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<h3 class="sectionHead"><span class="titlemark"><span
class="cmr-12">5 </span></span> <a
id="x17-160005"></a><span
class="cmr-12">Getting Started</span></h3>
<!--l. 5--><p class="noindent" ><span
class="cmr-12">We describe the basics for building and applying MLD2P4 one-level and multilevel</span>
<span
class="cmr-12">(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS </span><span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli4.html#XPSBLASGUIDE"><span
class="cmr-12">13</span></a><span
class="cmr-12">]</span></span><span
class="cmr-12">. The</span>
<span
class="cmr-12">following steps are required:</span>
<ol class="enumerate1" >
<li
class="enumerate" id="x17-16002x1"><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"><span class="verb"><span
class="cmtt-12">mld_</span></span></span><span
class="cmti-12">x</span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">prec_</span></span></span> <span class="obeylines-h"><span class="verb"><span
class="cmtt-12">type</span></span></span><span
class="cmr-12">, where </span><span
class="cmti-12">x </span><span
class="cmr-12">may be </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">s</span></span></span><span
class="cmr-12">, </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">d</span></span></span><span
class="cmr-12">, </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">c</span></span></span> <span
class="cmr-12">or </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">z</span></span></span><span
class="cmr-12">, according to the basic</span>
<span
class="cmr-12">data type of the sparse matrix (</span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">s</span></span></span> <span
class="cmr-12">= real single precision; </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">d</span></span></span> <span
class="cmr-12">= real double</span>
<span
class="cmr-12">precision; </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">c</span></span></span> <span
class="cmr-12">= complex single precision; </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">z</span></span></span> <span
class="cmr-12">= complex double precision). This</span>
<span
class="cmr-12">data structure is accessed by the user only through the MLD2P4 routines,</span>
<span
class="cmr-12">following an object-oriented approach.</span>
</li>
<li
class="enumerate" id="x17-16004x2"><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>
<span class="obeylines-h"><span class="verb"><span
class="cmtt-12">init</span></span></span><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"> </span><a
href="#x17-160151"><span
class="cmr-12">1</span><!--tex4ht:ref: tab:precinit --></a><span
class="cmr-12">, where the strings used by </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">init</span></span></span> <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="x17-16006x3"><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><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">set</span></span></span><span
class="cmr-12">. This routine must be</span>
<span
class="cmr-12">called only 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><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">set</span></span></span> <span
class="cmr-12">are given in Section</span><span
class="cmr-12"> </span><a
href="userhtmlsu9.html#x18-170005.1"><span
class="cmr-12">5.1</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"> </span><a
href="userhtmlse6.html#x19-180006"><span
class="cmr-12">6</span><!--tex4ht:ref: sec:userinterface --></a><span
class="cmr-12">, Tables</span><span
class="cmr-12"> </span><a
href="userhtmlsu11.html#x21-200092"><span
class="cmr-12">2</span><!--tex4ht:ref: tab:p_cycle --></a><span
class="cmr-12">-</span><a
href="userhtmlsu11.html#x21-200158"><span
class="cmr-12">8</span><!--tex4ht:ref: tab:p_smoother_1 --></a><span
class="cmr-12">.</span>
</li>
<li
class="enumerate" id="x17-16008x4"><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="x17-16009x0"><span
class="cmti-12">Build the aggregation hierarchy for a given matrix. </span><span
class="cmr-12">This is performed</span>
<span
class="cmr-12">by the routine </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">hierarchy_build</span></span></span><span
class="cmr-12">.</span>
</li>
<li
class="enumerate" id="x17-16010x0"><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><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">smoothers_build</span></span></span><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><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">bld</span></span></span><span
class="cmr-12">.</span>
</li>
<li
class="enumerate" id="x17-16012x5"><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><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">apply</span></span></span><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><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">apply</span></span></span> <span
class="cmr-12">is called by the PSBLAS routine</span>
