution computed by the smoother, and of using the solution of this equation to correct the previous approximation. The transfer of information between the original (fine) space and the coarse one is performed by using suitable restriction and prolongation operators. The construction of the coarse space and the corresponding transfer operators is carried out by applying a so-called coarsening algorithm to the system matrix. Two main approaches can be used to perform coarsening: the geometric approach, which exploits the knowledge of some physical grid associated with the matrix and requires the user to define transfer operators from the fine to the coarse level and vice versa, and the algebraic approach, which builds the coarse-space correction and the associate transfer operators using only matrix information. The first approach may be difficult when the system comes from discretizations on complex geometries; furthermore, ad hoc one-level smoothers may be required to get an efficient interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients are considered. The second approach performs a fully automatic coarsening and enforces the interplay between fine and coarse level by suitably choosing the coarse space and the coarse-to-fine interpolation (see, e.g., [2,27,25] for details.) MLD2P4 uses a pure algebraic approach, based on the smoothed aggregation algorithm [1,29], for building the sequence of coarse matrices and transfer operators, starting from the original one. A decoupled version of this algorithm is implemented, where the smoothed aggregation is applied locally to each submatrix [28]. A brief description of the AMG preconditioners implemented in MLD2P4 is given in Sections 4.1-4.3. For further details the reader is referred to [3,4,5,9]. We note that optimal multigrid preconditioners do not necessarily correspond to minimum execution times in a parallel setting. Indeed, to obtain effective parallel multigrid preconditioners, a tradeoff between the optimality and the cost of building and applying the smoothers and the coarse-space corrections must be achieved. Effective parallel preconditioners require algorithmic scalability to be coupled with implementation scalability, i.e., a computational cost per iteration which remains (almost) constant as the number of parallel processors increases.
    + HREF="node26.html#Briggs2000">3,23,21] for details.) MLD2P4 uses a pure algebraic approach, based on the smoothed aggregation algorithm [2,25], for building the sequence of coarse matrices and transfer operators, starting from the original one. A decoupled version of this algorithm is implemented, where the smoothed aggregation is applied locally to each submatrix [24]. A brief description of the AMG preconditioners implemented in MLD2P4 is given in Sections 4.1-4.3. For further details the reader is referred to [4,5,7,8]. We note that optimal multigrid preconditioners do not necessarily correspond to minimum execution times in a parallel setting. Indeed, to obtain effective parallel multigrid preconditioners, a tradeoff between the optimality and the cost of building and applying the smoothers and the coarse-space corrections must be achieved. Effective parallel preconditioners require algorithmic scalability to be coupled with implementation scalability, i.e., a computational cost per iteration which remains (almost) constant as the number of parallel processors increases.
    Subsections
    - next - up - previous - contents
    - Next: Next: AMG preconditioners - Up: Up: userhtml - Previous: Previous: Example and test programs -   Contents diff --git a/docs/html/node12.html b/docs/html/node12.html index c284507f..13e20b5e 100644 --- a/docs/html/node12.html +++ b/docs/html/node12.html @@ -25,26 +25,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Getting Started - Up: Up: Multigrid Background - Previous: Previous: Multigrid Background -   Contents

    @@ -115,7 +115,7 @@ by using the information contained in $\mathbb{R}^{n_{k}}$ is associated with $\Omega^k$. For all $k < nlev$, a restriction operator and a prolongation one are built, which connect two levels \begin{displaymath} P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad R^k \in \mathbb{R}^{n_{k+1}\times n_k}; \end{displaymath} diff --git a/docs/html/node13.html b/docs/html/node13.html index 1229c8ec..b7109d40 100644 --- a/docs/html/node13.html +++ b/docs/html/node13.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
    - Next: Next: Examples - Up: Up: userhtml - Previous: Previous: AMG preconditioners -   Contents

    @@ -59,7 +59,7 @@ Getting Started

    We describe the basics for building and applying MLD2P4 one-level and multi-level (i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS [17]. + HREF="node26.html#PSBLASGUIDE">13]. The following steps are required:

      @@ -122,7 +122,7 @@ Examples showing the basic use of MLD2P4 are reported in Section  +
      + the corresponding entry of the preconditioner is set to 1. @@ -192,32 +192,32 @@ on parallel computers. Subsections
      - next - up - previous - contents
      - Next: Next: Examples - Up: Up: userhtml - Previous: Previous: AMG preconditioners -   Contents diff --git a/docs/html/node14.html b/docs/html/node14.html index 6ecc3856..3602b1ff 100644 --- a/docs/html/node14.html +++ b/docs/html/node14.html @@ -25,26 +25,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: User Interface - Up: Up: Getting Started - Previous: Previous: Getting Started -   Contents

      @@ -78,7 +78,7 @@ Section 3.5). A sample test probl input data is available in examples/fileread/runs. For details on the use of the PSBLAS routines, see the PSBLAS User's Guide [17]. + HREF="node26.html#PSBLASGUIDE">13].

      The setup and application of the default multi-level preconditioner @@ -89,7 +89,7 @@ the corresponding codes are available in examples/fileread/.

      -

      +
      Table 1: Preconditioner types, corresponding strings and default choices. @@ -143,7 +143,7 @@ Preconditioner types, corresponding strings and default choices.
      'DIAG' or 'JACOBI' Diagonal preconditioner. For any zero diagonal entry of the matrix to be preconditioned, - the corresponding entry of he preconditioner is set to 1.
      Block Jacobi 'BJAC'
      - @@ -98,7 +97,7 @@ contained in val. - @@ -241,7 +240,7 @@ solver is changed to the default sequential solver.


      -
      +
      Figure 2: setup and application of the default multi-level preconditioner (example 1). @@ -127,11 +127,11 @@ setup and application of the default multi-level preconditioner (example 1). ! with basic smoothed aggregation, 1 hybrid forward/backward ! GS sweep as pre/post-smoother and UMFPACK as coarsest-level ! solver - call P%init(P,'ML',info) + call P%init('ML',info) ! ! build the preconditioner - call P%hierarchy_build(A,desc_A,P,info) - call P%smoothers_build(A,desc_A,P,info) + call P%hierarchy_build(A,desc_A,info) + call P%smoothers_build(A,desc_A,info) ! ! set the solver parameters and the initial guess @@ -142,7 +142,7 @@ setup and application of the default multi-level preconditioner (example 1). ... ... ! ! deallocate the preconditioner - call P%free(P,info) + call P%free(info) ! ! deallocate other data structures ... ... @@ -198,7 +198,7 @@ boundary conditions are also available in the directory examples/pdegen -
      +
      @@ -211,12 +211,12 @@ setup of a multi-level preconditioner ! build a V-cycle preconditioner with 1 block-Jacobi sweep (with ! ILU(0) on the blocks) as pre- and post-smoother, and 8 block-Jacobi ! sweeps (with ILU(0) on the blocks) as coarsest-level solver - call P%init(P,'ML',info) - call_P%set(P,'SMOOTHER_TYPE','BJAC',info) - call P%set(P,'COARSE_SOLVE','BJAC',info) - call P%set(P,'COARSE_SWEEPS',8,info) - call P%hierarchy_build(A,desc_A,P,info) - call P%smoothers_build(A,desc_A,P,info) + call P%init('ML',info) + call_P%set('SMOOTHER_TYPE','BJAC',info) + call P%set('COARSE_SOLVE','BJAC',info) + call P%set('COARSE_SWEEPS',8,info) + call P%hierarchy_build(A,desc_A,info) + call P%smoothers_build(A,desc_A,info) ... ... @@ -231,7 +231,7 @@ setup of a multi-level preconditioner

      -

      +
      Figure 3: setup of a multi-level preconditioner
      @@ -244,17 +244,17 @@ setup of a multi-level preconditioner ! build a W-cycle preconditioner with 2 Gauss-Seidel sweeps as ! post-smoother (and no pre-smoother), a distributed coarsest ! matrix, and MUMPS as coarsest-level solver - call P%init(P,'ML',info) + call P%init('ML',info) call P%set('ML_TYPE','WCYCLE',info) call P%set('SMOOTHER_TYPE','GS',info) call P%set('SMOOTHER_SWEEPS',0,info,pos='PRE') call P%set('SMOOTHER_SWEEPS',2,info,pos='POST') call P%set('COARSE_SOLVE','MUMPS',info) call P%set('COARSE_MAT','DIST',info) - call P%hierarchy_build(A,desc_A,P,info) - call P%smoothers_build(A,desc_A,P,info) + call P%hierarchy_build(A,desc_A,info) + call P%smoothers_build(A,desc_A,info) ... ... -! solve Ax=b with preconditioned CG +! solve Ax=b with preconditioned BiCGSTAB call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info) @@ -267,7 +267,7 @@ setup of a multi-level preconditioner

      -

      +
      Figure 4: setup of a multi-level preconditioner
      @@ -278,9 +278,9 @@ setup of a one-level Schwarz preconditioner.
       ... ...
 ! set RAS with overlap 2 and ILU(0) on the local blocks
-  call P%init(P,'AS',info)
-  call P%set(P,'SUB_OVR',2,info)
-  call P%bld(A,desc_A,P,info)
+  call P%init('AS',info)
+  call P%set('SUB_OVR',2,info)
+  call P%bld(A,desc_A,info)
 ... ...
      @@ -294,26 +294,26 @@ setup of a one-level Schwarz preconditioner.

      - next - up - previous - contents
      - Next: Next: User Interface - Up: Up: Getting Started - Previous: Previous: Getting Started -   Contents diff --git a/docs/html/node15.html b/docs/html/node15.html index f56a15b5..b7be0de3 100644 --- a/docs/html/node15.html +++ b/docs/html/node15.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Subroutine init - Up: Up: userhtml - Previous: Previous: Examples -   Contents

      @@ -89,13 +89,13 @@ i.e., WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="img14.png" ALT="$v$"> and $w$ involved in the preconditioner application $w=M^{-1}v$ must be of type + ALT="$w=B^{-1}v$"> must be of type psb_xvect_type with x = s, d, c, z, in a manner completely analogous to the sparse matrix type; @@ -115,46 +115,46 @@ A description of each routine is given in the remainder of this section. Subsections
      - next - up - previous - contents
      - Next: Next: Subroutine init - Up: Up: userhtml - Previous: Previous: Examples -   Contents diff --git a/docs/html/node16.html b/docs/html/node16.html index cd013670..2de4734e 100644 --- a/docs/html/node16.html +++ b/docs/html/node16.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Subroutine set - Up: Up: User Interface - Previous: Previous: User Interface -   Contents

      @@ -86,7 +86,7 @@ This routine allocates and initializes the preconditioner - +
      Figure 5: setup of a one-level Schwarz preconditioner.
      integer, intent(out).
       Error code. If no error, 0 is returned. See Section 8 for details.Error code. If no error, 0 is returned. See Section 7 for details.
      diff --git a/docs/html/node17.html b/docs/html/node17.html index a06b7c1d..dcd5e947 100644 --- a/docs/html/node17.html +++ b/docs/html/node17.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Subroutine build - Up: Up: User Interface - Previous: Previous: Subroutine init -   Contents

      @@ -77,9 +77,8 @@ contained in val.
      character(len=*).
       The parameter to be set. It can be specified by - a predefined constant, or through its name; the string - is case-insensitive. See also +The parameter to be set. It can be specified through its name; + the string is case-insensitive. See Tables 2-8.
      val integer, intent(out).
       Error code. If no error, 0 is returned. See Section 8 +Error code. If no error, 0 is returned. See Section 7 for details.
      ilev
      - + - + @@ -105,8 +105,8 @@ and hence it is completely transparent to the user. @@ -122,40 +122,40 @@ and hence it is completely transparent to the user. - + + WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" + SRC="img36.png" + ALT="$B^{-1})$">. @@ -183,26 +183,26 @@ as follows:

      - next - up - previous - contents
      - Next: Next: Subroutine free - Up: Up: User Interface - Previous: Previous: Subroutine smoothers_build -   Contents diff --git a/docs/html/node22.html b/docs/html/node22.html index ff93d7a4..0d55daa6 100644 --- a/docs/html/node22.html +++ b/docs/html/node22.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Subroutine descr - Up: Up: User Interface - Previous: Previous: Subroutine apply -   Contents

      @@ -75,7 +75,7 @@ This routine deallocates the preconditioner data structure p. - +
      Table 2: Parameters defining the multi-level cycle and the number of cycles to @@ -256,7 +255,7 @@ be applied.
      DEFAULT COMMENTS
      ML_CYCLE
      'ML_CYCLE' character(len=*) 'VCYCLE'

      @@ -274,7 +273,7 @@ be applied. Note that hybrid Multiplicative Schwarz is equivalent to V-cycle and is included for compatibility with previous versions of MLD2P4.
      OUTER_SWEEPS
      'OUTER_SWEEPS' integer Any integer

      @@ -294,7 +293,7 @@ number

      -
      +
      - + - + - + - + @@ -374,16 +373,16 @@ Currently, only the SRC="img22.png" ALT="$A+A^T$">. - + + HREF="node26.html#VANEK_MANDEL_BREZINA">25]. - + @@ -409,7 +408,7 @@ of levels.


