based on PSBLAS}) is a package of parallel algebraic multilevel preconditioners.
based on PSBLAS}) is a package of parallel algebraic multilevel preconditioners included in the PSCToolkit (Parallel Sparse Computation Toolkit) software framework.
The first release of MLD2P4 made available multilevel additive and hybrid Schwarz
It is a progress of a software development project started in 2007, named MLD2P4, which implemented a multilevel version of some domain decomposition preconditioners of additive-Schwarz type and was based on a parallel decoupled version of the well known smoothed
preconditioners, as well as one-level additive Schwarz preconditioners. The package
aggregation method to generate the multilevel hierarchy of coarser matrices. In the last years, within the context of the EU-H2020 EoCoE project (Energy Oriented Center of Excellence), the package was extended including new algorithms and functionalities for setup and application of new AMG preconditioners with the final aims of improving efficiency and scalability when tens of thousands cores are
has been extended to include further multilevel cycles and smoothers widely used in
used and of boosting reliability in dealing with general symmetric positive definite linear systems. Due to the significant number of changes and the increase in scope, we decided to rename the package as AMG4PSBLAS.
multigrid methods. In the multilevel case, a purely algebraic approach is applied to
generate coarse-level corrections, so that no geometric background is needed
concerning the matrix to be preconditioned. The matrix is assumed to be square,
real or complex.
MLD2P4 has been designed to provide scalable and easy-to-use preconditioners
AMG4PSBLAS has been designed to provide scalable and easy-to-use preconditioners
in the context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms)
in the context of the PSBLAS (Parallel Sparse Basic Linear Algebra Subprograms)
computational framework and can be used in conjuction with the Krylov solvers
computational framework and can be used in conjuction with the Krylov solvers
available in this framework. MLD2P4 enables the user to easily specify different
available in this framework.
features of an algebraic multilevel preconditioner, thus allowing to search
Our package is based on a completely algebraic approach and users level interfaces
for the ``best'' preconditioner for the problem at hand.
assume that the system matrix and preconditioners are represented as PSBLAS
distributed sparse matrices.
AMG4PSBLAS enables the user to easily specify different
features of an algebraic multilevel preconditioner, thus allowing to experiment
with different preconditioners for the problem and parallel computers at hand.
The package employs object-oriented design techniques in
The package employs object-oriented design techniques in
Fortran~2003, with interfaces to additional third party libraries
Fortran~2003, with interfaces to additional third party libraries
@ -27,4 +27,4 @@ paradigm; the inter-process communication is based on MPI and
is managed mainly through PSBLAS.
is managed mainly through PSBLAS.
This guide provides a brief description of the functionalities and
This guide provides a brief description of the functionalities and
to be used in the iterative solution of linear systems,
to be used in the iterative solution of linear systems,
\begin{equation}
\begin{equation}
Ax=b,
Ax=b,
\label{system1}
\label{system1}
\end{equation}
\end{equation}
where $A$ is a square, real or complex, sparse matrix.
where $A$ is a square, real or complex, sparse symmetric positive definite (s.p.d) matrix.
%
%
%\textbf{NOTA: Caso non simmetrico, aggregazione con $(A+A^T)$ fatta!
%\textbf{NOTA: Caso non simmetrico, aggregazione con $(A+A^T)$ fatta!
%Dovremmo implementare uno smoothed prolongator
%Dovremmo implementare uno smoothed prolongator
%adeguato e fare qualcosa di consistente anche con 1-lev Schwarz.}
%adeguato e fare qualcosa di consistente anche con 1-lev Schwarz.}
%
%
The name of the package comes from its original implementation, containing
multilevel additive and hybrid Schwarz preconditioners, as well as one-level additive
Schwarz preconditioners. The current version extends the original plan by including
multilevel cycles and smoothers widely used in multigrid methods.
