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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"http://www.w3.org/TR/html4/loose.dtd">
<html >
<head><title>Getting Started</title>
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<!--l. 1--><div class="crosslinks"><p class="noindent"><span
class="cmr-12">[</span><a
href="userhtmlse5.html" ><span
class="cmr-12">next</span></a><span
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href="userhtml.html#userhtmlse4.html" ><span
class="cmr-12">up</span></a><span
class="cmr-12">] </span></p></div>
<h3 class="sectionHead"><span class="titlemark"><span
class="cmr-12">4 </span></span> <a
id="x14-130004"></a><span
class="cmr-12">Getting Started</span></h3>
<!--l. 5--><p class="noindent" ><span
class="cmr-12">This section describes the basics for building and applying AMG4PSBLAS one-level</span>
<span
class="cmr-12">and multilevel (i.e., AMG) preconditioners with the Krylov solvers included in</span>
<span
class="cmr-12">PSBLAS</span><span
class="cmr-12">&#x00A0;</span><span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli5.html#XPSBLASGUIDE"><span
class="cmr-12">21</span></a><span
class="cmr-12">]</span></span><span
class="cmr-12">.</span>
<!--l. 9--><p class="indent" > <span
class="cmr-12">The following steps are required:</span>
<ol class="enumerate1" >
<li
class="enumerate" id="x14-13002x1">
<!--l. 11--><p class="noindent" ><span
class="cmti-12">Declare the preconditioner data structure</span><span
class="cmr-12">. It is a derived data type,</span>
<span class="obeylines-h"><code class="verb">amg_</code></span><span
class="cmti-12">x</span><span class="obeylines-h"><code class="verb">prec_</code></span> <span class="obeylines-h"><code class="verb">type</code></span><span
class="cmr-12">, where </span><span
class="cmti-12">x </span><span
class="cmr-12">may be </span><span class="obeylines-h"><code class="verb">s</code></span><span
class="cmr-12">, </span><span class="obeylines-h"><code class="verb">d</code></span><span
class="cmr-12">, </span><span class="obeylines-h"><code class="verb">c</code></span> <span
class="cmr-12">or </span><span class="obeylines-h"><code class="verb">z</code></span><span
class="cmr-12">, according to the basic data</span>
<span
class="cmr-12">type of the sparse matrix (</span><span class="obeylines-h"><code class="verb">s</code></span> <span
class="cmr-12">= real single precision; </span><span class="obeylines-h"><code class="verb">d</code></span> <span
class="cmr-12">= real double precision;</span>
<span class="obeylines-h"><code class="verb">c</code></span> <span
class="cmr-12">= complex single precision; </span><span class="obeylines-h"><code class="verb">z</code></span> <span
class="cmr-12">= complex double precision). This data</span>
<span
class="cmr-12">structure is accessed by the user only through the AMG4PSBLAS routines,</span>
<span
class="cmr-12">following an object-oriented approach.</span>
</li>
<li
class="enumerate" id="x14-13004x2">
<!--l. 19--><p class="noindent" ><span
class="cmti-12">Allocate and initialize the preconditioner data structure, according to a</span>
<span
class="cmti-12">preconditioner type chosen by the user</span><span
class="cmr-12">. This is performed by the routine</span>
<code class="lstinline"><span style="color:#000000">init</span></code><span
class="cmr-12">, which also sets defaults for each preconditioner type selected by</span>
<span
class="cmr-12">the user. The preconditioner types and the defaults associated with them</span>
<span
class="cmr-12">are given in Table</span><span
class="cmr-12">&#x00A0;</span><a
href="#x14-13015r1"><span
class="cmr-12">1</span><!--tex4ht:ref: tab:precinit --></a><span
class="cmr-12">, where the strings used by </span><code class="lstinline"><span style="color:#000000">init</span></code> <span
class="cmr-12">to identify the</span>
<span
class="cmr-12">preconditioner types are also given. Note that these strings are valid also if</span>
<span
class="cmr-12">uppercase letters are substituted by corresponding lowercase ones.</span>
</li>
<li
class="enumerate" id="x14-13006x3">
<!--l. 29--><p class="noindent" ><span
class="cmti-12">Modify the selected preconditioner type, by properly setting preconditioner</span>
<span
class="cmti-12">parameters. </span><span
class="cmr-12">This is performed by the routine </span><code class="lstinline"><span style="color:#000000">set</span></code><span
class="cmr-12">. This routine must be</span>
<span
class="cmr-12">called if the user wants to modify the default values of the parameters</span>
<span
class="cmr-12">associated with the selected preconditioner type, to obtain a variant of that</span>