<span
class="cmr-12">implementing the Krylov solver (</span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">psb_krylov</span></span></span><span
class="cmr-12">).</span>
</li>
<li
class="enumerate" id="x17-16014x6"><span
class="cmti-12">Free the preconditioner data structure</span><span
class="cmr-12">. This is performed by the routine </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">free</span></span></span><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"> </span><a
href="userhtmlse6.html#x19-180006"><span
class="cmr-12">6</span><!--tex4ht:ref: sec:userinterface --></a><span
class="cmr-12">. Examples showing the basic use of</span>
<span
class="cmr-12">MLD2P4 are reported in Section</span><span
class="cmr-12"> </span><a
href="userhtmlsu9.html#x18-170005.1"><span
class="cmr-12">5.1</span><!--tex4ht:ref: sec:examples --></a><span
class="cmr-12">.</span>
<div class="table">
<!--l. 63--><p class="indent" > <a
id="x17-160151"></a><hr class="float"><div class="float"
>
<div class="center"
>
<!--l. 64--><p class="noindent" >
<div class="tabular"> <table id="TBL-1" class="tabular"
cellspacing="0" cellpadding="0"
><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:wrap; 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:wrap; 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:wrap; text-align:left;" id="TBL-1-2-2"
class="td11"><!--l. 69--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">’NONE’</span></span></span> </td><td style="white-space:wrap; 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:wrap; text-align:left;" id="TBL-1-3-2"
class="td11"><!--l. 71--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">’DIAG’</span></span></span> or
<span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">’JACOBI’</span></span></span> </td><td style="white-space:wrap; 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 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:wrap; text-align:left;" id="TBL-1-4-2"
class="td11"><!--l. 74--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">’GS’</span></span></span> </td><td style="white-space:wrap; 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:wrap; text-align:left;" id="TBL-1-5-2"
class="td11"><!--l. 77--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">’FBGS’</span></span></span> </td><td style="white-space:wrap; 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:wrap; text-align:left;" id="TBL-1-6-2"
class="td11"><!--l. 80--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">’BJAC’</span></span></span> </td><td style="white-space:wrap; 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:wrap; text-align:left;" id="TBL-1-7-2"
class="td11"><!--l. 81--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">’AS’</span></span></span> </td><td style="white-space:wrap; text-align:left;" id="TBL-1-7-3"
class="td11"><!--l. 81--><p class="noindent" >Additive Schwarz (AS), with overlap 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:wrap; text-align:left;" id="TBL-1-8-2"
class="td11"><!--l. 83--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-10x-x-109">’ML’</span></span></span> </td><td style="white-space:wrap; 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, basic
smoothed aggregation as coarsening algorithm,
and LU (plus triangular solve) as coarsest-level
solver. See the default values in Tables <a
href="userhtmlsu11.html#x21-200092">2<!--tex4ht:ref: tab:p_cycle --></a>-<a
href="userhtmlsu11.html#x21-200158">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 1: </span><span
class="content">Preconditioner types, corresponding strings and default choices. </span></div><!--tex4ht:label?: x17-160151 -->
</div>
</div><hr class="endfloat" />
</div>
<!--l. 98--><p class="indent" > <span
class="cmr-12">Note that the module </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">mld_prec_mod</span></span></span><span
class="cmr-12">, containing the definition of the preconditioner</span>
<span
class="cmr-12">data type and the interfaces to the routines of MLD2P4, must be used in any program</span>
<span
class="cmr-12">calling such routines. The modules </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">psb_base_mod</span></span></span><span
class="cmr-12">, for the sparse matrix and</span>
<span
class="cmr-12">communication descriptor data types, and </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">psb_krylov_mod</span></span></span><span
class="cmr-12">, for interfacing with the</span>
<span
class="cmr-12">Krylov solvers, must be also used (see Section</span><span
class="cmr-12"> </span><a
href="userhtmlsu9.html#x18-170005.1"><span
class="cmr-12">5.1</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 smallest execution time</span>
<span
class="cmr-12">on parallel computers.</span>
<div class="subsectionTOCS">
<span
class="cmr-12"> </span><span class="subsectionToc" ><span
class="cmr-12">5.1 </span><a
href="userhtmlsu9.html#x18-170005.1"><span
class="cmr-12">Examples</span></a></span>
</div>
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