This commit is contained in:
Salvatore Filippone
2021-03-10 17:25:08 +01:00
parent e786f26a05
commit 7f2145a6d4
29 changed files with 4128 additions and 4815 deletions
+79 -355
View File
@@ -1,7 +1,7 @@
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"http://www.w3.org/TR/html4/loose.dtd">
<html >
<head><title>Smoothed Aggregation</title>
<head><title>Method init</title>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
<meta name="generator" content="TeX4ht (https://tug.org/tex4ht/)">
<meta name="originator" content="TeX4ht (https://tug.org/tex4ht/)">
@@ -10,386 +10,110 @@
<link rel="stylesheet" type="text/css" href="userhtml.css">
</head><body
>
<!--l. 133--><div class="crosslinks"><p class="noindent"><span
<!--l. 43--><div class="crosslinks"><p class="noindent"><span
class="cmr-12">[</span><a
href="userhtmlsu8.html" ><span
class="cmr-12">next</span></a><span
class="cmr-12">] [</span><a
href="userhtmlsu6.html" ><span
class="cmr-12">prev</span></a><span
class="cmr-12">] [</span><a
href="userhtmlsu6.html#tailuserhtmlsu6.html" ><span
class="cmr-12">prev-tail</span></a><span
class="cmr-12">] [</span><a
href="#tailuserhtmlsu7.html"><span
class="cmr-12">tail</span></a><span
class="cmr-12">] [</span><a
href="userhtmlse4.html#userhtmlsu7.html" ><span
href="userhtmlse5.html#userhtmlsu7.html" ><span
class="cmr-12">up</span></a><span
class="cmr-12">] </span></p></div>
<h4 class="subsectionHead"><span class="titlemark"><span
class="cmr-12">4.2 </span></span> <a
id="x15-140004.2"></a><span
class="cmr-12">Smoothed Aggregation</span></h4>
<!--l. 135--><p class="noindent" ><span
class="cmr-12">In order to define the prolongator </span><span
class="cmmi-12">P</span><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">, used to compute the coarse-level matrix </span><span
class="cmmi-12">A</span><sup><span
class="cmmi-8">k</span><span
class="cmr-8">+1</span></sup><span
class="cmr-12">,</span>
class="cmr-12">5.1 </span></span> <a
id="x16-150005.1"></a><span
class="cmr-12">Method init</span></h4>
<div class="center"
>
<!--l. 45--><p class="noindent" >
<!--l. 46--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">call</span><span
class="cmtt-12">&#x00A0;p%init(icontx,ptype,info)</span></span></span></div>
<!--l. 49--><p class="noindent" ><span
class="cmr-12">This method allocates and initializes the preconditioner </span><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">p</span></span></span><span
class="cmr-12">, according to the</span>
<span
class="cmr-12">MLD2P4 uses the smoothed aggregation algorithm described in </span><span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli4.html#XBREZINA_VANEK"><span
class="cmr-12">2</span></a><span
class="cmr-12">,</span><span
class="cmr-12">preconditioner type chosen by the user.</span>
<!--l. 53--><p class="noindent" ><span
class="cmbx-12">Arguments</span>
<div class="tabular"> <table id="TBL-2" class="tabular"
cellspacing="0" cellpadding="0"
><colgroup id="TBL-2-1g"><col
id="TBL-2-1"><col
id="TBL-2-2"></colgroup><tr
style="vertical-align:baseline;" id="TBL-2-1-"><td style="white-space:normal; text-align:left;" id="TBL-2-1-1"
class="td11"><!--l. 57--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">icontxt</span></span></span> </td> <td style="white-space:normal; text-align:left;" id="TBL-2-1-2"
class="td11"><!--l. 57--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">integer,</span><span
class="cmtt-12">&#x00A0;intent(in)</span></span></span><span
class="cmr-12">.</span> </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-2-2-"><td style="white-space:normal; text-align:left;" id="TBL-2-2-1"
class="td11"><!--l. 58--><p class="noindent" > </td><td style="white-space:normal; text-align:left;" id="TBL-2-2-2"
class="td11"><!--l. 58--><p class="noindent" ><span
class="cmr-12">The communication context.</span> </td></tr><tr
style="vertical-align:baseline;" id="TBL-2-3-"><td style="white-space:normal; text-align:left;" id="TBL-2-3-1"
class="td11"><!--l. 59--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">ptype</span></span></span> </td> <td style="white-space:normal; text-align:left;" id="TBL-2-3-2"
class="td11"><!--l. 59--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">character(len=*),</span><span