<span
class="cmr-12">preconditioner. Examples of use of </span><code class="lstinline"><span style="color:#000000">set</span></code> <span
class="cmr-12">are given in Section</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlsu6.html#x15-140004.1"><span
class="cmr-12">4.1</span><!--tex4ht:ref: sec:examples --></a><span
class="cmr-12">; a complete</span>
<span
class="cmr-12">list of all the preconditioner parameters and their allowed and default values</span>
<span
class="cmr-12">is provided in Section</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlse5.html#x17-160005"><span
class="cmr-12">5</span><!--tex4ht:ref: sec:userinterface --></a><span
class="cmr-12">, Tables</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlsu9.html#x19-18009r2"><span
class="cmr-12">2</span><!--tex4ht:ref: tab:p_cycle --></a><span
class="cmr-12">-</span><a
href="userhtmlsu9.html#x19-18015r8"><span
class="cmr-12">8</span><!--tex4ht:ref: tab:p_smoother_1 --></a><span
class="cmr-12">.</span>
</li>
<li
class="enumerate" id="x14-13008x4">
<!--l. 40--><p class="noindent" ><span
class="cmti-12">Build the preconditioner for a given matrix</span><span
class="cmr-12">. If the selected preconditioner is</span>
<span
class="cmr-12">multilevel, then two steps must be performed, as specified next.</span>
<ol class="enumerate2" >
<li
class="enumerate" id="x14-13009x0">
<!--l. 43--><p class="noindent" ><span
class="cmti-12">Build the AMG hierarchy for a given matrix. </span><span
class="cmr-12">This is performed by the</span>
<span
class="cmr-12">routine </span><code class="lstinline"><span style="color:#000000">hierarchy_build</span></code><span
class="cmr-12">.</span>
</li>
<li
class="enumerate" id="x14-13010x0">
<!--l. 45--><p class="noindent" ><span
class="cmti-12">Build the preconditioner for a given matrix. </span><span
class="cmr-12">This is performed by the</span>
<span
class="cmr-12">routine </span><code class="lstinline"><span style="color:#000000">smoothers_build</span></code><span
class="cmr-12">.</span></li></ol>
<!--l. 48--><p class="noindent" ><span
class="cmr-12">If the selected preconditioner is one-level, it is built in a single step, performed by</span>
<span
class="cmr-12">the routine </span><code class="lstinline"><span style="color:#000000">bld</span></code><span
class="cmr-12">.</span>
</li>
<li
class="enumerate" id="x14-13012x5">
<!--l. 50--><p class="noindent" ><span
class="cmti-12">Apply the preconditioner at each iteration of a Krylov solver. </span><span
class="cmr-12">This is performed by</span>
<span
class="cmr-12">the method </span><code class="lstinline"><span style="color:#000000">apply</span></code><span
class="cmr-12">. When using the PSBLAS Krylov solvers, this step is</span>
<span
class="cmr-12">completely transparent to the user, since </span><code class="lstinline"><span style="color:#000000">apply</span></code> <span
class="cmr-12">is called by the PSBLAS routine</span>
<span
class="cmr-12">implementing the Krylov solver (</span><code class="lstinline"><span style="color:#000000">psb_krylov</span></code><span
class="cmr-12">).</span>
</li>
<li
class="enumerate" id="x14-13014x6">
<!--l. 54--><p class="noindent" ><span
class="cmti-12">Free the preconditioner data structure</span><span
class="cmr-12">. This is performed by the routine </span><code class="lstinline"><span style="color:#000000">free</span></code><span
class="cmr-12">.</span>
<span
class="cmr-12">This step is complementary to step 1 and should be performed when the</span>
<span
class="cmr-12">preconditioner is no more used.</span></li></ol>
<!--l. 59--><p class="indent" > <span
class="cmr-12">All the previous routines are available as methods of the preconditioner object. A</span>
<span
class="cmr-12">detailed description of them is given in Section</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlse5.html#x17-160005"><span
class="cmr-12">5</span><!--tex4ht:ref: sec:userinterface --></a><span
class="cmr-12">. Examples showing the basic use of</span>
<span
class="cmr-12">AMG4PSBLAS are reported in Section</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlsu6.html#x15-140004.1"><span
class="cmr-12">4.1</span><!--tex4ht:ref: sec:examples --></a><span
class="cmr-12">.</span>
<div class="table">
<!--l. 63--><p class="indent" > <a
id="x14-13015r1"></a><hr class="float"><div class="float"
>
<div class="center"
>
<!--l. 64--><p class="noindent" >
<div class="tabular"> <table id="TBL-1" class="tabular"
><colgroup id="TBL-1-1g"><col
id="TBL-1-1"></colgroup><colgroup id="TBL-1-2g"><col
id="TBL-1-2"></colgroup><colgroup id="TBL-1-3g"><col
id="TBL-1-3"></colgroup><tr