      -
      +
      Table 3: Parameters defining the aggregation algorithm. @@ -308,7 +307,7 @@ Parameters defining the aggregation algorithm.
      DEFAULT COMMENTS
      MIN_COARSE_SIZE
      'MIN_COARSE_SIZE' integer Any number

      @@ -333,7 +332,7 @@ Parameters defining the aggregation algorithm. is lower than or equal to this threshold (see Note).
      MIN_CR_RATIO
      'MIN_CR_RATIO' real Any number

      @@ -347,7 +346,7 @@ Parameters defining the aggregation algorithm. at two consecutive levels is lower than or equal to this threshold (see Note).
      MAX_LEVS
      'MAX_LEVS' integer Any integer

      @@ -359,7 +358,7 @@ number Maximum number of levels. The aggregation stops if the number of levels reaches this value (see Note).
      PAR_AGGR
      'PAR_AGGR' character(len=*) 'DEC', 'SYMDEC' 'DEC'
      AGGR_TYPE
      'AGGR_TYPE' character(len=*) 'VMB' 'VMB' Type of aggregation algorithm: currently, the scalar aggregation algorithm by Vanek, Mandel and Brezina is implemented [29].
      AGGR_PROL
      'AGGR_PROL' character(len=*) 'SMOOTHED', 'UNSMOOTHED' 'SMOOTHED'
      - + - + - - - - - - - - - - - - - - - - - - + ALT="$\theta$"> in the aggregation algorithm, + see (3) in Section 4.2. + See also the note at the bottom of this table. - + + prolongator: filtered or unfiltered (see (4) in Section 4.2).


      -
      +
      Table 4: Parameters defining the aggregation algorithm (continued). @@ -423,7 +422,7 @@ Parameters defining the aggregation algorithm (continued).
      DEFAULT COMMENTS
      AGGR_ORD
      'AGGR_ORD' character(len=*) 'NATURAL'

      @@ -434,7 +433,7 @@ Parameters defining the aggregation algorithm (continued). descending degrees of the nodes in the matrix graph.
      AGGR_THRESH
      'AGGR_THRESH' real(kind_parameter) Any real

      @@ -446,93 +445,22 @@ number The threshold $\theta$ in the aggregation algorithm - (see Note).

      AGGR_OMEGA_ALGcharacter(len=*)'EIG_EST' -

      -'USER_CHOICE'

      		'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.
      AGGR_EIGcharacter(len=*)'A_NORMI''A_NORMI'How to estimate the spectral radius of $D^{-1}A$. - Currently only the infinity norm estimate - is available.
      AGGR_OMEGA_VALreal(kind_parameter)Any real -

      -number $> 0$

      		 -$4/(3\rho(D^{-1}A))$Damping parameter $\omega$ in the smoothed aggregation algorithm. - It must be set by the user if - USER_CHOICE was specified for - mld_aggr_omega_alg_, - otherwise it is computed by the library, using the - selected estimate of the spectral radius $\rho(D^{-1}A)$ of - $D^{-1}A$.
      AGGR_FILTER
      'AGGR_FILTER' character(len=*) 'FILTER'

      'NOFILTER'

      		 'NOFILTER' Matrix used in computing the smoothed - prolongator: filtered or unfiltered.
      Note. Different thresholds at different levels, such as those used in [29, Section 5.1], can be easily set by + HREF="node26.html#VANEK_MANDEL_BREZINA">25, Section 5.1], can be easily set by invoking the rou-
      tine set with @@ -547,7 +475,7 @@ the parameter ilev.
      - + - + - + + HREF="node26.html#PSBLASGUIDE">13]. @@ -91,13 +91,13 @@ hierarchy produced by a previous call to hierarchy_build + HREF="node26.html#PSBLASGUIDE">13]. - +
      Table 5: Parameters defining the coarse-space correction at the coarsest @@ -561,7 +489,7 @@ level.
      DEFAULT COMMENTS
      COARSE_MAT
      'COARSE_MAT' character(len=*) 'DIST'

      @@ -570,7 +498,7 @@ level.

      		Coarsest matrix layout: distributed among the processes or replicated on each of them.
      COARSE_SOLVE
      'COARSE_SOLVE' character(len=*) 'MUMPS'

      @@ -604,7 +532,7 @@ Note that UMF and SLU require the coarsest value UMFPACK and SuperLU_Dist are available only in double precision.
      COARSE_SUBSOLVE
      'COARSE_SUBSOLVE' character(len=*) 'ILU'

      @@ -622,14 +550,14 @@ Note that UMF and SLU require the coarsest 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 (plus triangular solve). Note that UMFPACK and SuperLU_Dist @@ -655,7 +583,7 @@ Note that UMF and SLU require the coarsest


      -
      +
      - + - + - +
      Table 6: Parameters defining the coarse-space correction at the coarsest @@ -669,7 +597,7 @@ level (continued).
      DEFAULT COMMENTS
      COARSE_SWEEPS
      'COARSE_SWEEPS' integer Any integer

      @@ -681,35 +609,35 @@ number Number of sweeps when JACOBI, GS or BJAC is chosen as coarsest-level solver.
      COARSE_FILLIN
      'COARSE_FILLIN' integer Any integer

      number $\ge 0$

      		 0 Fill-in level $p$ of the ILU factorizations.
      COARSE_ILUTHRS
      'COARSE_ILUTHRS' real(kind_parameter) Any real

      number $\ge 0$

      		 0 Drop tolerance $t$ in the ILU($p,t$) factorization.
      @@ -721,7 +649,7 @@ number

      -
      +
      - + - + - + - + + HREF="node26.html#PSBLASGUIDE">13]. @@ -90,13 +90,13 @@ single/double precision version of MLD2P4 under use. + HREF="node26.html#PSBLASGUIDE">13]. - +
      Table 7: Parameters defining the smoother or the details of the one-level preconditioner. @@ -737,7 +665,7 @@ Parameters defining the smoother or the details of the one-level preconditioner.
      COMMENTS
      SMOOTHER_TYPE
      'SMOOTHER_TYPE' character(len=*) 'JACOBI' @@ -760,7 +688,7 @@ Parameters defining the smoother or the details of the one-level preconditioner.

      It is ignored by one-level preconditioners.

      SUB_SOLVE
      'SUB_SOLVE' character(len=*) 'JACOBI' @@ -792,26 +720,26 @@ Parameters defining the smoother or the details of the one-level preconditioner. hybrid (forward) Gauss-Seidel, hybrid backward Gauss-Seidel, ILU($p$), ILU($p,t$), MILU($p$), LU from MUMPS, SuperLU or UMFPACK (plus triangular solve). See Note for details on hybrid Gauss-Seidel.
      SMOOTHER_SWEEPS
      'SMOOTHER_SWEEPS' integer Any integer

      number $\ge 0$

      		1 @@ -822,14 +750,14 @@ Parameters defining the smoother or the details of the one-level preconditioner. together with pos='PRE' or pos='POST, respectively.
      SUB_OVR
      'SUB_OVR' integer Any integer

      number $\ge 0$

      		1 @@ -844,7 +772,7 @@ Parameters defining the smoother or the details of the one-level preconditioner.


      -
      +
      - + - + - + - + @@ -944,26 +872,26 @@ Parameters defining the smoother or the details of the one-level preconditioner

      - next - up - previous - contents
      - Next: Next: Subroutine build - Up: Up: User Interface - Previous: Previous: Subroutine init -   Contents diff --git a/docs/html/node18.html b/docs/html/node18.html index 6e9e191a..48134560 100644 --- a/docs/html/node18.html +++ b/docs/html/node18.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Subroutine hierarchy_build - Up: Up: User Interface - Previous: Previous: Subroutine set -   Contents

      @@ -81,7 +81,7 @@ made by the user through the routines init and set matrix to be preconditioned. Note that x must be chosen according to the real/complex, single/double precision version of MLD2P4 under use. See the PSBLAS User's Guide for details [17]. + HREF="node26.html#PSBLASGUIDE">13]. @@ -89,13 +89,13 @@ made by the user through the routines init and set + HREF="node26.html#PSBLASGUIDE">13]. - +
      Table 8: Parameters defining the smoother or the details of the one-level preconditioner @@ -860,7 +788,7 @@ Parameters defining the smoother or the details of the one-level preconditioner
      COMMENTS
      SUB_RESTR
      'SUB_RESTR' character(len=*) 'HALO' @@ -876,7 +804,7 @@ Parameters defining the smoother or the details of the one-level preconditioner Note that HALO must be chosen for the classical Addditive Schwarz smoother and its RAS variant.
      SUB_PROL
      'SUB_PROL' character(len=*) 'SUM' @@ -892,39 +820,39 @@ Parameters defining the smoother or the details of the one-level preconditioner Note that SUM must be chosen for the classical Additive Schwarz smoother, and NONE for its RAS variant.
      SUB_FILLIN
      'SUB_FILLIN' integer Any integer

      number $\ge 0$

      		0 Fill-in level $p$ of the incomplete LU factorizations.
      SUB_ILUTHRS
      'SUB_ILUTHRS' real(kind_parameter) Any real number $\ge 0$ 0 Drop tolerance $t$ in the ILU($p,t$) factorization.
      desc_a type(psb_desc_type), intent(in).
        The communication descriptor of a. See the PSBLAS User's Guide for details [17].
      info integer, intent(out).
       Error code. If no error, 0 is returned. See Section 8 for details.Error code. If no error, 0 is returned. See Section 7 for details.
      @@ -117,26 +117,26 @@ In this case, the routine can be used to build multi-level preconditioners too.

      - next - up - previous - contents
      - Next: Next: Subroutine hierarchy_build - Up: Up: User Interface - Previous: Previous: Subroutine set -   Contents diff --git a/docs/html/node19.html b/docs/html/node19.html index 35bd97df..ff0a8458 100644 --- a/docs/html/node19.html +++ b/docs/html/node19.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Subroutine smoothers_build - Up: Up: User Interface - Previous: Previous: Subroutine build -   Contents

      @@ -82,7 +82,7 @@ made by the user through the routines init and set. to the real/complex, single/double precision version of MLD2P4 under use. See the PSBLAS User's Guide for details [17].
      desc_a type(psb_desc_type), intent(in).
        The communication descriptor of a. See the PSBLAS User's Guide for details [17].
      info integer, intent(out).
       Error code. If no error, 0 is returned. See Section 8 for details.Error code. If no error, 0 is returned. See Section 7 for details.
      diff --git a/docs/html/node2.html b/docs/html/node2.html index 581ee848..a9ba2cc0 100644 --- a/docs/html/node2.html +++ b/docs/html/node2.html @@ -26,21 +26,21 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous
      - Next: Next: General Overview - Up: Up: userhtml - Previous: Previous: Abstract

      @@ -53,68 +53,66 @@ Contents diff --git a/docs/html/node20.html b/docs/html/node20.html index 2c381342..6d41f54f 100644 --- a/docs/html/node20.html +++ b/docs/html/node20.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Subroutine apply - Up: Up: User Interface - Previous: Previous: Subroutine hierarchy_build -   Contents

      @@ -83,7 +83,7 @@ hierarchy produced by a previous call to hierarchy_build matrix to be preconditioned. Note that x must be chosen according to the real/complex, single/double precision version of MLD2P4 under use. See the PSBLAS User's Guide for details [17].
      desc_a type(psb_desc_type), intent(in).
        The communication descriptor of a. See the PSBLAS User's Guide for details [17].
      info integer, intent(out).
       Error code. If no error, 0 is returned. See Section 8 for details.Error code. If no error, 0 is returned. See Section 7 for details.
      diff --git a/docs/html/node21.html b/docs/html/node21.html index 65974c7c..b3ccbba6 100644 --- a/docs/html/node21.html +++ b/docs/html/node21.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Subroutine free - Up: Up: User Interface - Previous: Previous: Subroutine smoothers_build -   Contents

      @@ -64,18 +64,18 @@ Subroutine apply

      This routine computes $y = op(M^{-1}) x$, where $M$ is a previously built + WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" + SRC="img29.png" + ALT="$y = op(B^{-1}) x$">, where $B$ is a previously built preconditioner, stored into p, and $op$ denotes the preconditioner itself or its transpose, according to the value of trans. @@ -94,8 +94,8 @@ and hence it is completely transparent to the user.

