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133 lines
6.6 KiB
TeX
133 lines
6.6 KiB
TeX
\section{Introduction}
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The PSBLAS library, developed with the aim to facilitate the
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parallelization of computationally intensive scientific applications,
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is designed to address parallel implementation of iterative solvers
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for sparse linear systems through the distributed memory paradigm. It
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includes routines for multiplying sparse matrices by dense matrices,
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solving block diagonal systems with triangular diagonal entries,
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preprocessing sparse matrices, and contains additional routines for
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dense matrix operations. The current implementation of PSBLAS
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addresses a distributed memory execution model operating with message
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passing. However, the overall design does not preclude different
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implementation paradigms, such as those based on a shared memory
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model.
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The PSBLAS library is internally implemented in a mixture of
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Fortran~77 and Fortran~95~\cite{metcalf} programming languages. A
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similar approach has been advocated by a number of authors,
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e.g.~\cite{machiels}. Moreover, the Fortran~95 facilities for dynamic
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memory management and interface overloading greatly enhance the usability of the PSBLAS
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subroutines. In this way, the library can take care of runtime memory
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requirements that are quite difficult or even impossible to predict at
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implementation or compilation time. The following presentation of the
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PSBLAS library follows the general structure of the proposal for
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serial Sparse BLAS~\cite{sblas97}, which in its turn is based on the
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proposal for BLAS on dense matrices~\cite{BLAS1,BLAS2,BLAS3}.
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The applicability of sparse iterative solvers to many different areas
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causes some terminology problems because the same concept may be
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denoted through different names depending on the application area. The
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PSBLAS features presented in this section will be discussed mainly in terms of finite
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difference discretizations of Partial Differential Equations (PDEs).
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However, the scope of the library is wider than that: for example, it
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can be applied to finite element discretizations of PDEs, and even to
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different classes of problems such as nonlinear optimization, for
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example in optimal control problems.
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The design of a solver for sparse linear systems is driven by many
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conflicting objectives, such as limiting occupation of storage
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resources, exploiting regularities in the input data, exploiting
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hardware characteristics of the parallel platform. To achieve an
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optimal communication to computation ratio on distributed memory
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machines it is essential to keep the {\em data locality} as high as
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possible; this can be done through an appropriate data allocation
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strategy. The choice of the preconditioner is another very important
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factor that affects efficiency of the implemented application. Optimal
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data distribution requirements for a given preconditioner may conflict
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with distribution requirements of the rest of the solver. Finding the
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optimal trade-off may be very difficult because it is application
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dependent. Possible solution to these problems and other important
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inputs to the development of the PSBLAS software package has come from
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an established experience in applying the PSBLAS solvers to
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computational fluid dynamics applications.
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\section{General overview}
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\label{sec:overview}
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The PSBLAS library is designed to handle the implementation of
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iterative solvers for sparse linear systems on distributed memory
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parallel computers. The system coefficient matrix $A$ must be square;
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it may be real or complex, nonsymmetric, and its sparsity pattern
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needs not to be symmetric. The serial computation parts are based on
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the serial sparse BLAS, so that any extension made to the data
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structures of the serial kernels is available to the parallel
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version. The overall design and parallelization strategy have been
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influenced by the structure of the ScaLAPACK parallel
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library~\cite{scalapack}. The layered structure of the PSBLAS library
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is shown in figure~\ref{fig:psblas} ; lower layers of the library
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indicate an encapsulation relationship with upper layers. The ongoing
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discussion focuses on the Fortran~95 layer immediately below the
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application layer; two examples of iterative solvers built through the
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PSBLAS routines, will be also given in Section~\ref{sec:itmethd}. The
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serial parts of the computation on each process are executed through
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calls to the serial sparse BLAS subroutines. In a similar way, the
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inter-process message exchanges are implemented through the Basic
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Linear Algebra Communication Subroutines (BLACS) library~\cite{BLACS}
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that guarantees a portable and efficient communication layer. The
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Message Passing Interface code is encapsulated within the BLACS
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layer. However, in some cases, MPI routines are directly used either
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to improve efficiency or to implement communication patterns for which
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the BLACS package doesn't provide any method.
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\begin{figure}[h] \begin{center}
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\includegraphics[scale=0.45]{figures/psblas}
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\end{center}
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\caption{PSBLAS library components hierarchy.\label{fig:psblas}}
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\end{figure}
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The PSBLAS library consists of two classes of subroutines that is, the
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{\em computational routines} and the {\em auxiliary routines}. The
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computational routine set includes:
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\begin{itemize}
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\item Sparse matrix by dense matrix product; \item Sparse triangular
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systems solution for block diagonal matrices;
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\item Vector and matrix norms;
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\item Dense matrix sums;
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\item Dot products.
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\end{itemize}
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The auxiliary routine set includes:
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\begin{itemize}
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\item Communication descriptors allocation;
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\item Dense and sparse matrix allocation;
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\item Dense and sparse matrix build and update;
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\item Sparse matrix and data distribution preprocessing.
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\end{itemize}
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The following naming scheme has been adopted for all the symbols
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internally defined in the PSBLAS software package:
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\begin{itemize}
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\item all the symbols (i.e. subroutine names, data types...) are
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prefixed by \verb|psb_|
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\item all the data type names are suffixed by \verb|_type|
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\item all the constant values are suffixed by \verb|_|
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\item all the subroutine names follow the rule \verb|psb_xxname| where
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\verb|xx| can be either:
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\begin{itemize}
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\item \verb|ds|: the routine is related to dense data,
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\item \verb|sp|: the routine is related to sparse data,
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\item \verb|cd|: the routine is related to communication descriptor (see~\ref{sec:datastruct}).
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\end{itemize}
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For example the \verb|psb_dsins|, \verb|psb_spins| and
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\verb|psb_cdins| perform the same action (see~\ref{sec:toolsrout}) on
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dense matrices, sparse matrices and communication descriptors
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respectively.
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Interface overloading allows the usage of the same subroutine
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interfaces for both real and complex data.
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\end{itemize}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "userguide"
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%%% End:
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