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<H2><A NAME="SECTION00033000000000000000">
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Application structure</A>
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</H2>
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<P>
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The main underlying principle of the PSBLAS library is that the
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library objects are created and exist with reference to a discretized
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space to which there corresponds an index space and a matrix sparsity
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pattern. As an example, consider a cell-centered finite-volume
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discretization of the Navier-Stokes equations on a simulation domain;
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the index space <IMG
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WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img14.png"
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ALT="$1\dots n$"> is isomorphic to the set of cell centers,
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whereas the pattern of the associated linear system matrix is
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isomorphic to the adjacency graph imposed on the discretization mesh
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by the discretization stencil.
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<P>
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Thus the first order of business is to establish an index space, and
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this is done with a call to <code>psb_cdall</code> in which we specify the
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size of the index space <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$n$"> and the allocation of the elements of the
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index space to the various processes making up the MPI (virtual)
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parallel machine.
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<P>
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The index space is partitioned among processes, and this creates a
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mapping from the ``global'' numbering <IMG
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WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img14.png"
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ALT="$1\dots n$"> to a numbering
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``local'' to each process; each process <IMG
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$i$"> will own a certain subset
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<!-- MATH
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$1\dots n_{\hbox{row}_i}$
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-->
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<IMG
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WIDTH="76" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img16.png"
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ALT="$1\dots n_{\hbox{row}_i}$">, each element of which corresponds to a certain
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element of <IMG
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WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img14.png"
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ALT="$1\dots n$">. The user does not set explicitly this mapping;
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when the application needs to indicate to which element of the index
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space a certain item is related, such as the row and column index of a
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matrix coefficient, it does so in the ``global'' numbering, and the
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library will translate into the appropriate ``local'' numbering.
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<P>
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For a given index space <IMG
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WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img14.png"
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ALT="$1\dots n$"> there are many possible associated
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topologies, i.e. many different discretization stencils; thus the
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description of the index space is not completed until the user has
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defined a sparsity pattern, either explicitly through <code>psb_cdins</code>
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or implicitly through <code>psb_spins</code>. The descriptor is finalized
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with a call to <code>psb_cdasb</code> and a sparse matrix with a call to
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<code>psb_spasb</code>. After <code>psb_cdasb</code> each process <IMG
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$i$"> will have
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defined a set of ``halo'' (or ``ghost'') indices
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<!-- MATH
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$n_{\hbox{row}_i}+1\dots n_{\hbox{col}_i}$
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-->
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<IMG
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WIDTH="131" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="$n_{\hbox{row}_i}+1\dots n_{\hbox{col}_i}$">, denoting elements of the index
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space that are <I>not</I> assigned to process <IMG
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WIDTH="9" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$i$">; however the
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variables associated with them are needed to complete computations
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associated with the sparse matrix <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1.png"
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ALT="$A$">, and thus they have to be
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fetched from (neighbouring) processes. The descriptor of the index
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space is built exactly for the purpose of properly sequencing the
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communication steps required to achieve this objective.
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<P>
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A simple application structure will walk through the index space
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allocation, matrix/vector creation and linear system solution as
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follows:
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<OL>
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<LI>Initialize parallel environment with <code>psb_init</code>
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</LI>
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<LI>Initialize index space with <code>psb_cdall</code>
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</LI>
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<LI>Allocate sparse matrix and dense vectors with <code>psb_spall</code>
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and <code>psb_geall</code>
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</LI>
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<LI>Loop over all local rows, generate matrix and vector entries,
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and insert them with <code>psb_spins</code> and <code>psb_geins</code>
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</LI>
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<LI>Assemble the various entities:
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<OL>
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<LI><code>psb_cdasb</code>
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</LI>
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<LI><code>psb_spasb</code>
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</LI>
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<LI><code>psb_geasb</code>
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</LI>
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</OL>
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</LI>
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<LI>Choose the preconditioner to be used with <code>psb_precset</code> and
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build it with <code>psb_precbld</code>
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</LI>
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<LI>Call the iterative method of choice, e.g. <code>psb_bicgstab</code>
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</LI>
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</OL>
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This is the structure of the sample program
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<code>test/pargen/ppde90.f90</code>.
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<P>
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For a simulation in which the same discretization mesh is used over
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multiple time steps, the following structure may be more appropriate:
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<OL>
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<LI>Initialize parallel environment with <code>psb_init</code>
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</LI>
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<LI>Initialize index space with <code>psb_cdall</code>
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</LI>
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<LI>Loop over the topology of the discretization mesh and build the
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descriptor with <code>psb_cdins</code>
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</LI>
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<LI>Assemble the descriptor with <code>psb_cdasb</code>
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</LI>
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<LI>Allocate the sparse matrices and dense vectors with
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<code>psb_spall</code> and <code>psb_geall</code>
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</LI>
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<LI>Loop over the time steps:
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<OL>
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<LI>If after first time step,
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reinitialize the sparse matrix with <code>psb_sprn</code>; also zero out
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the dense vectors;
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</LI>
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<LI>Loop over the mesh, generate the coefficients and insert/update
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them with <code>psb_spins</code> and <code>psb_geins</code>
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</LI>
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<LI>Assemble with <code>psb_spasb</code> and <code>psb_geasb</code>
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</LI>
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<LI>Choose and build preconditioner with <code>psb_precset</code> and
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<code>psb_precbld</code>
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</LI>
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<LI>Call the iterative method of choice, e.g. <code>psb_bicgstab</code>
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</LI>
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</OL>
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</LI>
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</OL>
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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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<P>
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In principle there is no specific order in the calls to
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<code>psb_spins</code>, nor is there a requirement to build a matrix row in
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its entirety before calling the routine; this allows the application
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programmer to walk through the discretization mesh element by element,
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generating the main part of a given matrix row but also contributions
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to the rows corresponding to neighbouring elements.
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<P>
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From a functional point of view it is even possible to execute one
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call for each nonzero coefficient; however this would have a
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substantial computational overhead. It is therefore advisable to pack
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a certain amount of data into each call to the insertion routine, say
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touching on a few tens of rows; the best performng value would depend
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on both the architecture of the computer being used and on the problem
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structure.
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At the opposite extreme, it would be possible to generate the entire
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part of a coefficient matrix residing on a process and pass it in a
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single call to <code>psb_spins</code>; this, however, would entail a
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doubling of memory occupation, and thus would be almost always far
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from optimal.
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<P>
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