        The local part of the vector $x$. Note that type and kind_parameter must be chosen according to the real/complex, single/double precision version of MLD2P4 under use.
        The local part of the vector $y$. Note that type and kind_parameter must be chosen according to the real/complex, single/double precision version of MLD2P4 under use.integer, intent(out).
       Error code. If no error, 0 is returned. See Section 8 for details.Error code. If no error, 0 is returned. See Section 7 for details.
      trans character(len=1), optional, intent(in).
        If trans = 'N','n' then $op(M^{-1}) = M^{-1}$; + WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" + SRC="img34.png" + ALT="$op(B^{-1}) = B^{-1}$">; if trans = 'T','t' then $op(M^{-1}) = M^{-T}$ + WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0" + SRC="img35.png" + ALT="$op(B^{-1}) = B^{-T}$"> (transpose of $M^{-1})$; if trans = 'C','c' then $op(M^{-1}) = M^{-C}$ + WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0" + SRC="img37.png" + ALT="$op(B^{-1}) = B^{-C}$"> (conjugate transpose of $M^{-1})$.
      work type(kind_parameter), dimension(:), optional, target. integer, intent(out).
       Error code. If no error, 0 is returned. See Section 8 for details.Error code. If no error, 0 is returned. See Section 7 for details.
      diff --git a/docs/html/node23.html b/docs/html/node23.html index 813d9ea4..9e89a6d8 100644 --- a/docs/html/node23.html +++ b/docs/html/node23.html @@ -25,26 +25,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Adding new smoothers and - Up: Next: Error Handling + Up: User Interface - Previous: Previous: Subroutine free -   Contents

      @@ -76,7 +76,7 @@ or build, have been called. integer, intent(out).   -Error code. If no error, 0 is returned. See Section 8 for details. +Error code. If no error, 0 is returned. See Section 7 for details. iout integer, intent(in), optional. @@ -99,7 +99,6 @@ as follows:

      -


      diff --git a/docs/html/node24.html b/docs/html/node24.html index 4b77d43e..1a52a490 100644 --- a/docs/html/node24.html +++ b/docs/html/node24.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Adding new smoothers and solvers to MLD2P4 - +Error Handling + @@ -26,137 +26,53 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Error Handling - Up: Next: License + Up: userhtml - Previous: Previous: Subroutine descr -   Contents

      -

      +


      -Adding new smoothers and solvers to MLD2P4 +Error Handling

      -Developers can add completely new smoother and/or solver classes -derived from the base objects in the library may be used without -recompiling the library itself. - -

      -To do so it is necessary first to select the base type to be extended; -in our experience, it is quite likely that the new application needs -only require the definition of a ``solver'' object, which is almost -always acting only on the local part of the distributed matrix. - -

      -The parallel actions required to connect the various solver objects -are most often already provided by the Block Jacobi or the Additive -Schwarz smoothers. To define a new solver, the developer will then -have to define its components and methods, 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 the multilevel preconditioners it is necessary to: - -

        -
      • Declare in the application program a variable of the new type; -
        • -
      • Pass that variable as the argument to the se routine as in the - following: -
        -call p%set(smoother,info [,ilev, ilmax,pos]) -
        call p%set(solver,info [,ilev, ilmax,pos]) - -
        -
        • -
      • Link into the application executable the code implementing the - various methods. -
        • -
      -The new solver object is then dynamically included in the -preconditioner structure, and will act as a mold to which the -preconditioner will conform, even though the MLD2P4 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 routines; if the library code does not recognize a keyword, -it passes it down the composition hierarchy (levels containing -smoothers containing solvers), so that it can be eventually caught by -the new solver. +The error handling in MLD2P4 is based on the PSBLAS (version 2) error +handling. Error conditions are signaled via an integer argument +info; whenever an error condition is detected, an error trace +stack is built by the library up to the top-level, user-callable +routine. This routine will then decide, according to the user +preferences, whether the error should be handled by terminating the +program or by returning the error condition to the user code, which +will then take action, and whether +an error message 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].

      -An example is contained in the source code distribution under the -folder tests/newslv. This example solver is simply the ILU(0) -solver under a new name, but it should give an idea of what needs to -be done. - -

      - -
      -

      - - - - - - - - - - - - - -
      smootherclass(mld_x_base_smoother_type)
       The user-defined new smoother to be employed in the - preconditioner.
      solverclass(mld_x_base_solver_type)
       The user-defined new solver to be employed in the - preconditioner.
      -

      -

      - - -next - -up - -previous - -contents -
      - Next: Error Handling - Up: userhtml - Previous: Subroutine descr -   Contents - +
      diff --git a/docs/html/node25.html b/docs/html/node25.html index a0e6da22..16114c93 100644 --- a/docs/html/node25.html +++ b/docs/html/node25.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Error Handling - +License + @@ -26,51 +26,77 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: License - Up: Next: Bibliography + Up: userhtml - Previous: Adding new smoothers and -   Previous: Error Handling +   Contents

      -

      +


      -Error Handling +License

      -The error handling in MLD2P4 is based on the PSBLAS (version 2) error -handling. Error conditions are signaled via an integer argument -info; whenever an error condition is detected, an error trace -stack is built by the library up to the top-level, user-callable -routine. This routine will then decide, according to the user -preferences, whether the error should be handled by terminating the -program or by returning the error condition to the user code, which -will then take action, and whether -an error message should be printed. These options may be set by using -the PSBLAS error handling routines; for further details see the PSBLAS -User's Guide [17]. +The MLD2P4 is freely distributable under the following copyright +terms: 

       
 
-
      
+ 
+                           MLD2P4  version 2.1
+  MultiLevel Domain Decomposition Parallel Preconditioners Package
+             based on PSBLAS (Parallel Sparse BLAS version 3.4)
+  
+  (C) Copyright 2008, 2010, 2012, 2017
+
+  Salvatore Filippone    Cranfield University, Cranfield, UK
+  Ambra Abdullahi Hassan University of Rome Tor Vergata, Rome, IT
+  Alfredo Buttari        CNRS-IRIT, Toulouse, FR
+  Pasqua D'Ambra         IAC-CNR, Naples, IT
+  Daniela di Serafino    University of Campania L. Vanvitelli, Caserta, IT
+
+  Redistribution and use in source and binary forms, with or without
+  modification, are permitted provided that the following conditions
+  are met:
+    1. Redistributions of source code must retain the above copyright
+       notice, this list of conditions and the following disclaimer.
+    2. Redistributions in binary form must reproduce the above copyright
+       notice, this list of conditions, and the following disclaimer in the
+       documentation and/or other materials provided with the distribution.
+    3. The name of the MLD2P4 group or the names of its contributors may
+       not be used to endorse or promote products derived from this
+       software without specific written permission.
+ 
+  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+  ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
+  TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
+  PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE MLD2P4 GROUP OR ITS CONTRIBUTORS
+  BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+  CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+  SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+  INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+  CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+  ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+  POSSIBILITY OF SUCH DAMAGE.
+

      diff --git a/docs/html/node26.html b/docs/html/node26.html index 0730f26e..6487cb01 100644 --- a/docs/html/node26.html +++ b/docs/html/node26.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -License - +Bibliography + @@ -26,78 +26,153 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Bibliography - Up: Next: About this document ... + Up: userhtml - Previous: Error Handling -   Previous: License +   Contents -
      -
      +

      - -

      -
      -License -

      - -

      -The MLD2P4 is freely distributable under the following copyright -terms: 

       
-
  
-                           MLD2P4  version 2.1
-  MultiLevel Domain Decomposition Parallel Preconditioners Package
-             based on PSBLAS (Parallel Sparse BLAS version 3.4)
-  
-  (C) Copyright 2008, 2010, 2012, 2017
+
      
+Bibliography
+
      
 
-  Salvatore Filippone    Cranfield University, Cranfield, UK
-  Ambra Abdullahi Hassan University of Rome Tor Vergata, Rome, IT
-  Alfredo Buttari        CNRS-IRIT, Toulouse, FR
-  Pasqua D'Ambra         IAC-CNR, Naples, IT
-  Daniela di Serafino    University of Campania L. Vanvitelli, Caserta, IT
+
      
+
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+P. R. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J. L'Excellent, C. Weisbecker,
+Improving multifrontal methods by means of block low-rank representations,
+SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452-A1474.
+See also http://mumps.enseeiht.fr.

      2
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+M. Brezina, P. Vanek,
+A Black-Box Iterative Solver Based on a Two-Level Schwarz Method,
+Computing, 63, 1999, 233-263.

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+
      
+W. L. Briggs, V. E. Henson, S. F. McCormick, 
+A Multigrid Tutorial, Second Edition,
+SIAM, 2000.

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+A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone,
+Extending PSBLAS to Build Parallel Schwarz Preconditioners,
+in J. Dongarra, K. Madsen, J. Wasniewski, editors,
+Proceedings of PARA 04 Workshop on State of the Art
+in Scientific Computing, Lecture Notes in Computer Science,
+Springer, 2005, 593-602.

      5
+
       
+A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone,
+2LEV-D2P4: a package of high-performance preconditioners
+for scientific and engineering applications,
+Applicable Algebra in Engineering, Communications and Computing, 
+18 (3) 2007, 223-239.

      6
+
      
+X. C. Cai, M. Sarkis,
+A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems,
+SIAM Journal on Scientific Computing, 21 (2), 1999, 792-797.

      7
+
      
+P. D'Ambra, S. Filippone,  D. di Serafino,
+On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners,
+Applied Numerical Mathematics, Elsevier Science, 
+57 (11-12), 2007, 1181-1196.

      8
+
       
+P. D'Ambra, D. di Serafino, S. Filippone,
+MLD2P4: a Package of Parallel Multilevel
+Algebraic Domain Decomposition Preconditioners
+in Fortran 95,  ACM Trans. Math. Softw., 37(3), 2010, art. 30.

      9
+
      
+T. A. Davis, 
+Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal
+Method with a Column Pre-ordering Strategy,
+ACM Transactions on Mathematical Software, 30, 2004, 196-199.
+(See also http://www.cise.ufl.edu/ davis/)

      10
+
      
+J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu,
+A supernodal approach to sparse partial pivoting,
+SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720-755.

      11
+
      
+J. J. Dongarra, J. Du Croz, I. S. Duff, S. Hammarling,
+A set of Level 3 Basic Linear Algebra Subprograms,
+ACM Transactions on Mathematical Software, 16 (1) 1990, 1-17.

      12
+
      
+J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson,
+An extended set of FORTRAN Basic Linear Algebra Subprograms,
+ACM Transactions on Mathematical Software, 14 (1) 1988, 1-17.

      13
+
      
+S. Filippone, A. Buttari, 
+PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library, 2012,
+available from http://www.ce.uniroma2.it/psblas/.

      14
+
      
+S. Filippone, A. Buttari,
+Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003.
+ACM Transactions on on Mathematical Software, 38 (4), 2012, art. 23.

      15
+
      
+S. Filippone, M. Colajanni, 
+PSBLAS: A Library for Parallel Linear Algebra
+Computation on Sparse Matrices,
+ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550.

      16
+
      
+W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir, 
+MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions,
+MIT Press, 1998.

      17
+
      
+C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh,
+Basic Linear Algebra Subprograms for FORTRAN usage,
+ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323.

      18
+
      
+X. S. Li, J. W. Demmel,
+SuperLU_DIST: A Scalable Distributed-memory
+Sparse Direct Solver for Unsymmetric Linear Systems,
+ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140.

      19
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+Y. Notay, P. S. Vassilevski,
+Recursive Krylov-based multigrid cycles,
+Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487. 

      20
+
      
+Y. Saad,
+Iterative methods for sparse linear systems, 2nd edition, SIAM, 2003.

      21
+
      
+B. Smith, P. Bjorstad, W. Gropp,
+Domain Decomposition: Parallel Multilevel Methods for Elliptic
+Partial Differential Equations,
+Cambridge University Press, 1996.

      22
+
      
+M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra,
+MPI: The Complete Reference. Volume 1 - The MPI Core, second edition,
+MIT Press, 1998.

      23
+
      
+K. Stüben,
+An Introduction to Algebraic Multigrid,
+in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid,
+Academic Press, 2001.

      24
+
      
+R. S. Tuminaro, C. Tong,
+Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.

      25
+
      
+P. Vanek, J. Mandel, M. Brezina,
+Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems,
+Computing, 56 (3) 1996, 179-196.
 
-  Redistribution and use in source and binary forms, with or without
-  modification, are permitted provided that the following conditions
-  are met:
-    1. Redistributions of source code must retain the above copyright
-       notice, this list of conditions and the following disclaimer.
-    2. Redistributions in binary form must reproduce the above copyright
-       notice, this list of conditions, and the following disclaimer in the
-       documentation and/or other materials provided with the distribution.
-    3. The name of the MLD2P4 group or the names of its contributors may
-       not be used to endorse or promote products derived from this
-       software without specific written permission.
- 
-  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
-  ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
-  TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
-  PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE MLD2P4 GROUP OR ITS CONTRIBUTORS
-  BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
-  CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
-  SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
-  INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
-  CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
-  ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
-  POSSIBILITY OF SUCH DAMAGE.
-
      +

      + +


      diff --git a/docs/html/node27.html b/docs/html/node27.html index 34989e95..12d71e5c 100644 --- a/docs/html/node27.html +++ b/docs/html/node27.html @@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds Jens Lippmann, Marek Rouchal, Martin Wilck and others --> -Bibliography - +About this document ... + @@ -18,175 +18,52 @@ original version by: Nikos Drakos, CBLU, University of Leeds - - - -next - + up - previous - contents
      - Next: About this document ... - Up: Up: userhtml - Previous: License -   Previous: Bibliography +   Contents -

      +
      +
      - -

      -Bibliography -

      +

      +About this document ... +

      +

      +This document was generated using the +LaTeX2HTML translator Version 2012 (1.2)

      -

      1 -
      -M. Brezina, P. Vanek, -A Black-Box Iterative Solver Based on a Two-Level Schwarz Method, -Computing, 63, 1999, 233-263.

      2 -
      -W. L. Briggs, V. E. Henson, S. F.  McCormick, -A Multigrid Tutorial, Second Edition, -SIAM, 2000.

      3 -
      -A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, -Extending PSBLAS to Build Parallel Schwarz Preconditioners, -in J. Dongarra, K. Madsen, J. Wasniewski, editors, -Proceedings of PARA 04 Workshop on State of the Art -in Scientific Computing, Lecture Notes in Computer Science, -Springer, 2005, 593-602.