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-1-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-1-1"
class="td11"> <span
class="cmcsc-10x-x-109"><span
class="small-caps">t</span><span
class="small-caps">y</span><span
class="small-caps">p</span><span
class="small-caps">e</span> </span></td><td style="white-space:normal; text-align:left;" id="TBL-1-1-2"
class="td11"> <!--l. 68--><p class="noindent" ><span
class="cmcsc-10x-x-109"><span
class="small-caps">s</span><span
class="small-caps">t</span><span
class="small-caps">r</span><span
class="small-caps">i</span><span
class="small-caps">n</span><span
class="small-caps">g</span></span> </td><td style="white-space:normal; text-align:left;" id="TBL-1-1-3"
class="td11"> <!--l. 68--><p class="noindent" ><span
class="cmcsc-10x-x-109"><span
class="small-caps">d</span><span
class="small-caps">e</span><span
class="small-caps">f</span><span
class="small-caps">a</span><span
class="small-caps">u</span><span
class="small-caps">l</span><span
class="small-caps">t</span> <span
class="small-caps">p</span><span
class="small-caps">r</span><span
class="small-caps">e</span><span
class="small-caps">c</span><span
class="small-caps">o</span><span
class="small-caps">n</span><span
class="small-caps">d</span><span
class="small-caps">i</span><span
class="small-caps">t</span><span
class="small-caps">i</span><span
class="small-caps">o</span><span
class="small-caps">n</span><span
class="small-caps">e</span><span
class="small-caps">r</span></span> </td></tr><tr
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-2-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-2-1"
class="td11">
No preconditioner </td><td style="white-space:normal; text-align:left;" id="TBL-1-2-2"
class="td11"> <!--l. 69--><p class="noindent" ><code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">NONE</span><span style="color:#000000">&#8217;</span></code>
</td><td style="white-space:normal; text-align:left;" id="TBL-1-2-3"
class="td11"> <!--l. 69--><p class="noindent" >Considered to use the PSBLAS Krylov
solvers with no preconditioner.
</td>
</tr><tr
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-3-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-3-1"
class="td11">
Diagonal </td><td style="white-space:normal; text-align:left;" id="TBL-1-3-2"
class="td11"> <!--l. 71--><p class="noindent" ><code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">DIAG</span><span style="color:#000000">&#8217;</span></code>,
<code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">JACOBI</span><span style="color:#000000">&#8217;</span></code>,
<code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">L1</span><span style="color:#000000">-</span><span style="color:#000000">JACOBI</span><span style="color:#000000">&#8217;</span></code>
</td><td style="white-space:normal; text-align:left;" id="TBL-1-3-3"
class="td11"> <!--l. 71--><p class="noindent" >Diagonal preconditioner. For any zero
diagonal entry of the matrix to be
preconditioned, the corresponding entry
of the preconditioner is set to&#x00A0;1.
</td>
</tr><tr
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-4-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-4-1"
class="td11">
Gauss-Seidel </td><td style="white-space:normal; text-align:left;" id="TBL-1-4-2"
class="td11"> <!--l. 74--><p class="noindent" ><code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">GS</span><span style="color:#000000">&#8217;</span></code>,
<code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">L1</span><span style="color:#000000">-</span><span style="color:#000000">GS</span><span style="color:#000000">&#8217;</span></code>
</td><td style="white-space:normal; text-align:left;" id="TBL-1-4-3"
class="td11"> <!--l. 74--><p class="noindent" >Hybrid Gauss-Seidel (forward), that is,
global block Jacobi with Gauss-Seidel as
local solver.
</td>
</tr><tr
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-5-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-5-1"
class="td11">
Symmetrized Gauss-Seidel </td><td style="white-space:normal; text-align:left;" id="TBL-1-5-2"
class="td11"> <!--l. 77--><p class="noindent" ><code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">FBGS</span><span style="color:#000000">&#8217;</span></code>,
<code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">L1</span><span style="color:#000000">-</span><span style="color:#000000">FBGS</span><span style="color:#000000">&#8217;</span></code>
</td><td style="white-space:normal; text-align:left;" id="TBL-1-5-3"
class="td11"> <!--l. 77--><p class="noindent" >Symmetrized hybrid Gauss-Seidel, that
is, forward Gauss-Seidel followed by
backward Gauss-Seidel.