class="cmtt-12">&#x00A0;intent(in)</span></span></span><span
class="cmr-12">.</span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-2-4-"><td style="white-space:normal; text-align:left;" id="TBL-2-4-1"
class="td11"><!--l. 60--><p class="noindent" > </td><td style="white-space:normal; text-align:left;" id="TBL-2-4-2"
class="td11"><!--l. 60--><p class="noindent" ><span
class="cmr-12">The type of preconditioner. Its values are specified in Table</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlli4.html#XVANEK_MANDEL_BREZINA"><span
class="cmr-12">29</span></a><span
class="cmr-12">]</span></span><span
class="cmr-12">. The basic idea</span>
<span
class="cmr-12">of this algorithm is to build a coarse set of indices &#x03A9;</span><sup><span
class="cmmi-8">k</span><span
class="cmr-8">+1</span></sup> <span
class="cmr-12">by suitably grouping the</span>
<span
class="cmr-12">indices of &#x03A9;</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">into disjoint subsets (aggregates), and to define the coarse-to-fine space</span>
<span
class="cmr-12">transfer operator </span><span
class="cmmi-12">P</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">by applying a suitable smoother to a simple piecewise constant</span>
<span
class="cmr-12">prolongation operator, with the aim of improving the quality of the coarse-space</span>
<span
class="cmr-12">correction.</span>
<!--l. 144--><p class="indent" > <span
class="cmr-12">Three main steps can be identified in the smoothed aggregation procedure:</span>
<ol class="enumerate1" >
<li
class="enumerate" id="x15-14002x1"><span
class="cmr-12">aggregation of the indices of &#x03A9;</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">to obtain &#x03A9;</span><sup><span
class="cmmi-8">k</span><span
class="cmr-8">+1</span></sup><span
class="cmr-12">;</span>
</li>
<li
class="enumerate" id="x15-14004x2"><span
class="cmr-12">construction of the prolongator </span><span
class="cmmi-12">P</span><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">;</span>
</li>
<li
class="enumerate" id="x15-14006x3"><span
class="cmr-12">application of </span><span
class="cmmi-12">P</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">and </span><span
class="cmmi-12">R</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">= (</span><span
class="cmmi-12">P</span><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">)</span><sup><span
class="cmmi-8">T</span> </sup> <span
class="cmr-12">to build </span><span
class="cmmi-12">A</span><sup><span
class="cmmi-8">k</span><span
class="cmr-8">+1</span></sup><span
class="cmr-12">.</span></li></ol>
<!--l. 151--><p class="indent" > <span
class="cmr-12">In order to perform the coarsening step, the smoothed aggregation algorithm</span>
<span
class="cmr-12">described in</span><span
class="cmr-12">&#x00A0;</span><span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli4.html#XVANEK_MANDEL_BREZINA"><span
class="cmr-12">29</span></a><span
class="cmr-12">]</span></span> <span
class="cmr-12">is used. In this algorithm, each index </span><span
class="cmmi-12">j </span><span
class="cmsy-10x-x-120">&#x2208; </span><span
class="cmr-12">&#x03A9;</span><sup><span
class="cmmi-8">k</span><span
class="cmr-8">+1</span></sup> <span
class="cmr-12">corresponds</span>
<span
class="cmr-12">to an aggregate &#x03A9;</span><sub><span
class="cmmi-8">j</span></sub><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">of &#x03A9;</span><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">, consisting of a suitably chosen index </span><span
class="cmmi-12">i </span><span
class="cmsy-10x-x-120">&#x2208; </span><span
class="cmr-12">&#x03A9;</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">and</span>
<span
class="cmr-12">indices that are (usually) contained in a strongly-coupled neighborood of </span><span
class="cmmi-12">i</span><span
class="cmr-12">,</span>
<span
class="cmr-12">i.e.,</span>
<table
class="equation"><tr><td>
<center class="math-display" >
<img
src="userhtml13x.png" alt=" { &#x2218; -------}
&#x03A9;kj &#x2282; N ki (&#x03B8;) = r &#x2208; &#x03A9;k : |akir| &#x003E; &#x03B8; |akiiakrr| &#x222A; {i} ,
" class="math-display" ><a
id="x15-14007r3"></a></center></td><td class="equation-label"><span
class="cmr-12">(3)</span></td></tr></table>
