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@ -26,12 +26,12 @@ 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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PSBLAS features presented in this document will be discussed referring
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to a finite difference discretization of a Partial Differential
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Equation (PDE). However, the scope of the library is wider than
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that: for example, it can be applied to finite element discretizations
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of PDEs, and even to different classes of problems such as nonlinear
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optimization, for 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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@ -75,6 +75,9 @@ 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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In any case we provide wrappers around the BLACS routines so that the
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user does not need to delve into their details (see Sec.~\ref{sec:toolsrout}).
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%% We assume that the user program has initialized a BLACS process grid
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%% with one column and as many rows as there are processes; the PSBLAS
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%% initialization routines will take the communication context for this
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@ -86,6 +89,121 @@ the BLACS package doesn't provide any method.
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\caption{PSBLAS library components hierarchy.\label{fig:psblas}}
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\end{figure}
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The type of linear system matrices that we address typically arise in the
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numerical solution of PDEs; in such a context,
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it is necessary to pay special attention to the
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structure of the problem from which the application originates.
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The nonzero pattern of a matrix arising from the
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discretization of a PDE is influenced by various factors, such as the
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shape of the domain, the discretization strategy, and
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the equation/unknown ordering. The matrix itself can be interpreted as
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the adjacency matrix of the graph associated with the discretization
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mesh.
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The distribution of the coefficient matrix for the linear system is
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based on the ``owner computes'' rule:
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the variable associated to each mesh point is assigned to a process
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that will own the corresponding row in the coefficient matrix and
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will carry out all related computations. This allocation strategy
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is equivalent to a partition of the discretization mesh into {\em
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sub-domains}.
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Our library supports any distribution that keeps together
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the coefficients of each matrix row; there are no other constraints on
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the variable assignment.
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This choice is consistent with data distributions commonly used in
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ScaLAPACK such as \verb|CYCLIC(N)| and \verb|BLOCK|,
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as well as completely arbitrary assignments of
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equation indices to processes. In particular it is consistent with the
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usage of graph partitioning tools commonly available in the
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literature, e.g. METIS~\cite{METIS}.
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Dense vectors conform to sparse
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matrices, that is, the entries of a vector follow the same distribution
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of the matrix rows.
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We assume that the sparse matrix is built in parallel, where each
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process generates its own portion. We never require that the entire
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matrix be available on a single node. However, it is possible
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to hold the entire matrix in one process and distribute it
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explicitly\footnote{In our prototype implementation we provide
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sample scatter/gather routines.}, even though the resulting
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bottleneck would make this option unattractive in most cases.
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\subsection{Basic Nomenclature}
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Our computational model implies that the data allocation on the
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parallel distributed memory machine is guided by the structure of the
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physical model, and specifically by the discretization mesh of the
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PDE.
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Each point of the discretization mesh will have (at least) one
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associated equation/variable, and therefore one index. We say that
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point $i$ {\em depends\/} on point $j$ if the equation for a
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variable associated with $i$ contains a term in $j$, or equivalently
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if $a_{ij} \ne0$.
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After the partition of the discretization mesh into {\em sub-domains\/}
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assigned to the parallel processes,
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we classify the points of a given sub-domain as following.
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\begin{description}
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\item[Internal.] An internal point of
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a given domain {\em depends} only on points of the
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same domain.
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If all points of a domain are assigned to one
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process, then a computational step (e.g., a
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matrix-vector product) of the
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equations associated with the internal points requires no data
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items from other domains and no communications.
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\item[Boundary.] A point of
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a given domain is a boundary point if it {\em depends} on points
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belonging to other domains.
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\item[Halo.] A halo point for a given domain is a point belonging to
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another domain such that there is a boundary point which {\em depends\/}
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on it. Whenever performing a computational step, such as a
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matrix-vector product, the values associated with halo points are
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requested from other domains. A boundary point of a given
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domain is a halo point for (at least) another domain; therefore
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the cardinality of the boundary points set denotes the amount of data
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sent to other domains.
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\item[Overlap.] An overlap point is a boundary point assigned to
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multiple domains. Any operation that involves an overlap point
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has to be replicated for each assignment.
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\end{description}
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Overlap points do not usually exist in the basic data
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distribution, but they are a feature of Domain Decomposition
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Schwarz preconditioners which we are in the process of including in
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our distribution~\cite{PARA04,APNUM06}.
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We denote the sets of internal, boundary and halo points for a given
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subdomain by $\cal I$, $\cal B$ and $\cal H$.
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Each subdomain is assigned to one process; each process usually
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owns one subdomain, although the user may choose to assign more than
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one subdomain to a process. If each process $i$ owns one
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subdomain, the number of rows in the local sparse matrix is
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$|{\cal I}_i| + |{\cal B}_i|$, and the number of local columns
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(i.e. those for which there exists at least one non-zero entry in the
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local rows) is $|{\cal I}_i| + |{\cal B}_i| +|{\cal H}_i|$.
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\begin{figure}[h]
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\begin{center}
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\leavevmode
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\rotatebox{-90}{\includegraphics[scale=0.45]{figures/points}}
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\end{center}
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\caption{Point classfication.\label{fig:points}}
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\end{figure}
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This classification of mesh points guides the naming scheme that we
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adopted in the library internals and in the data structures. We
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explicitly note that ``Halo'' points are also often called ``ghost''
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points in the literature.
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\subsection{Library contents}
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The PSBLAS library consists of various classes of subroutines:
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\begin{description}
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\item[Computational routines] comprising:
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@ -231,8 +349,8 @@ multiple time steps, the following structure may be more appropriate:
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\item Call the iterative method of choice, e.g. \verb|psb_bicgstab|
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\end{enumerate}
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\end{enumerate}
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The insertion routines will be called as many times as needed; it is
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clear that they only need be called on the data that is actually
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The insertion routines will be called as many times as needed;
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they only need to be called on the data that is actually
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allocated to the current process, i.e. each process generates its own
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data.
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Salvatore Filippone
19 years ago