      4 -
      -A. Buttari, P. D'Ambra, D. di Serafino, S. Filippone, -2LEV-D2P4: a package of high-performance preconditioners -for scientific and engineering applications, -Applicable Algebra in Engineering, Communications and Computing, -18 (3) 2007, 223-239.

      5 -
      P. D'Ambra, S. Filippone, D. di Serafino, -On the Development of PSBLAS-based Parallel Two-level Schwarz Preconditioners, -Applied Numerical Mathematics, Elsevier Science, -57 (11-12), 2007, 1181-1196.

      6 -
      -X. C. Cai, M. Sarkis, -A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems, -SIAM Journal on Scientific Computing, 21 (2), 1999, 792-797.

      7 -
      -X. C. Cai, O. B. Widlund, -Domain Decomposition Algorithms for Indefinite Elliptic Problems, -SIAM Journal on Scientific and Statistical Computing, 13 (1), 1992, 243-258.

      8 -
      -T. Chan and T. Mathew, -Domain Decomposition Algorithms, -in A. Iserles, editor, Acta Numerica 1994, 61-143. -Cambridge University Press.

      9 -
      -P. D'Ambra, D. di Serafino, S. Filippone, -MLD2P4: a Package of Parallel Multilevel -Algebraic Domain Decomposition Preconditioners -in Fortran 95, ACM Trans. Math. Softw., 37(3), 2010, art. 30.

      10 -
      -T.A. Davis, -Algorithm 832: UMFPACK - an Unsymmetric-pattern Multifrontal -Method with a Column Pre-ordering Strategy, -ACM Transactions on Mathematical Software, 30, 2004, 196-199. -(See also http://www.cise.ufl.edu/ davis/)

      11 -
      -P.R. Amestoy, C. Ashcraft, O. Boiteau, A. Buttari, J. L'Excellent, C. Weisbecker -Improving multifrontal methods by means of block low-rank representations, -SIAM Journal on Scientific Computing, volume 37 (3), 2015, A1452-A1474. -See also http://mumps.enseeiht.fr.

      12 -
      -J.W. Demmel, S.C. Eisenstat, J.R. Gilbert, X.S. Li and J.W.H. Liu, -A supernodal approach to sparse partial pivoting, -SIAM Journal on Matrix Analysis and Applications, 20 (3), 1999, 720-755.

      13 -
      -J. J. Dongarra, J. Du Croz, I. S. Duff, S. Hammarling, -A set of Level 3 Basic Linear Algebra Subprograms, -ACM Transactions on Mathematical Software, 16 (1) 1990, 1-17.

      14 -
      -J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson, -An extended set of FORTRAN Basic Linear Algebra Subprograms, -ACM Transactions on Mathematical Software, 14 (1) 1988, 1-17.

      15 -
      -J. J. Dongarra and R. C. Whaley, -A User's Guide to the BLACS v. 1.1, -Lapack Working Note 94, Tech. Rep. UT-CS-95-281, University of -Tennessee, March 1995 (updated May 1997).

      16 -
      -E. Efstathiou, J. G. Gander, -Why Restricted Additive Schwarz Converges Faster than Additive Schwarz, -BIT Numerical Mathematics, 43 (5), 2003, 945-959.

      17 -
      -S. Filippone, A. Buttari, -PSBLAS-3.0 User's Guide. A Reference Guide for the Parallel Sparse BLAS Library, 2012, -available from http://www.ce.uniroma2.it/psblas/.

      18 -
      -Salvatore Filippone and Alfredo Buttari. -Object-Oriented Techniques for Sparse Matrix Computations in Fortran 2003. -ACM Transactions on on Mathematical Software, 38 (4), 2012, art. 23.

      19 -
      -S. Filippone, M. Colajanni, -PSBLAS: A Library for Parallel Linear Algebra -Computation on Sparse Matrices, -ACM Transactions on Mathematical Software, 26 (4), 2000, 527-550.

      20 -
      -W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, M. Snir, -MPI: The Complete Reference. Volume 2 - The MPI-2 Extensions, -MIT Press, 1998.

      21 -
      -C. L. Lawson, R. J. Hanson, D. Kincaid, F. T. Krogh, -Basic Linear Algebra Subprograms for FORTRAN usage, -ACM Transactions on Mathematical Software, 5 (3), 1979, 308-323.

      22 -
      -X. S. Li, J. W. Demmel, SuperLU_DIST: A Scalable Distributed-memory -Sparse Direct Solver for Unsymmetric Linear Systems, -ACM Transactions on Mathematical Software, 29 (2), 2003, 110-140.

      23 -
      -Y. Notay, P. S. Vassilevski, Recursive Krylov-based multigrid cycles, -Numerical Linear Algebra with Applications, 15 (5), 2008, 473-487.

      24 -
      -Y. Saad, -Iterative methods for sparse linear systems, 2nd edition, SIAM, 2003.

      25 -
      -B. Smith, P. Bjorstad, W. Gropp, -Domain Decomposition: Parallel Multilevel Methods for Elliptic -Partial Differential Equations, -Cambridge University Press, 1996.

      26 -
      -M. Snir, S. Otto, S. Huss-Lederman, D. Walker, J. Dongarra, -MPI: The Complete Reference. Volume 1 - The MPI Core, second edition, -MIT Press, 1998.

      27 -
      -K. Stüben, -An Introduction to Algebraic Multigrid, -in A. Schüller, U. Trottenberg, C. Oosterlee, Multigrid, -Academic Press, 2001.

      28 -
      -R. S. Tuminaro, C. Tong, -Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines, in J. Donnelley, editor, Proceedings of SuperComputing 2000, Dallas, 2000.

      29 -
      -P. Vanek, J. Mandel and M. Brezina, -Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems, -Computing, 56 (3) 1996, 179-196. +Copyright © 1993, 1994, 1995, 1996, +Nikos Drakos, +Computer Based Learning Unit, University of Leeds. +
      +Copyright © 1997, 1998, 1999, +Ross Moore, +Mathematics Department, Macquarie University, Sydney.

      -

      - +The command line arguments were:
      + latex2html -local_icons -noaddress -dir ../../html userhtml.tex

      +The translation was initiated by Salvatore Filippone on 2017-07-17

      diff --git a/docs/html/node3.html b/docs/html/node3.html index c71fd541..7cdd18e4 100644 --- a/docs/html/node3.html +++ b/docs/html/node3.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Code Distribution - Up: Up: userhtml - Previous: Previous: Contents -   Contents

      @@ -60,9 +60,9 @@ General Overview The MULTI-LEVEL DOMAIN DECOMPOSITION PARALLEL PRECONDITIONERS PACKAGE BASED ON PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid (AMG) and Domain Decomposition preconditioners (see, e.g., [2,27,25]), + HREF="node26.html#Briggs2000">3,23,21]), to be used in the iterative solution of linear systems,
      @@ -95,17 +95,17 @@ multi-level cycles and smoothers widely used in multigrid methods. The multi-level preconditioners implemented in MLD2P4 are obtained by combining AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and K-cycles [2,23] are available, which allow to define + HREF="node26.html#Briggs2000">3,19] are available, which allow to define almost all the preconditioners in the package, including the multi-level hybrid -Schwarz ones; a specific cycle is implemented to obained multi-level additive +Schwarz ones; a specific cycle is implemented to obtain multi-level additive Schwarz preconditioners. The Jacobi, hybridforward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz methods are available as smoothers. An algebraic approach is used to generate a hierarchy of coarse-level matrices and operators, without explicitly using any information on the geometry of the original problem, e.g., the discretization of a PDE. To this end, the smoothed aggregation technique [1,29] + HREF="node26.html#BREZINA_VANEK">2,25] 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, @@ -126,8 +126,8 @@ interface. MLD2P4 has been designed to implement scalable and easy-to-use multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS) computational framework [19,18]. PSBLAS provides basic linear algebra + HREF="node26.html#psblas_00">15,14]. 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 mainly motivated by the need of having @@ -150,14 +150,14 @@ few black-box routines at the upper 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 smoothers and solvers for building new versions of the preconditioners (see -Section 7). +Section [*]).

      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 multi-level hierarchy from the construction of the smoothers and solvers, and to allow for more flexibility at each level. The software architecture described in [9] has significantly + HREF="node26.html#MLD2P4_TOMS">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. @@ -171,34 +171,34 @@ preconditioners with the Krylov solvers implemented in PSBLAS are reported in Section 5, where the Fortran codes of a few sample programs are also shown. A reference guide for the user interface routines is provided in Section 6. Information on the extension of the package -through the addition of new smoothers and solvers is reported in Section 7. +through the addition of new smoothers and solvers is reported in Section [*]. The error handling mechanism used by the package -is briefly described in Section 8. The copyright terms concerning the -distribution and modification of MLD2P4 are reported in Appendix A. +is briefly described in Section 7. The copyright terms concerning the +distribution and modification of MLD2P4 are reported in Appendix A.

      - next - up - previous - contents
      - Next: Next: Code Distribution - Up: Up: userhtml - Previous: Previous: Contents -   Contents diff --git a/docs/html/node4.html b/docs/html/node4.html index d4084a4c..17c69ee8 100644 --- a/docs/html/node4.html +++ b/docs/html/node4.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Configuring and Building MLD2P4 - Up: Up: userhtml - Previous: Previous: General Overview -   Contents

      @@ -63,11 +63,10 @@ MLD2P4 is available from the web site where contact points for further information can be also found. -Passiamo subito a GitHub?

      The software is available under a modified BSD license, as specified -in Appendix A; please note that some of the optional +in Appendix A; 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. diff --git a/docs/html/node5.html b/docs/html/node5.html index 41b1e2f3..b37cef8f 100644 --- a/docs/html/node5.html +++ b/docs/html/node5.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Prerequisites - Up: Up: userhtml - Previous: Previous: Code Distribution -   Contents

      @@ -73,25 +73,57 @@ 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. +

      +Building MLD2P4 requires some base libraries (see Section 3.1); +interfaces to optional third-party libraries, which extend the functionalities of MLD2P4 +(see Section 3.2), 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. +


      Subsections -
      +
      + + +next + +up + +previous + +contents +
      + Next: Prerequisites + Up: userhtml + Previous: Code Distribution +   Contents + diff --git a/docs/html/node6.html b/docs/html/node6.html index a4b459c5..3a44d332 100644 --- a/docs/html/node6.html +++ b/docs/html/node6.html @@ -26,33 +26,34 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Optional third party libraries - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Configuring and Building MLD2P4 -   Contents

      -

      -Prerequisites +

      +
      +Prerequisites

      @@ -60,9 +61,9 @@ The following base libraries are needed:

      BLAS
      [13,14,21] Many vendors provide optimized versions + HREF="node26.html#blas3">11,12,17] Many vendors provide optimized versions of BLAS; if no vendor version is available for a given platform, the ATLAS software (
      MPI
      [20,26] A version of MPI is available on most + HREF="node26.html#MPI2">16,22] A version of MPI is available on most high-performance computing systems.
      PSBLAS
      [17,19] Parallel Sparse BLAS (PSBLAS) is + HREF="node26.html#PSBLASGUIDE">13,15] Parallel Sparse BLAS (PSBLAS) is available from www.ce.uniroma2.it/psblas; version 3.5.0 (or later) is required. Indeed, all the prerequisites @@ -108,26 +109,26 @@ compiler as MLD2P4.

      - next - up - previous - contents
      - Next: Next: Optional third party libraries - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Configuring and Building MLD2P4 -   Contents diff --git a/docs/html/node7.html b/docs/html/node7.html index 7c5a9b49..b0ee2a68 100644 --- a/docs/html/node7.html +++ b/docs/html/node7.html @@ -26,32 +26,32 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Configuration options - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Prerequisites -   Contents

      -

      +


      Optional third party libraries

      @@ -65,7 +65,7 @@ for multi-level preconditioners may change to reflect their presence.
      UMFPACK
      [10] + HREF="node26.html#UMFPACK">9] A sparse LU factorization package included in the SuiteSparse library, available from faculty.cse.tamu.edu/davis/suitesparse.html; @@ -76,31 +76,31 @@ for multi-level preconditioners may change to reflect their presence.
      MUMPS
      [11] + HREF="node26.html#MUMPS">1] A sparse LU factorization package available from mumps.enseeiht.fr; it provides sequential and parallel factorizations and triangular system solution for single and double precision, real and complex data. - We tested versions 4.10.0 and version 5.0.1. + We tested versions 4.10.0 and 5.0.1.
      SuperLU
      [12] + HREF="node26.html#SUPERLU">10] A sparse LU factorization package available from crd.lbl.gov/~xiaoye/SuperLU/; it provides sequential factorization and triangular system solution for single and double precision, - real and complex data. We tested version 4.3 and 5.0. If you installed BLAS from + real and complex data. We tested versions 4.3 and 5.0. If you installed BLAS from ATLAS, remember to define the BLASLIB variable in the make.inc file.
      SuperLU_Dist
      [22] + HREF="node26.html#SUPERLUDIST">18] A sparse LU factorization package available from the same site as SuperLU; it provides parallel factorization and triangular system solution for double precision real and complex data. - We tested version 3.3 and 4.2. If you installed BLAS from + We tested versions 3.3 and 4.2. If you installed BLAS from ATLAS, remember to define the BLASLIB variable in the make.inc file and to add the -std=c99 option to the C compiler options. Note that this library requires the ParMETIS @@ -113,26 +113,26 @@ for multi-level preconditioners may change to reflect their presence.