The multilevel preconditioners implemented in MLD2P4 are obtained by combining
The preconditioners implemented in AMG4PSBLAS are obtained by combining
AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and
3 different types of AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and a version of a Krylov-type cycle (K-cycle)~\cite{Briggs2000,Notay2008} are available, which can be combined with weighted versions of Jacobi, hybrid
K-cycles~\cite{Briggs2000,Notay2008} are available, which allow to define
almost all the preconditioners in the package, including the multilevel hybrid
Schwarz ones; a specific cycle is implemented to obtain multilevel additive
Schwarz preconditioners. The Jacobi, hybrid
%\footnote{see Note 2 in Table~\ref{tab:p_coarse}, p.~28.}
%\footnote{see Note 2 in Table~\ref{tab:p_coarse}, p.~28.}
forward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz methods
forward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz smoothers. An algebraic approach is used to generate a hierarchy of
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
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,
geometry of the original problem, e.g., the discretization of a PDE. To this end,
the smoothed aggregation technique~\cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}
two different coarsening strategies, based on aggregation, are available:
is applied. Either exact or approximate solvers can be used on the coarsest-level
\begin{itemize}
system. Specifically, different sparse LU factorizations from external
\item a decoupled version of the well known smoothed aggregation procedure proposed in~\cite{BREZINA_VANEK,VANEK_MANDEL_BREZINA}, and already included in the previous versions of the package~\cite{BDDF2007,MLD2P4_TOMS};
packages, and native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel,
\item the first parallel implementation of a coupled version of Coarsening based on Compatible Weighted Matching introduced in~\cite{DV2013,DFV2018} and described in details in~\cite{DDF2020};
and block-Jacobi solvers are available. All smoothers can be also exploited as one-level
\end{itemize}
Either exact or approximate solvers can be used on the coarsest-level system. Specifically, different sparse LU factorizations from external
packages, native incomplete LU factorizations, weighted Jacobi, hybrid Gauss-Seidel,
and block-Jacobi solvers are available. All the smoothers can be also exploited as one-level
preconditioners.
preconditioners.
MLD2P4 is written in Fortran~2003, following an
AMG4PSBLAS is written in Fortran~2003, following an
object-oriented design through the exploitation of features
object-oriented design through the exploitation of features
such as abstract data type creation, type extension, functional overloading, and
such as abstract data type creation, type extension, functional overloading, and
dynamic memory management. %\textbf{Va bene cos\'{i} o \`e meglio
dynamic memory management.
% fare riferimento alle classi?}
The parallel implementation is based on a Single Program Multiple Data
The parallel implementation is based on a Single Program Multiple Data
(SPMD) paradigm. Single and
(SPMD) paradigm. Single and
double precision implementations of MLD2P4 are available for both the
double precision implementations of AMG4PSBLAS are available for both the
real and the complex case, which can be used through a single
real and the complex case, which can be used through a single
interface.
interface.
MLD2P4 has been designed to implement scalable and easy-to-use
AMG4PSBLAS has been designed to implement scalable and easy-to-use
multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS)
multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS)
computational framework~\cite{psblas_00,PSBLAS3}. PSBLAS provides basic linear algebra
computational framework~\cite{psblas_00,PSBLAS3}. PSBLAS provides basic linear algebra
operators and data management facilities for distributed sparse matrices,
operators and data management facilities for distributed sparse matrices,
as well as parallel Krylov solvers which can be used with the MLD2P4 preconditioners.
as well as parallel Krylov solvers which can be used with the AMG4PSBLAS preconditioners.
The choice of PSBLAS has been mainly motivated by the need of having
The choice of PSBLAS has been mainly motivated by the need of having
a portable and efficient software infrastructure implementing ``de facto'' standard
a portable and efficient software infrastructure implementing ``de facto'' standard
parallel sparse linear algebra kernels, to pursue goals such as performance,
parallel sparse linear algebra kernels, to pursue goals such as performance,
portability, modularity ed extensibility in the development of the preconditioner
portability, modularity ed extensibility in the development of the preconditioner
package. On the other hand, the implementation of MLD2P4 has led to some
package. On the other hand, the implementation of AMG4PSBLAS, which was driven by the need to face the exascale challenge, has led to some important revisions and extentions of the PSBLAS infrastructure.
revisions and extentions of the original PSBLAS kernels.