</td>
</tr><tr
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-6-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-6-1"
class="td11">
Block Jacobi </td><td style="white-space:normal; text-align:left;" id="TBL-1-6-2"
class="td11"> <!--l. 80--><p class="noindent" ><code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">BJAC</span><span style="color:#000000">&#8217;</span></code>,
<code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">L1</span><span style="color:#000000">-</span><span style="color:#000000">BJAC</span><span style="color:#000000">&#8217;</span></code>
</td><td style="white-space:normal; text-align:left;" id="TBL-1-6-3"
class="td11"> <!--l. 80--><p class="noindent" >Block-Jacobi with ILU(0) on the local
blocks.
</td>
</tr><tr
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-7-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-7-1"
class="td11">
Additive Schwarz </td><td style="white-space:normal; text-align:left;" id="TBL-1-7-2"
class="td11"> <!--l. 81--><p class="noindent" ><code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">AS</span><span style="color:#000000">&#8217;</span></code>
</td><td style="white-space:normal; text-align:left;" id="TBL-1-7-3"
class="td11"> <!--l. 81--><p class="noindent" >Additive Schwarz (AS), with overlap&#x00A0;1
and ILU(0) on the local blocks.
</td>
</tr><tr
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-8-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-8-1"
class="td11">
Multilevel </td><td style="white-space:normal; text-align:left;" id="TBL-1-8-2"
class="td11"> <!--l. 83--><p class="noindent" ><code class="lstinline"><span style="color:#000000">&#8217;</span><span style="color:#000000">ML</span><span style="color:#000000">&#8217;</span></code>
</td><td style="white-space:normal; text-align:left;" id="TBL-1-8-3"
class="td11"> <!--l. 83--><p class="noindent" >V-cycle with one hybrid
forward Gauss-Seidel (GS) sweep as
pre-smoother and one hybrid backward
GS sweep as post-smoother, decoupled
smoothed aggregation as coarsening
algorithm, and LU (plus triangular solve)
as coarsest-level solver. See the default
values in Tables&#x00A0;<a
href="userhtmlsu9.html#x19-18009r2">2<!--tex4ht:ref: tab:p_cycle --></a>-<a
href="userhtmlsu9.html#x19-18015r8">8<!--tex4ht:ref: tab:p_smoother_1 --></a> for further details of
the preconditioner.
</td>
</tr><tr
class="hline"><td><hr></td><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-1-9-"><td style="white-space:nowrap; text-align:left;" id="TBL-1-9-1"
class="td11"> </td></tr></table> </div>
<br /> <div class="caption"
><span class="id">Table&#x00A0;1: </span><span
class="content">Preconditioner types, corresponding strings and default choices. </span></div><!--tex4ht:label?: x14-13015r1 -->
</div>
</div><hr class="endfloat" />
</div>
<!--l. 98--><p class="indent" > <span
class="cmr-12">Note that the module </span><code class="lstinline"><span style="color:#000000">amg_prec_mod</span></code><span
class="cmr-12">, containing the definition of the preconditioner</span>
<span
class="cmr-12">data type and the interfaces to the routines of AMG4PSBLAS, must be used</span>
<span
class="cmr-12">in any program calling such routines. The modules </span><code class="lstinline"><span style="color:#000000">psb_base_mod</span></code><span
class="cmr-12">, for the</span>
<span
class="cmr-12">sparse matrix and communication descriptor data types, and </span><code class="lstinline"><span style="color:#000000">psb_krylov_mod</span></code><span
class="cmr-12">,</span>
<span
class="cmr-12">for interfacing with the Krylov solvers, must be also used (see Section</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlsu6.html#x15-140004.1"><span
class="cmr-12">4.1</span><!--tex4ht:ref: sec:examples --></a><span
class="cmr-12">).</span>
<br
class="newline" />
<!--l. 105--><p class="indent" > <span
class="cmbx-12">Remark 1. </span><span
class="cmr-12">Coarsest-level solvers based on the LU factorization, such as those</span>
<span
class="cmr-12">implemented in UMFPACK, MUMPS, SuperLU, and SuperLU</span><span
class="cmr-12">_Dist, usually lead to</span>
<span
class="cmr-12">smaller numbers of preconditioned Krylov iterations than inexact solvers, when the</span>
<span
class="cmr-12">linear system comes from a standard discretization of basic scalar elliptic PDE</span>
<span
class="cmr-12">problems. However, this does not necessarily correspond to the shortest execution time</span>
<span
class="cmr-12">on parallel</span><span
class="cmr-12">&#x00A0;computers.</span>
<div class="subsectionTOCS">
<span class="subsectionToc" ><span
class="cmr-12">4.1 </span><a
href="userhtmlsu6.html#x15-140004.1"><span
class="cmr-12">Examples</span></a></span>
<br /> <span class="subsectionToc" ><span
class="cmr-12">4.2 </span><a
href="userhtmlsu7.html#x16-150004.2"><span
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