      - next - up - previous - contents
      - Next: Next: Configuration options - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Prerequisites -   Contents diff --git a/docs/html/node8.html b/docs/html/node8.html index 354f39e5..40db8486 100644 --- a/docs/html/node8.html +++ b/docs/html/node8.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Bug reporting - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Optional third party libraries -   Contents

      @@ -247,7 +247,7 @@ install directory under the name Make.inc.MLD2P4. 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 --ldmumps -lsmumps -lzmumps -mumps_common -lpord. +-ldmumps -lsmumps -lzmumps -lcmumps -mumps_common -lpord. 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 --with-extra-libs configure option. @@ -260,35 +260,28 @@ make followed (optionally) by 
       make install
-
      -Many Linux distributions (e.g. Ubuntu, Fedora, CentOS) provide -precompiled packages for the prerequisite softwares; in many cases the -software packages are split between a runtime part and a ``developer'' -part, to rebuild MLD2P4 you'll need both. - -

      -

      +
      - next - up - previous - contents
      - Next: Next: Bug reporting - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Optional third party libraries -   Contents diff --git a/docs/html/node9.html b/docs/html/node9.html index 3a523221..3ba9fb3f 100644 --- a/docs/html/node9.html +++ b/docs/html/node9.html @@ -26,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents
      - Next: Next: Example and test programs - Up: Up: Configuring and Building MLD2P4 - Previous: Previous: Configuration options -   Contents

      @@ -54,15 +54,13 @@ original version by: Nikos Drakos, CBLU, University of Leeds

      Bug reporting

      -If you find any bugs in our codes, please let us know at - - -bugreport@mld2p4.it +If you find any bugs in our codes, please send an email to
      pasqua.dambra@cnr.it +
      daniela.diserafino@unicampania.it +
      salvatore.filippone@cranfield.ac.uk
      +You should be aware that the amount of information needed to reproduce a problem +in a parallel program may vary quite a lot. -       -; be aware that -the amount of information needed to reproduce a problem in a parallel -program may vary quite a lot. +


