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<H2><A NAME="SECTION00033000000000000000">
Application structure</A>
</H2>

<P>
The main underlying principle of the PSBLAS library is that the
library objects are created and exist with reference to a discretized
space to which there corresponds an index space and a matrix sparsity
pattern. As an example, consider a cell-centered finite-volume
discretization of  the Navier-Stokes equations on a simulation domain;
the index space <IMG
 WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img12.png"
 ALT="$1\dots n$"> is isomorphic to the set of cell centers,
whereas the pattern of the associated linear system matrix is
isomorphic to the adjacency graph imposed on the discretization mesh
by the discretization stencil. 

<P>
Thus the first order of business is to establish an index space, and
this is done with a call to  <code>psb_cdall</code> in which we specify the
size of the index space <IMG
 WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.png"
 ALT="$n$"> and the allocation of the elements of the
index space to the various processes making up the MPI (virtual)
parallel machine. 

<P>
The index space is partitioned among processes, and this creates a
mapping from the ``global'' numbering <IMG
 WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img12.png"
 ALT="$1\dots n$"> to a numbering
``local'' to each process; each process <IMG
 WIDTH="10" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img3.png"
 ALT="$i$"> will own a certain subset
<!-- MATH
 $1\dots n_{\hbox{row}_i}$
 -->
<IMG
 WIDTH="76" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img14.png"
 ALT="$1\dots n_{\hbox{row}_i}$">, each element of which corresponds to a certain
element of <IMG
 WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img12.png"
 ALT="$1\dots n$">. The user does not set explicitly this mapping;
when the application needs to indicate to which element of the index
space a certain item is related, such as the row and column index of a
matrix coefficient, it does so in the ``global'' numbering, and the
library will translate into the appropriate ``local'' numbering. 

<P>
For  a given index space <IMG
 WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img12.png"
 ALT="$1\dots n$"> there are many possible associated
topologies, i.e. many different discretization stencils; thus the
description of the index space is not completed until the user has
defined a sparsity pattern, either explicitly through <code>psb_cdins</code>
or implicitly through <code>psb_spins</code>. The descriptor is finalized
with a call to <code>psb_cdasb</code> and a sparse matrix with a call to
<code>psb_spasb</code>. After <code>psb_cdasb</code> each process <IMG
 WIDTH="10" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img3.png"
 ALT="$i$"> will have
defined a set of ``halo'' (or ``ghost'') indices
<!-- MATH
 $n_{\hbox{row}_i}+1\dots n_{\hbox{col}_i}$
 -->
<IMG
 WIDTH="131" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.png"
 ALT="$n_{\hbox{row}_i}+1\dots n_{\hbox{col}_i}$">, denoting elements of the index
space that are <I>not</I> assigned to process <IMG
 WIDTH="10" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img3.png"
 ALT="$i$">; however the
variables associated with them are needed to complete computations
associated with the sparse matrix <IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img1.png"
 ALT="$A$">, and thus they have to be
fetched from (neighbouring) processes. The descriptor of the index
space is built exactly for the purpose of properly sequencing the
communication steps required to achieve this objective. 

<P>
A simple application structure will walk through the index space
allocation, matrix/vector creation and linear system solution as
follows:

<OL>
<LI>Initialize parallel environment with <code>psb_init</code>
</LI>
<LI>Initialize index space with <code>psb_cdall</code>
</LI>
<LI>Allocate sparse matrix and dense vectors with <code>psb_spall</code>
  and <code>psb_geall</code>
</LI>
<LI>Loop over all local rows, generate matrix and vector entries,
  and insert them with <code>psb_spins</code> and <code>psb_geins</code>
</LI>
<LI>Assemble the various entities: 

<OL>
<LI><code>psb_cdasb</code>
</LI>
<LI><code>psb_spasb</code>
</LI>
<LI><code>psb_geasb</code>
</LI>
</OL>
</LI>
<LI>Choose the preconditioner to be used with <code>psb_precset</code> and
  build it with <code>psb_precbld</code>
</LI>
<LI>Call the iterative method of choice, e.g. <code>psb_bicgstab</code>
</LI>
</OL>
This is the structure of the sample program
<code>test/pargen/ppde90.f90</code>. 

<P>
For a simulation in which the same discretization mesh is used over
multiple time steps, the following structure may be more appropriate:

<OL>
<LI>Initialize parallel environment with <code>psb_init</code>
</LI>
<LI>Initialize index space with <code>psb_cdall</code>
</LI>
<LI>Loop over the topology of the discretization mesh and build the
  descriptor with <code>psb_cdins</code>
</LI>
<LI>Assemble the descriptor with <code>psb_cdasb</code>
</LI>
<LI>Allocate the sparse matrices and dense vectors with
  <code>psb_spall</code> and <code>psb_geall</code>
</LI>
<LI>Loop over the time steps: 

<OL>
<LI>If after first time step, 
  reinitialize the sparse matrix with <code>psb_sprn</code>; also zero out
  the dense vectors;
</LI>
<LI>Loop over the mesh, generate the coefficients and insert/update
  them with <code>psb_spins</code> and <code>psb_geins</code>
</LI>
<LI>Assemble with <code>psb_spasb</code> and <code>psb_geasb</code>
</LI>
<LI>Choose and build preconditioner with <code>psb_precset</code> and
  <code>psb_precbld</code>
</LI>
<LI>Call the iterative method of choice, e.g. <code>psb_bicgstab</code>
 
</LI>
</OL>
</LI>
</OL>
The insertion routines will be called as many times as needed; 
they only need to  be called on the data that is actually
allocated to the current process, i.e. each process generates its own
data. 

<P>
In principle there is no specific order in the calls to
<code>psb_spins</code>, nor is there a requirement to build a matrix row in
its entirety before calling the routine; this allows the application
programmer to walk through the discretization mesh element by element,
generating the main part of a given matrix row but also contributions
to the rows corresponding to neighbouring elements. 

<P>
From a functional point of view it is even possible to execute one
call for each nonzero coefficient; however this would have a
substantial computational overhead. It is therefore advisable to pack
a certain amount of data into each call to the insertion routine, say
touching on a few tens of rows; the best performng value would depend
on both the architecture of the computer being used and on the problem
structure. 
At the opposite extreme, it would be possible to generate the entire
part of a coefficient matrix residing on a process and pass it in a
single call to <code>psb_spins</code>; this, however, would entail a
doubling of memory occupation, and thus would be almost always far
from optimal. 

<P>
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