The inter-process comunication required by AMG4PSBLAS is encapsulated
The inter-process comunication required by MLD2P4 is encapsulated
in the PSBLAS routines;
in the PSBLAS routines;
% , except few cases where MPI~\cite{MPI1} is explicitly called.
therefore, AMG4PSBLAS can be run on any parallel machine where PSBLAS
therefore, MLD2P4 can be run on any parallel machine where PSBLAS
implementations are available.
implementations are available.
MLD2P4 has a layered and modular software architecture where three main layers can be
AMG4PSBLAS has a layered and modular software architecture where three main layers can be
identified. The lower layer consists of the PSBLAS kernels, the middle one implements
identified. The lower layer consists of the PSBLAS kernels, the middle one implements
the construction and application phases of the preconditioners, and the upper one
the construction and application phases of the preconditioners, and the upper one
provides a uniform interface to all the preconditioners.
provides a uniform interface to all the preconditioners.
This architecture allows for different levels of use of the package:
This architecture allows for different levels of use of the package:
few black-box routines at the upper layer allow all users to easily
few black-box routines at the upper layer allow all users to easily
build and apply any preconditioner available in MLD2P4;
build and apply any preconditioner available in AMG4PSBLAS;
facilities are also available allowing expert users to extend the set of smoothers
facilities are also available allowing expert users to extend the set of smoothers
and solvers for building new versions of the preconditioners (see
and solvers for building new versions of the preconditioners (see
Section~\ref{sec:adding}).
Section~\ref{sec:adding}).
We note that the user interface of MLD2P4 2.1 has been extended with respect to the
previous versions in order to separate the construction of the multilevel hierarchy from
the construction of the smoothers and solvers, and to allow for more flexibility
at each level. The software architecture described in~\cite{MLD2P4_TOMS} has significantly
evolved too, in order to fully exploit the Fortran~2003 features implemented in PSBLAS 3.
However, compatibility with previous versions has been preserved.
This guide is organized as follows. General information on the distribution of the source
This guide is organized as follows. General information on the distribution of the source
code is reported in Section~\ref{sec:distribution}, while details on the configuration
code is reported in Section~\ref{sec:distribution}, while details on the configuration
and installation of the package are given in Section~\ref{sec:building}. A short description
and installation of the package are given in Section~\ref{sec:building}. The basics for building and applying the
of the preconditioners implemented in MLD2P4 is provided in Section~\ref{sec:background},
to help the users in choosing among them. The basics for building and applying the
preconditioners with the Krylov solvers implemented in PSBLAS are reported
preconditioners with the Krylov solvers implemented in PSBLAS are reported
in~Section~\ref{sec:started}, where the Fortran codes of a few sample programs
in~Section~\ref{sec:started}, where the Fortran codes of a few sample programs
are also shown. A reference guide for the user interface routines is provided
are also shown. A reference guide for the user interface routines is provided
@ -97,7 +79,7 @@ in Section~\ref{sec:userinterface}. Information on the extension of the package
through the addition of new smoothers and solvers is reported in Section~\ref{sec:adding}.
through the addition of new smoothers and solvers is reported in Section~\ref{sec:adding}.
The error handling mechanism used by the package
The error handling mechanism used by the package
is briefly described in Section~\ref{sec:errors}. The copyright terms concerning the
is briefly described in Section~\ref{sec:errors}. The copyright terms concerning the
distribution and modification of MLD2P4 are reported in Appendix~\ref{sec:license}.
distribution and modification of AMG4PSBLAS are reported in Appendix~\ref{sec:license}.
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