      diff --git a/docs/html/userhtml.html b/docs/html/userhtml.html index bb6796be..a627bdd3 100644 --- a/docs/html/userhtml.html +++ b/docs/html/userhtml.html @@ -136,15 +136,13 @@ Feb. 28, 2017
    1. Adding new smoothers and solvers to MLD2P4 + HREF="node24.html">Error Handling
    2. Error Handling + HREF="node25.html">License
    3. License + HREF="node26.html">Bibliography
    4. Bibliography -
    5. About this document ... + HREF="node27.html">About this document ...
      diff --git a/docs/mld2p4-2.1-guide.pdf b/docs/mld2p4-2.1-guide.pdf index 4e53a168..b0be6e94 100644 --- a/docs/mld2p4-2.1-guide.pdf +++ b/docs/mld2p4-2.1-guide.pdf @@ -1,6 +1,6 @@ %PDF-1.5 % -121 0 obj +117 0 obj << /Length 1216 >> @@ -31,7 +31,7 @@ ET endstream endobj -130 0 obj +126 0 obj << /Length 32 >> @@ -43,7 +43,7 @@ stream endstream endobj -134 0 obj +130 0 obj << /Length 3902 >> @@ -66,7 +66,7 @@ ET endstream endobj -140 0 obj +136 0 obj << /Length 79 >> @@ -81,9 +81,9 @@ ET endstream endobj -172 0 obj +167 0 obj << -/Length 7176 +/Length 7025 >> stream 0 g 0 G @@ -119,7 +119,7 @@ BT 0 g 0 G [-711(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-499(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ 0 g 0 G - [-1781(4)]TJ + [-1781(5)]TJ 0 g 0 G 0 0 1 rg 0 0 1 RG 0 -13.549 Td [(3.3)-1022(Con\014guration)-333(options)]TJ @@ -253,21 +253,17 @@ BT [-1281(38)]TJ 0 g 0 G 0 0 1 rg 0 0 1 RG -/F43 10.9091 Tf -16.364 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[(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 -244.258 -19.516 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.885 -3.958 Td [(=)-437(\050)]TJ/F22 10.9091 Tf 17.493 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 11.282 -3.958 Td [(in)-429(the)-428(c)-1(u)1(rren)27(t)-428(implemen)27(tation)-428(of)-429(MLD2P4.)-731(A)-428(smo)-28(other)-429(with)-429(iteration)]TJ -64.908 -13.55 Td [(matrix)]TJ/F22 10.9091 Tf 35.437 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.708 -3.959 Td [(is)-329(set)-329(up)-329(at)-329(eac)28(h)-329(lev)28(el)]TJ/F22 10.9091 Tf 106.2 0 Td [(k)-309(<)-278(nl)-20(ev)]TJ/F15 10.9091 Tf 41.345 0 Td [(,)-330(and)-329(a)-329(solv)28(er)-329(is)-329(set)-329(up)-329(at)-329(the)-329(coarsest)-329(lev)28(el,)]TJ -203.463 -13.549 Td [(so)-397(that)-396(they)-397(are)-397(ready)-396(for)-397(application)-397(\050for)-396(example,)-413(setting)-396(up)-397(a)-397(solv)28(er)-397(based)-396(on)-397(the)]TJ/F22 10.9091 Tf 0 -13.549 Td [(LU)]TJ/F15 10.9091 Tf 19.321 0 Td [(factorization)-299(means)-298(computing)-299(and)-299(storing)-298(the)]TJ/F22 10.9091 Tf 224.886 0 Td [(L)]TJ/F15 10.9091 Tf 10.683 0 Td [(and)]TJ/F22 10.9091 Tf 20.834 0 Td [(U)]TJ/F15 10.9091 Tf 11.897 0 Td [(factors\051.)-433(The)-299(construction)]TJ -287.621 -13.549 Td [(of)-306(the)-306(hierac)28(h)28(y)-306(of)-306(AMG)-305(c)-1(omp)-27(onen)28(ts)-306(describ)-28(ed)-306(so)-306(far)-306(corresp)-27(onds)-306(to)-306(the)-306(so-called)-306(build)]TJ 0 -13.549 Td [(phase)-333(of)-334(the)-333(preconditioner.)]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(.,)-311(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 + -398.511 -19.516 Td [(where)]TJ/F22 10.9091 Tf 32.496 0 Td [(A)]TJ/F15 10.9091 Tf 12.796 0 Td [(=)-423(\050)]TJ/F22 10.9091 Tf 17.342 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 -1.636 Td [(ij)]TJ/F15 10.9091 Tf 7.265 1.636 Td [(\051)]TJ/F25 10.9091 Tf 8.857 0 Td [(2)]TJ/F34 10.9091 Tf 11.888 0 Td [(R)]TJ/F23 7.9701 Tf 7.878 3.959 Td [(n)]TJ/F26 7.9701 Tf 5.139 0 Td [(\002)]TJ/F23 7.9701 Tf 6.586 0 Td [(n)]TJ/F15 10.9091 Tf 10.223 -3.959 Td [(is)-421(a)-420(nonsingular)-420(sparse)-421(matrix;)-464(for)-420(ease)-421(of)-420(presen)28(tation)-421(w)28(e)]TJ -126.237 -13.549 Td [(assume)]TJ/F22 10.9091 Tf 37.697 0 Td [(A)]TJ/F15 10.9091 Tf 11.818 0 Td [(is)-333(real,)-334(but)-333(the)-333(results)-334(are)-333(v)56(alid)-334(for)-333(the)-333(complex)-334(case)-333(as)-333(w)27(ell.)]TJ -32.579 -13.549 Td [(Let)-497(us)-498(assume)-497(as)-497(\014nest)-498(in)1(dex)-498(space)-497(the)-497(set)-498(of)-497(ro)28(w)-497(\050column\051)-498(ind)1(ice)-1(s)-497(of)]TJ/F22 10.9091 Tf 361.461 0 Td [(A)]TJ/F15 10.9091 Tf 8.182 0 Td [(,)-538(i.e.,)]TJ -386.579 -13.55 Td [(\012)-377(=)]TJ/F25 10.9091 Tf 24.593 0 Td [(f)]TJ/F15 10.9091 Tf 5.454 0 Td [(1)]TJ/F22 10.9091 Tf 5.455 0 Td [(;)]TJ/F15 10.9091 Tf 4.848 0 Td [(2)]TJ/F22 10.9091 Tf 5.455 0 Td [(;)-167(:)-166(:)-167(:)-167(;)-166(n)]TJ/F25 10.9091 Tf 30.79 0 Td [(g)]TJ/F15 10.9091 Tf 5.455 0 Td [(.)-623(An)28(y)-393(algebraic)-393(m)27(ultilev)28(el)-393(preconditioners)-393(implemen)28(ted)-393(in)-393(MLD2P4)]TJ -82.05 -13.549 Td [(generates)-333(a)-334(hierarc)28(h)28(y)-333(of)-334(index)-333(spaces)-333(and)-334(a)-333(corresp)-28(onding)-333(hierarc)28(h)28(y)-334(of)-333(matrices,)]TJ 81.377 -23.491 Td [(\012)]TJ/F20 7.9701 Tf 7.879 4.505 Td [(1)]TJ/F25 10.9091 Tf 7.762 -4.505 Td [(\021)]TJ/F15 10.9091 Tf 11.515 0 Td [(\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 -244.258 -19.516 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.885 -3.958 Td [(=)-437(\050)]TJ/F22 10.9091 Tf 17.493 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 11.282 -3.958 Td [(in)-429(the)-428(c)-1(u)1(rren)27(t)-428(implemen)27(tation)-428(of)-429(MLD2P4.)-731(A)-428(smo)-28(other)-429(with)-429(iteration)]TJ -64.908 -13.55 Td [(matrix)]TJ/F22 10.9091 Tf 35.437 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.708 -3.959 Td [(is)-329(set)-329(up)-329(at)-329(eac)28(h)-329(lev)28(el)]TJ/F22 10.9091 Tf 106.2 0 Td [(k)-309(<)-278(nl)-20(ev)]TJ/F15 10.9091 Tf 41.345 0 Td [(,)-330(and)-329(a)-329(solv)28(er)-329(is)-329(set)-329(up)-329(at)-329(the)-329(coarsest)-329(lev)28(el,)]TJ -203.463 -13.549 Td [(so)-280(that)-280(they)-281(are)-280(ready)-280(for)-280(application)-280(\050for)-280(example,)-291(setting)-280(up)-280(a)-281(solv)28(er)-280(based)-280(on)-280(the)]TJ/F22 10.9091 Tf 396.389 0 Td [(LU)]TJ/F15 10.9091 Tf -396.389 -13.549 Td [(factorization)-353(means)-353(computing)-353(and)-353(storing)-352(the)]TJ/F22 10.9091 Tf 228.435 0 Td [(L)]TJ/F15 10.9091 Tf 11.275 0 Td [(and)]TJ/F22 10.9091 Tf 21.426 0 Td [(U)]TJ/F15 10.9091 Tf 12.488 0 Td [(factors\051.)-503(The)-353(construction)-353(of)]TJ -273.624 -13.549 Td [(the)-366(hierarc)28(h)28(y)-366(of)-366(AMG)-366(comp)-27(onen)27(ts)-365(desc)-1(r)1(ib)-28(ed)-366(so)-366(far)-366(corresp)-27(onds)-366(to)-366(the)-366(so-called)-366(build)]TJ 0 -13.549 Td [(phase)-333(of)-334(the)-333(preconditioner.)]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(.,)-311(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(withi)1(n)-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 - [(24)]TJ + [(20)]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 @@ -1902,19 +1910,19 @@ BT 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 - [(2)]TJ + [(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 - [(23)]TJ + [(19)]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)-375(t)1(he)-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 - [(1)]TJ + [(2)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-319(29)]TJ + [-319(25)]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 @@ -1923,9 +1931,9 @@ ET endstream endobj -342 0 obj +338 0 obj << -/Length 10301 +/Length 11453 >> stream 0 g 0 G @@ -1963,40 +1971,48 @@ 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 -47.165 Td [(op)-28(erator)]TJ/F22 10.9091 Tf 42.856 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 7.611 -3.959 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 -13.549 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 -14.651 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 + -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 -26.924 Td [(1.)]TJ + -3.603 -23.466 Td [(1.)]TJ 0 g 0 G - [-500(aggregation)-333(of)-334(t)1(he)-334(indices)-333(set)-334(\012)]TJ/F23 7.9701 Tf 164.697 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.959 Td [(,)-333(to)-334(obtain)-333(\012)]TJ/F23 7.9701 Tf 61.819 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 11.319 -3.959 Td [(;)]TJ + [-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 - -247.575 -26.924 Td [(2.)]TJ + -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 -26.923 Td [(3.)]TJ + -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 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.756 -3.959 Td [(and)]TJ/F22 10.9091 Tf 21.212 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.15 -3.959 Td [(=)-278(\050)]TJ/F22 10.9091 Tf 15.757 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(\051)]TJ/F23 7.9701 Tf 4.242 3.959 Td [(T)]TJ/F15 10.9091 Tf 10.241 -3.959 Td [(to)-333(build)]TJ/F22 10.9091 Tf 41.212 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 -243.343 -26.924 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)-242(in)-243([)]TJ + [-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 - [(29)]TJ + [(25)]TJ 0 g 0 G - [(])-242(is)-243(used.)-414(In)-242(this)-243(algorithm,)-260(eac)27(h)-242(index)-243(in)-242(\012)]TJ/F23 7.9701 Tf 262.99 3.959 Td [(k)]TJ/F20 7.9701 Tf 4.621 0 Td [(+1)]TJ/F15 10.9091 Tf 13.964 -3.959 Td [(corresp)-28(onds)-242(to)-243(an)-242(aggregate)]TJ -281.575 -13.549 Td [(of)-301(\012)]TJ/F23 7.9701 Tf 19.952 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.959 Td [(,)-308(consisting)-301(of)-301(a)-301(suitably)-301(c)28(hosen)-301(index)]TJ/F22 10.9091 Tf 182.094 0 Td [(j)]TJ/F15 10.9091 Tf 8.402 0 Td [(and)-301(of)-301(the)-301(indices)]TJ/F22 10.9091 Tf 86.836 0 Td [(i)]TJ/F15 10.9091 Tf 7.043 0 Td [(that)-301(are)-301(strongly)-301(cou-)]TJ -309.446 -13.549 Td [(pled)-333(to)]TJ/F22 10.9091 Tf 36.97 0 Td [(j)]TJ/F15 10.9091 Tf 5.117 0 Td [(,)-333(i.e.,)]TJ/F25 10.9091 Tf 121.576 -19.945 Td [(j)]TJ/F22 10.9091 Tf 3.03 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 4.504 Td [(k)]TJ 0 -7.201 Td [(ij)]TJ/F25 10.9091 Tf 7.265 2.697 Td [(j)]TJ/F22 10.9091 Tf 6.061 0 Td [(>)-278(\022)]TJ/F28 10.9091 Tf 16.939 12.648 Td [(q)]TJ + [(])-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 1 307.234 255.295 cm -[]0 d 0 J 0.436 w 0 0 m 32.124 0 l S +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 307.234 242.428 Td [(j)]TJ/F22 10.9091 Tf 3.03 0 Td [(a)]TJ/F23 7.9701 Tf 5.767 3.758 Td [(k)]TJ 0 -7.015 Td [(ii)]TJ/F22 10.9091 Tf 6.264 3.257 Td [(a)]TJ/F23 7.9701 Tf 5.767 3.758 Td [(k)]TJ 0 -7.015 Td [(j)-58(j)]TJ/F25 10.9091 Tf 8.266 3.257 Td [(j)]TJ/F22 10.9091 Tf 3.03 0 Td [(;)]TJ/F15 10.9091 Tf -245.758 -26.063 Td [(for)-331(a)-331(giv)28(en)]TJ/F22 10.9091 Tf 54.197 0 Td [(\022)]TJ/F25 10.9091 Tf 8.455 0 Td [(2)]TJ/F15 10.9091 Tf 10.303 0 Td [([0)]TJ/F22 10.9091 Tf 8.484 0 Td [(;)]TJ/F15 10.9091 Tf 4.849 0 Td [(1].)-444(Since)-331(this)-331(algorith)1(m)-332(h)1(as)-332(a)-331(sequen)28(tial)-331(nature,)-331(a)-331(decoupled)-331(v)28(ersion)]TJ -86.288 -13.549 Td [(of)-306(it)-305(is)-306(applied,)-311(where)-306(eac)28(h)-306(pro)-28(cessor)]TJ/F22 10.9091 Tf 177.316 0 Td [(i)]TJ/F15 10.9091 Tf 7.094 0 Td [(indep)-28(enden)28(tly)-305(e)-1(xecutes)-305(the)-306(algorithm)-306(on)-305(the)-306(set)]TJ -184.41 -13.549 Td [(of)-341(i)1(ndices)-341(assigned)-341(to)-340(it)-341(in)-340(the)-341(initi)1(al)-341(data)-340(distribution.)-466(This)-341(v)28(ersion)-341(i)1(s)-341(em)28(barrassingly)]TJ 0 -13.549 Td [(parallel,)-506(since)-472(it)-472(do)-28(es)-472(not)-472(requ)1(ire)-472(an)28(y)-472(data)-472(comm)28(unication.)-860(On)-472(the)-472(other)-472(hand,)-506(it)]TJ 0 -13.55 Td [(ma)28(y)-394(pro)-27(duce)-394(some)-393(non-uniform)-393(aggre)-1(gat)1(e)-1(s)-393(and)-393(is)-394(strongly)-393(dep)-28(enden)28(t)-393(on)-394(the)-393(n)28(um)28(b)-28(er)]TJ 0 -13.549 Td [(of)-358(p)1(ro)-28(cessors)-358(and)-357(on)-358(the)-357(initial)-358(partitioning)-357(of)-358(the)-357(matrix)]TJ/F22 10.9091 Tf 280.52 0 Td [(A)]TJ/F15 10.9091 Tf 8.181 0 Td [(.)-517(Nev)28(ertheless,)-364(this)-357(parall)]TJ -288.701 -13.549 Td [(algorithm)-248(has)-249(b)-27(een)-249(c)28(hosen)-248(for)-249(MLD2P4,)-265(since)-248(it)-249(has)-248(b)-28(een)-248(sho)28(wn)-249(to)-248(pro)-28(duce)-248(go)-28(o)-28(d)-248(results)]TJ 0 -13.549 Td [(in)-333(practice)-334([)]TJ +/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 - [(4)]TJ + [(25)]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(5)]TJ + [-333(7)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-333(28)]TJ + [-334(24)]TJ 0 g 0 G [(].)]TJ 0 g 0 G @@ -2005,49 +2021,49 @@ ET endstream endobj -356 0 obj +353 0 obj << -/Length 14031 +/Length 15415 >> stream 0 g 0 G BT /F15 10.9091 Tf 86.4 740.002 Td [(14)]TJ/F41 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/F15 10.9091 Tf -290.076 -20.871 Td [(where)-357(\012)]TJ/F23 7.9701 Tf 39.687 3.958 Td [(k)]TJ 0 -7.014 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 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.019 -3.958 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 3.958 Td [(k)]TJ/F20 7.9701 Tf 4.622 0 Td [(+1)]TJ/F15 10.9091 Tf 11.318 -3.958 Td [(.)]TJ/F22 10.9091 Tf 8.669 0 Td [(P)]TJ/F23 7.9701 Tf 8.519 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.02 -3.958 Td [(is)-357(obtained)]TJ -360.006 -15.096 Td [(b)28(y)-333(applying)-334(to)]TJ 75.471 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/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 -24.189 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 -24.189 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)-266(impro)27(v)28(e)-266(the)-267(con)28(v)28(ergence)-267(pr)1(op)-28(erties)-267(of)-266(the)-267(m)28(ulti-lev)28(el)-266(metho)-28(d)-267([)]TJ +/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/F15 10.9091 Tf -290.076 -19.797 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)-266(impro)27(v)28(e)-266(the)-267(con)28(v)28(ergence)-267(pr)1(op)-28(erties)-267(of)-266(the)-267(m)28(ulti-lev)28(el)-266(metho)-28(d)-267([)]TJ 1 0 0 rg 1 0 0 RG - [(1)]TJ + [(2)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-266(27)]TJ + [-266(23)]TJ +0 g 0 G + [(].)-422(A)-266(simple)-267(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 - [(].)-422(A)-266(simple)-267(c)28(hoice)]TJ 0 -13.55 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 119.024 -24.189 Td [(S)]TJ/F23 7.9701 Tf 7.319 4.504 Td [(k)]TJ/F15 10.9091 Tf 8.15 -4.504 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 7.181 4.504 Td [(k)]TJ/F15 10.9091 Tf 5.12 -4.504 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(D)]TJ/F23 7.9701 Tf 9.334 4.504 Td [(k)]TJ/F15 10.9091 Tf 5.12 -4.504 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 [(A)]TJ/F23 7.9701 Tf 8.182 4.504 Td [(k)]TJ/F22 10.9091 Tf 5.12 -4.504 Td [(;)]TJ/F15 10.9091 Tf -257.625 -24.189 Td [(where)]TJ/F22 10.9091 Tf 32.246 0 Td [(D)]TJ/F23 7.9701 Tf 9.335 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.457 -3.959 Td [(is)-398(the)-397(diagonal)-398(matrix)-397(with)-398(the)-397(sam)-1(e)-397(diagonal)-398(en)28(tries)-397(as)]TJ/F22 10.9091 Tf 278.83 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.959 Td [(,)-414(and)]TJ/F22 10.9091 Tf 29.456 0 Td [(!)]TJ/F23 7.9701 Tf 7.182 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.457 -3.959 Td [(is)-398(an)]TJ -389.265 -13.549 Td [(appro)28(ximation)-371(of)-370(4)]TJ/F22 10.9091 Tf 91.45 0 Td [(=)]TJ/F15 10.9091 Tf 5.454 0 Td [(\0503)]TJ/F22 10.9091 Tf 9.697 0 Td [(\032)]TJ/F23 7.9701 Tf 5.64 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.958 Td [(\051,)-380(where)]TJ/F22 10.9091 Tf 43.37 0 Td [(\032)]TJ/F23 7.9701 Tf 5.64 3.958 Td [(k)]TJ/F15 10.9091 Tf 9.163 -3.958 Td [(is)-371(the)-370(sp)-28(ectral)-371(radiu)1(s)-371(of)-371(\050)]TJ/F22 10.9091 Tf 122.276 0 Td [(D)]TJ/F23 7.9701 Tf 9.335 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.958 Td [(\051)]TJ/F26 7.9701 Tf 4.243 3.958 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.733 -3.958 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.958 Td [(k)]TJ/F15 10.9091 Tf 5.12 -3.958 Td [(.)-556(computed)-371(b)28(y)]TJ -341.127 -13.55 Td [(using)-333(some)-334(estimate)-333(of)-333(the)-334(sp)-28(ectral)-333(radius)-333(of)-333(\050)]TJ/F22 10.9091 Tf 221.273 0 Td [(D)]TJ/F23 7.9701 Tf 9.335 3.959 Td [(k)]TJ/F15 10.9091 Tf 5.119 -3.959 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 [(A)]TJ/F23 7.9701 Tf 8.181 3.959 Td [(k)]TJ/F15 10.9091 Tf 8.757 -3.959 Td [([)]TJ +/F15 10.9091 Tf 39.721 0 Td [(\0504\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 - [(1)]TJ + [(2)]TJ 0 g 0 G - [(].)]TJ/F17 11.9552 Tf -268.227 -29.162 Td [(4.3)-1125(Smo)-31(others)-375(and)-375(coarsest-lev)31(el)-375(solv)31(ers)]TJ/F15 10.9091 Tf 0 -20.595 Td [(The)-294(smo)-28(others)-294(implemen)28(ted)-295(i)1(n)-295(MLD2P)1(4)-295(includ)1(e)-295(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(ers)-1(i)1(on)-344(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 -13.549 Td [(Sc)28(h)28(w)27(arz)-333(\050AS\051)-333(ones)-334(\050see,)-333(e.g.,)-333([)]TJ + [(].)]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 - [(24)]TJ + [(20)]TJ 0 g 0 G [(,)]TJ 1 0 0 rg 1 0 0 RG - [-334(25)]TJ -0 g 0 G - [(]\051.)]TJ 16.936 -13.55 Td [(The)-481(h)28(ybrid)-480(Gauss-Seidel)-481(v)28(ersion)-481(i)1(s)-481(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)-451(its)-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 -13.549 Td [(with)-333(other)-334(pr)1(o)-28(cesses)-334(b)-28(efore)-333(the)-333(b)-28(eginning)-333(of)-333(the)-334(curren)28(t)-333(iteration.)]TJ 16.936 -13.55 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 33.938 0 Td [(n)]TJ/F23 7.9701 Tf 6.548 -1.777 Td [(k)-27(;i)]TJ/F15 10.9091 Tf 10.356 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 178.984 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(k)]TJ -0.084 -7.014 Td [(i)]TJ/F25 10.9091 Tf 10.026 3.055 Td [(2)]TJ/F34 10.9091 Tf 12.096 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 6.234 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.111 -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 6.234 3.959 Td [(k)]TJ 0 -7.015 Td [(i)]TJ/F15 10.9091 Tf 9.111 3.056 Td [(made)-366(of)-366(the)-365(comp)-28(onen)28(ts)-366(of)]TJ/F22 10.9091 Tf 134.559 0 Td [(x)]TJ/F23 7.9701 Tf 6.235 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 5.119 3.056 Td [(,)-301(and)-292(the)-293(prolongation)-292(op)-28(erator)]TJ/F22 10.9091 Tf 152.202 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 3.959 Td [(k)]TJ 0 -7.015 Td [(i)]TJ/F15 10.9091 Tf 8.557 3.056 Td [(=)]TJ/F22 10.9091 Tf 11.922 0 Td [(R)]TJ/F23 7.9701 Tf 8.368 3.959 Td [(k)]TJ -0.084 -7.015 Td [(i)]TJ/F22 10.9091 Tf 5.204 3.056 Td [(A)]TJ/F23 7.9701 Tf 8.181 3.959 Td [(k)]TJ/F22 10.9091 Tf 5.12 -3.959 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 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 3.959 Td [(k)]TJ/F15 10.9091 Tf 9.001 -3.959 Td [(to)-356(the)-355(index)-356(space)-356(\012)]TJ/F23 7.9701 Tf 99.523 3.959 Td [(k)]TJ 0 -7.015 Td [(i)]TJ/F15 10.9091 Tf 5.119 3.056 Td [(.)-512(The)-355(class)-1(ical)-355(AS)]TJ -324.744 -13.549 Td [(preconditioner)]TJ/F22 10.9091 Tf 72.182 0 Td [(M)]TJ/F23 7.9701 Tf 11.773 3.958 Td [(k)]TJ -1.19 -7.191 Td [(AS)]TJ/F15 10.9091 Tf 16.073 3.233 Td [(is)-333(de\014ned)-334(as)]TJ 38.284 -33.465 Td [(\050)]TJ/F22 10.9091 Tf 4.243 0 Td [(M)]TJ/F23 7.9701 Tf 11.772 4.505 Td [(k)]TJ -1.189 -7.201 Td [(AS)]TJ/F15 10.9091 Tf 12.437 2.696 Td [(\051)]TJ/F26 7.9701 Tf 4.242 4.505 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F15 10.9091 Tf 7.762 -4.505 Td [(=)]TJ/F23 7.9701 Tf 13.363 14.073 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 1.026 -23.451 Td [(i)]TJ/F20 7.9701 Tf 2.884 0 Td [(=1)]TJ/F22 10.9091 Tf 13.665 13.087 Td [(P)]TJ/F23 7.9701 Tf 8.519 4.505 Td [(k)]TJ -1.515 -7.201 Td [(i)]TJ/F15 10.9091 Tf 6.635 2.696 Td [(\050)]TJ/F22 10.9091 Tf 4.242 0 Td [(A)]TJ/F23 7.9701 Tf 8.182 4.505 Td [(k)]TJ 0 -7.201 Td [(i)]TJ/F15 10.9091 Tf 5.12 2.696 Td [(\051)]TJ/F26 7.9701 Tf 4.242 4.505 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F22 10.9091 Tf 4.732 -4.505 Td [(R)]TJ/F23 7.9701 Tf 8.368 4.505 Td [(k)]TJ -0.085 -7.201 Td [(i)]TJ/F22 10.9091 Tf 5.204 2.696 Td [(;)]TJ/F15 10.9091 Tf -272.298 -35.308 Td [(where)]TJ/F22 10.9091 Tf 31.036 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 8.247 3.055 Td [(is)-287(supp)-27(osed)-287(to)-287(b)-27(e)-287(nonsingular.)-429(W)84(e)-287(observ)28(e)-287(that)-287(an)-286(appro)28(ximate)-287(in)28(v)28(e)-1(r)1(s)-1(e)-286(of)]TJ/F22 10.9091 Tf 351.684 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 -407.331 -10.494 Td [(is)-366(usually)-365(c)-1(on)1(s)-1(id)1(e)-1(r)1(e)-1(d)-365(instead)-366(of)-366(\050)]TJ/F22 10.9091 Tf 157.8 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 5.12 3.055 Td [(\051)]TJ/F26 7.9701 Tf 4.242 3.959 Td [(\000)]TJ/F20 7.9701 Tf 6.587 0 Td [(1)]TJ/F15 10.9091 Tf 4.732 -3.959 Td [(.)-542(The)-366(setup)-365(of)]TJ/F22 10.9091 Tf 74.003 0 Td [(S)]TJ/F23 7.9701 Tf 7.318 3.959 Td [(k)]TJ -0.628 -7.192 Td [(AS)]TJ/F15 10.9091 Tf 16.427 3.233 Td [(during)-366(the)-365(m)27(ultil)1(e)-1(v)28(el)-366(bu)1(ild)]TJ -283.783 -13.549 Td [(phase)-333(in)28(v)27(olv)28(es)]TJ + [-334(21)]TJ 0 g 0 G -/F25 10.9091 Tf 16.364 -22.25 Td [(\017)]TJ -0 g 0 G -/F15 10.9091 Tf 10.909 0 Td [(the)-383(de\014n)1(ition)-383(of)-383(th)1(e)-383(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)-382(of)-383(the)-383(corresp)-27(onding)-383(op)-28(erators)]TJ/F22 10.9091 Tf 173.536 0 Td [(R)]TJ/F23 7.9701 Tf 8.367 3.959 Td [(k)]TJ -0.084 -7.014 Td [(i)]TJ/F15 10.9091 Tf -379.974 -10.494 Td [(\050and)]TJ/F22 10.9091 Tf 25.454 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 6.635 3.056 Td [(\051;)]TJ + [(]\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 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[(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 -363 0 obj +361 0 obj << -/Length 5723 +/Length 7392 >> stream 0 g 0 G @@ -2057,12 +2073,17 @@ BT 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 -385.178 -35.866 Td [(\017)]TJ +/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 3.959 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.12 3.055 Td [(;)]TJ +/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 -22.515 Td [(\017)]TJ +/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 @@ -2072,26 +2093,26 @@ BT 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 3.958 Td [(k)]TJ -0.48 -7.014 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 3.958 Td [(k)]TJ 0 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.119 3.056 Td [(\051)]TJ/F26 7.9701 Tf 4.243 3.958 Td [(\000)]TJ/F20 7.9701 Tf 6.586 0 Td [(1)]TJ/F22 10.9091 Tf 4.733 -3.958 Td [(w)]TJ/F23 7.9701 Tf 8.103 3.958 Td [(k)]TJ -0.293 -7.014 Td [(i)]TJ/F15 10.9091 Tf 5.413 3.056 Td [(;)]TJ +/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 -3.019 Td [(m)]TJ/F24 5.9776 Tf 7.491 -1.405 Td [(k)]TJ/F23 7.9701 Tf -7.491 -7.015 Td [(i)]TJ/F20 7.9701 Tf 2.883 0 Td [(=1)]TJ/F22 10.9091 Tf 13.137 3.257 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 -13.55 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 -13.549 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 +/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 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[(Direct)-418(solv)28(e)-1(rs)-418(based)-418(on)-418(sparse)-419(LU)-418(factorizations,)-439(implemen)27(ted)-418(in)-418(the)-418(third)-418(part)27(y)]TJ -16.937 -13.549 Td [(libraries)-374(rep)-27(orted)-374(in)-374(Section)]TJ + [(].)]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 [(Nativ)28(e)-313(inexact)-314(solv)28(ers)-313(based)-313(on)-313(incomplete)-314(LU)-313(factorizations,)-317(as)-313(w)28(ell)-314(as)-313(Jacobi,)-317(h)28(ybrid)]TJ 0 -13.549 Td 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[(solv)28(ers)-279(usually)-278(lead)-279(to)-278(more)-279(e\013ectiv)28(e)-279(preconditi)1(oners)-279(in)-278(terms)-279(of)-278(algorithmic)-279(scalabilit)28(y;)]TJ 0 -13.549 Td [(ho)28(w)28(ev)27(er,)-333(this)-333(do)-28(es)-333(not)-334(guaran)28(tee)-333(parallel)-334(e\016ciency)84(.)]TJ 0 g 0 G 0 g 0 G ET endstream endobj -376 0 obj +374 0 obj << /Length 8463 >> @@ -2102,7 +2123,7 @@ BT 0 g 0 G /F17 14.3462 Tf -203.265 -35.866 Td [(5)-1125(Getting)-375(Started)]TJ/F15 10.9091 Tf 0 -24.802 Td [(W)83(e)-441(describ)-28(e)-441(the)-442(basics)-441(for)-441(building)-442(an)1(d)-442(applying)-441(MLD2P4)-441(one-lev)27(el)-441(and)-441(m)27(u)1(lti-lev)27(el)]TJ 0 -13.549 Td [(\050i.e.,)-468(AMG\051)-441(precondition)1(e)-1(rs)-441(with)-440(the)-441(Krylo)27(v)-441(solv)28(ers)-441(included)-441(in)-441(P)1(SBLAS)-441([)]TJ 1 0 0 rg 1 0 0 RG - [(17)]TJ + [(13)]TJ 0 g 0 G [(].)-768(The)]TJ 0 -13.549 Td [(follo)28(wing)-333(steps)-334(are)-333(required:)]TJ 0 g 0 G @@ -2163,9 +2184,9 @@ ET endstream endobj -395 0 obj +393 0 obj << -/Length 9271 +/Length 9282 >> stream 0 g 0 G @@ -2269,7 +2290,7 @@ q []0 d 0 J 0.398 w 0 0 m 0 40.648 l S Q BT -/F15 10.9091 Tf 263.982 609.608 Td [(Diagonal)-423(preconditioner.)-714(F)83(or)-423(an)28(y)-423(zero)-424(diagon)1(al)]TJ 0 -13.549 Td [(en)28(try)-275(of)-275(the)-275(matrix)-275(to)-276(b)-27(e)-276(p)1(rec)-1(on)1(ditioned,)-287(the)-275(cor-)]TJ 0 -13.55 Td [(resp)-28(onding)-326(en)28(try)-327(of)-326(he)-327(preconditioner)-326(is)-327(set)-327(to)-326(1.)]TJ +/F15 10.9091 Tf 263.982 609.608 Td [(Diagonal)-423(preconditioner.)-714(F)83(or)-423(an)28(y)-423(zero)-424(diagon)1(al)]TJ 0 -13.549 Td [(en)28(try)-275(of)-275(the)-275(matrix)-275(to)-276(b)-27(e)-276(p)1(rec)-1(on)1(ditioned,)-287(the)-275(cor-)]TJ 0 -13.55 Td [(resp)-28(onding)-278(en)28(try)-278(of)-278(the)-278(pr)1(e)-1(cond)1(itioner)-278(is)-278(set)-278(to)-278(1.)]TJ ET q 1 0 0 1 502.4 578.445 cm @@ -2405,7 +2426,7 @@ ET endstream endobj -409 0 obj +407 0 obj << /Length 7782 >> @@ -2420,7 +2441,7 @@ BT 0 g 0 G [(\051.)-518(A)-358(sample)-358(tes)-1(t)]TJ -101.269 -13.549 Td [(problem)-375(along)-374(with)-375(the)-375(relev)56(an)28(t)-375(input)-374(data)-375(is)-375(a)28(v)56(ailable)-375(in)]TJ/F44 10.9091 Tf 283.422 0 Td [(examples/fileread/runs)]TJ/F15 10.9091 Tf 125.999 0 Td [(.)]TJ -409.421 -13.549 Td [(F)83(or)-333(details)-333(on)-334(th)1(e)-334(use)-333(of)-334(th)1(e)-334(PSBLAS)-333(routines,)-333(see)-334(the)-333(PSBLAS)-333(User's)-334(Guide)-333([)]TJ 1 0 0 rg 1 0 0 RG - [(17)]TJ + [(13)]TJ 0 g 0 G [(].)]TJ 16.936 -13.55 Td [(The)-497(setup)-496(and)-496(application)-497(of)-496(the)-496(default)-497(m)28(ulti-lev)28(el)-497(preconditioner)-496(for)-496(the)-497(real)]TJ -16.936 -13.549 Td [(single)-253(precision)-253(and)-253(the)-253(complex,)-269(single)-253(and)-253(double)-253(precision,)-269(v)27(ersions)-253(are)-253(obtained)-253(with)]TJ 0 -13.549 Td [(straigh)28(tforw)28(ard)-229(mo)-28(di\014cations)-229(of)-229(the)-228(previous)-229(example)-229(\050see)-229(Section)]TJ 0 0 1 rg 0 0 1 RG @@ -2457,9 +2478,9 @@ ET endstream endobj -417 0 obj +415 0 obj << -/Length 3327 +/Length 3319 >> stream 0 g 0 G @@ -2474,7 +2495,7 @@ BT 0 g 0 G 0 g 0 G 0 g 0 G -/F44 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(multi-level)-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 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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/F15 10.9091 Tf -31.085 -21.354 Td [(Figure)-331(2:)-443(setup)-331(and)-331(application)-331(of)-331(the)-331(default)-331(m)28(ulti-lev)28(el)-331(preconditioner)-331(\050example)-331(1\051.)]TJ +/F44 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(multi-level)-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/F15 10.9091 Tf -31.085 -21.354 Td [(Figure)-331(2:)-443(setup)-331(and)-331(application)-331(of)-331(the)-331(default)-331(m)28(ulti-lev)28(el)-331(preconditioner)-331(\050example)-331(1\051.)]TJ 0 g 0 G 0 g 0 G 0 g 0 G @@ -2482,9 +2503,9 @@ ET endstream endobj -422 0 obj +420 0 obj << -/Length 3309 +/Length 3291 >> stream 0 g 0 G @@ -2496,7 +2517,7 @@ BT 0 g 0 G 0 g 0 G 0 g 0 G -/F44 9.9626 Tf -182.641 -45.351 Td [(...)-525(...)]TJ 0 -11.956 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 -11.955 Td [(call)-525(P%init\050P,'ML',info\051)]TJ 0 -11.955 Td [(call_P%set\050P,'SMOOTHER_TYPE','BJAC',info\051)]TJ 0 -11.955 Td [(call)-525(P%set\050P,'COARSE_SOLVE','BJAC',info\051)]TJ 0 -11.956 Td [(call)-525(P%set\050P,'COARSE_SWEEPS',8,info\051)]TJ 0 -11.955 Td [(call)-525(P%hierarchy_build\050A,desc_A,P,info\051)]TJ 0 -11.955 Td [(call)-525(P%smoothers_build\050A,desc_A,P,info\051)]TJ -10.46 -11.955 Td [(...)-525(...)]TJ +/F44 9.9626 Tf -182.641 -45.351 Td [(...)-525(...)]TJ 0 -11.956 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 -11.955 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 -11.956 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 -11.955 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 75.298 -18.535 Td [(Figure)-333(3:)-445(setup)-333(of)-333(a)-334(m)28(ulti-lev)28(el)-333(preconditioner)]TJ 0 g 0 G @@ -2506,7 +2527,7 @@ BT 0 g 0 G 0 g 0 G 0 g 0 G -/F44 9.9626 Tf -75.298 -61.69 Td [(...)-525(...)]TJ 0 -11.956 Td 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:-uv=Cw-zgH\%v(TX <8qiWy^O'U !gޢؐ&7gδlM\p.e.YSA<S>iNИ ,Z~gyS!WW#!: <8 zi+ڇ߂q<ױ'q LҒ&̑hȁA%o'W<<ָ.O&XfQoK *o վ"Sغ-so ,*xE+EU^[4Ka +2}&[$f *Ґ ѭV`Ag#K(Y_>Z_\f^S ojtPUރγiEPCswNSz܇$K4 )P#z1xTZR \?0 IQEi<ׂ2Yqv;@ҁ.'PQG= / +hNP`z< a'q U+~. :5,ߘ$LF2kLLSwVetũk/s1ߧ~,J=j8JpIajo.[r,YN@:m*C8#5f8ʯl rQgcC,N T[!Њ}뫀DUwwSu3rH vT +66 a,v{,/TbyEC& ֣toq)A5RR1DucݣXF:-t[řLgڦ4'ϼՋ*YdhY4"לoci/Af3TŒos?Ɯmv1,J\R=pV082K5]MZD&b I"%oST l%9:R%^%<ÙyH@Εe'sͩ%_T6j<+'pa4-}mQ-Rk)&pN>Rw6?V Ⱥ*/Eg޷xJ=d +?]<ˇʡsYCG@U#iCZ<4ʾaE'&1_ʫ+ | *xY˱3KDyf%=-\3V`.̢,(ob6e +0TF p=s@KJd >-g7Qٳ(f昂* ->/NC*J4\\?3՟,?p2VJʝSW} |b,1}b-^/gɤ)uF4sHu#8{ļ,z٪\J÷ .4?ao *gA*#^+YwWyF(T[yvP +Qc$Q lϩgoj#͡!ȃJ0Cw{&'H4D +f-R'"SGCed|ȇ5K 1, Vҡ6@h({ 5˟ޝ endstream endobj -609 0 obj +597 0 obj << -/Length1 1512 -/Length2 6732 +/Length1 1539 +/Length2 6948 /Length3 0 -/Length 8244 +/Length 8487 >> stream %!PS-AdobeFont-1.0: CMEX10 003.002 @@ -7057,7 +6847,7 @@ FontDirectory/CMEX10 known{/CMEX10 findfont dup/UniqueID known{dup 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def -/FontName /FQXIBK+CMEX10 def +/FontName /YJJMFD+CMEX10 def /FontBBox {-24 -2960 1454 772 }readonly def /PaintType 0 def /FontInfo 9 dict dup begin @@ -7074,6 +6864,7 @@ end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 26 /braceleftbigg put +dup 27 /bracerightbigg put dup 0 /parenleftbig put dup 1 /parenrightbig put dup 113 /radicalBig put @@ -7104,20 +6895,19 @@ WVV 5 ~7dh֫]V1Ԅ2D(W[/tG W ( zK֍3Z!>LWaF;w0cA܂LcEa; *DG5HX&sw*;s(#PBپ hoOt%J u.:ۃ &essE3a̕*Pv*k]5D6?x3{%ЯEiVjӨPNx[ ЩnP1#WvA8޿6dzLtGb5Pw,.#gF:4:uBV@aO2zn! `pgTwaG7= bzsФY@}0{9B6$ʗԧЋ>n4=#|h~ޕ4R_= # I쏿p1u@{i\_>2~XՀL-4n΄,ytQ - LrtKE {VSL>z ۭBLLQfq>>4+[ ( 7 9>0'mly?Wq;w(sebl?tʜ mxĔzvy\l] βEѩ +`'-LҖg5b@KJH5β -[,s)s!tw›HR7(~ ߉)iy-EG+-8fE/>Q^i;7}"t^i)Ĭy`pTN `yeNl,65hCgQߠ0a]%jr㓀s4f?uN+msYř:T6CtO!t'5z|t?%ҽ"!Kr,j3$NZǨKxub>A}%ݤt.QH6hZn+WIlQzu _z V֨r &"&oA)4A;Y8HK#?xzz=c_v^37@7Kz]pq}i8mrB=k*o2h)d72$,Xs/ -:hzDI>~iYSh$Y-r/ʅD^f$4|OQmZi&(,fXҘߏ085?U;G>6.jր~1vnd^B!KgIbsPRt -Յt7dO -TI3[Q|BNGށ~qF,;-Z뷊[U>ux.˅$uTQeCm0lK #.zY6*k>)4<ꞓ%UǤN*DG5HX&sw*;s(#PBپ hoOt%J u.:ۃ &essE3a̕*Pv*k]5D6?x3{%ЯEiVjӨPNx[ ЩnP1#WvA8޿6dzLtGb5Pw,.#gF:4:uBV@aO2zn! `pgTwaG7= bzsФY@}0{9B6$ʗԧЋ>n4=#|h~ޕ4R_= zY,97jn9Vp 6 :Ŋ_ GdLu l;>[Lמk'Dtp\XL>Dgb\/1}E(Íy Ma2rQCZԖ:g:=%Wgve󬓱: "ICqighʕicK1Tv͡OiuI;뜿W3;1t$HSVӪX1T~Bh] +*` ;1^`WǛ``Itb7'[zUYc +ܣN)V~Y54- mc a,j󊩰uR iF|_"KBͤנ>Sͯ\]oN݈?%$щA?- ڑ+"ͅnI-ht_MM?9-?e=?3=jYmҰC28qTLpXyl5ƒmzwa1DCg2;[E6s0t|D;-yKA3Q\}4 ;/}{>3"^0Z{>.N3aN"CB?dټ_aj%W",Y*8uu_;řü,| _Nқ4޵8QYgE +)tCdR 8ߊN:2Lyo(9*u]s`r_=ڈ`M~iao/P8'WsK$dl:3 ?4n?Q*OB>HLC7jf:uY8$64{"ff8oCl$G)«"Imto)q" |a2a?1j|_Z=Cd(P [uY7$)ƋsGPZK5PIRmv/z2n!*iF[^xTX&+E?~F&myN]=_ޘu0yrqe,"C#K P >'O=A}w>jYӪ.r!QDÃ1\1Z͝}R*'D!0טv96Y]u"9O(c +KdN늦)^bI|_|Kż7mf:="<dڄ`Pc9$++wgD' endstream endobj -611 0 obj +599 0 obj << -/Length1 1931 -/Length2 12907 +/Length1 1946 +/Length2 13085 /Length3 0 -/Length 14838 +/Length 15031 >> stream %!PS-AdobeFont-1.0: CMMI10 003.002 @@ -7137,7 +6927,7 @@ FontDirectory/CMMI10 known{/CMMI10 findfont dup/UniqueID known{dup 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def -/FontName /LRYROC+CMMI10 def +/FontName /VFYBNB+CMMI10 def /FontBBox {-32 -250 1048 750 }readonly def /PaintType 0 def /FontInfo 10 dict dup begin @@ -7180,6 +6970,7 @@ dup 111 /o put dup 33 /omega put dup 112 /p put dup 58 /period put +dup 114 /r put dup 26 /rho put dup 61 /slash put dup 116 /t put @@ -7219,45 +7010,42 @@ j Xw5/"|k(e$/JG 0Mo#hpT%-R(>* Ҩ25h|I.5u`QPւ)`QQǶ(45^ބM߯o5G*`?RFM Àgpfn51dK[L yq B$@'6!W ΐEI6,IT$yهHpiVch]l&ߡF:O$#Ȋ!D:%)aMx"_L?d*:02]7x(4p P8*.v5sb<]LQ0*K+@>e.-oӏCƀl\~R+$voDgya|n)>1_|hvD5"Pt#'徹.GESJW˕̪N AZr!o"XF-)$^Y|i RA5=OgCna8z*2e6]Ԃpc'S\I |,qnL %HTz88Wqs.op ^nCz\M@ς+<.(]wvڐ($hyB[٣'&k~Q 7ۣ%R*>t[ a\ -^{*yi|rdI'8 v#8sP67 is5ajABl нaxQB)߄<2\eg8kvƥa83m!f-x$5behRFۃiDPW^NS2eGaCG -RI%]O[>O*/|u5Tyܫ)1F!d*1Inb,pc뮬f$t+6Fkv9%'y+Qņ8$DH*1#JVOR$@۬~Z02O@8o߮pRxO}# SBB&%YƔxm' (U' K5_<{֐ -`|2&4\uܪL9 -03FaU5 ql2u 0<Ϛ'|Ry uX+kyDחFVWY:i|ctb } '>4W%u{%B -1c0Msuky-8Ҽ@]޲P\ڳͥ>ŕ Eas~ N=_S3;R53؎!A>{Mt'-,"U~ør%`MChEb{0a ~ҕْȐ;+U d=V~iϯTqYt 9'gv0\xcS:m_e]u֭0}Veҕh("! e݃wIt#uHtڣL٘qSc8\ BbMW)dufQMqWڷЎDfݨZ`TGJa3T+q]"bɛȻJETuwy*Gsrm{{-#!tm:iT; -VyTmI2;@MT!͕V#5|b%49a?ic9vkPGZ1ݼ|E2]wRek*l[ZiG0 UH +YXͱp@"+v -zoA - oɠK wIx>\VVy=j^uGN$[4LAɪq,_ -޻h;v@f - Fi C +@&/8CBa2]g*hT_P=4=[3ʉ<"Ss vbK^r$07”/*VaC("3+WSTJں;G.g.`j?Ӭ6,꬀LlSRG?KHhE+j".D.+n"j F(`5+G(yН44Nmz`϶= --\+sHA\: U#@4V]{GӒ v CT'9T(;ϜYыˡZq\$Ryд3ղC3,NG#7^YU&O^8bBqJ4N :@V$ 4بljfJG %ʔ)D43hVTdYH,覑#fAf]ԍ-_ gڄL0~BOUfk5.шm9-%cE%(YEEK! -_'+Zc3[,_=RzfNPlĂʮZnն R^ "(`cZ&szYHbFzqtz3ITBOEJDԄ6E jMfqnd<3'ϐS/mi޴[`,ywTКq΍Fx}/ΠD~h'R*^eZd BؘqL4tvp8Kbk2Ps)e`^Qj˱˥L? 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ZlMTX| ϰR1m%Z LX5(-NEHyˢՑՂu)D F:F*m"az|}tA]He xTBp)L gRpfY2RȳkYd`Sܜy5T:@^ELrʻNXK +QMp&TRKUó.c݈U"q*|2A='ks?[ts?^~Ľs̀,p(vϤCv:\d*1!1ﺞ-[98U}bW’X`;Ea'`?gsg?A\1%f`vnͺgA }avošo!),鰬r )mGA?zAӀow #qz2kO}&o/q̷dg'vSo\C@'6C/'vNm*1G(-x1tO%,м'~uMe jy강Nzlqz F"'(P}am"ӳܿId~H8!%FL#;]n<]%tX(5f*f4VէBa,XtV]Wc?hk(>JPV(5ĵvLN)*5~,i>( @}z)\! (Y\(8dвPPg^ +f)C?|a:zGNl'}z[pOq:M)Le$]QMXSiE^RsZ GA9E9m4u-6 e%}9M7~?Oݗ{9&;_1%[<~`a6߻U35$_XکO~^苨zuQ}.\vz0|"*(;,l< ;^u0þ>* {hE"Heh@rj*ubX +gT.~|v 4Bp$^,%k諸5d Nb #lbY)ȍMN<>džDcKWa׏@*)ʎ{ጌ)Ʉ#3i{Iw Q $vp\p8"PY?j!Sdž*LjkL:y&dp4rж:BTa)񁦜mmz"Sw*|gJYa_ي@ٷ~gRxzyG#+qz,ErA<| }G) +XטּX & m+[WNn +\1>=A)JL罈fpDVZ]|s1H-'&`b m sW W'B71n +nP{Wc `@_P@WU]ns^_d[ +,I8Zo (t5Mi ѩPpUq-;GwLTkz(ZKaΑnk)a|_Q"cB? &cG8|6`Yك_$P \\E<Իʋٹ]Ԏ 5jGJDos$c(-|h}^ôH7^*vEpX#CHBFԬh./4|%@lhyK7)o +EmPiџPuwq0Ikfa\څWŹD<)+ﳌY-S%dZcy$i;?||Pԋɘ$r=djQdCuwCɻF Tk*$QG>B_/ 4JʙVv۲7QrBwc_H&zR(Eqf\ @g'ӃsOQG]THdcR?x0oT"cVR4uaf!Xo)baR7eg8v@Lp΂Z &Nu~[O+O 7{١? 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MLD2P4. -\subsection{Optional third party libraries\label{sec:third_party}} +\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