href="userhtmlse4.html#x13-120151"><span
class="cmr-12">1</span><!--tex4ht:ref: tab:precinit --></a><span
class="cmr-12">.</span> </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-2-5-"><td style="white-space:normal; text-align:left;" id="TBL-2-5-1"
class="td11"><!--l. 62--><p class="noindent" > </td><td style="white-space:normal; text-align:left;" id="TBL-2-5-2"
class="td11"><!--l. 62--><p class="noindent" ><span
class="cmr-12">Note that the strings are case insensitive.</span> </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-2-6-"><td style="white-space:normal; text-align:left;" id="TBL-2-6-1"
class="td11"><!--l. 63--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">info</span></span></span> </td><td style="white-space:normal; text-align:left;" id="TBL-2-6-2"
class="td11"><!--l. 63--><p class="noindent" ><span class="obeylines-h"><span class="verb"><span
class="cmtt-12">integer,</span><span
class="cmtt-12">&#x00A0;intent(out)</span></span></span><span
class="cmr-12">.</span> </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-2-7-"><td style="white-space:normal; text-align:left;" id="TBL-2-7-1"
class="td11"><!--l. 64--><p class="noindent" > </td><td style="white-space:normal; text-align:left;" id="TBL-2-7-2"
class="td11"><!--l. 64--><p class="noindent" ><span
class="cmr-12">Error code. If no error, 0 is returned. See Section</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlse7.html#x26-300007"><span
class="cmr-12">7</span><!--tex4ht:ref: sec:errors --></a> <span
class="cmr-12">for details.</span> </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-2-8-"><td style="white-space:normal; text-align:left;" id="TBL-2-8-1"
class="td11"> </td> </tr></table></div>
<!--l. 160--><p class="nopar" >
<span
class="cmr-12">for a given threshold </span><span
class="cmmi-12">&#x03B8; </span><span
class="cmsy-10x-x-120">&#x2208; </span><span
class="cmr-12">[0</span><span
class="cmmi-12">, </span><span
class="cmr-12">1] (see</span><span
class="cmr-12">&#x00A0;</span><span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli4.html#XVANEK_MANDEL_BREZINA"><span
class="cmr-12">29</span></a><span
class="cmr-12">]</span></span> <span
class="cmr-12">for the details). Since this algorithm has a</span>
<span
class="cmr-12">sequential nature, a decoupled version of it is applied, where each processor</span>
<span
class="cmr-12">independently executes the algorithm on the set of indices assigned to it in the initial</span>
<span
class="cmr-12">data distribution. This version is embarrassingly parallel, since it does not require any</span>
<span
class="cmr-12">data communication. On the other hand, it may produce some nonuniform aggregates</span>
<span
class="cmr-12">and is strongly dependent on the number of processors and on the initial</span>
<span
class="cmr-12">partitioning of the matrix </span><span
class="cmmi-12">A</span><span
class="cmr-12">. Nevertheless, this parallel algorithm has been chosen</span>
<span
class="cmr-12">for MLD2P4, since it has been shown to produce good results in practice</span>
<span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli4.html#Xaaecc_07"><span
class="cmr-12">5</span></a><span
class="cmr-12">,</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlli4.html#Xapnum_07"><span
class="cmr-12">8</span></a><span
class="cmr-12">,</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlli4.html#XTUMINARO_TONG"><span
class="cmr-12">28</span></a><span
class="cmr-12">]</span></span><span
class="cmr-12">.</span>
<!--l. 172--><p class="indent" > <span
class="cmr-12">The prolongator </span><span
class="cmmi-12">P</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">is built starting from a tentative prolongator</span> <span class="bar-css"><span
class="cmmi-12">P</span></span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmsy-10x-x-120">&#x2208; </span><span
class="msbm-10x-x-120">&#x211D;</span><sup><span
class="cmmi-8">n</span><sub><span
class="cmmi-6">k</span></sub><span
class="cmsy-8">&#x00D7;</span><span
class="cmmi-8">n</span><sub><span
class="cmmi-6">k</span><span
class="cmr-6">+1</span></sub></sup><span
class="cmr-12">,</span>
<span
class="cmr-12">defined as</span>
<table
class="equation"><tr><td>
<center class="math-display" >
<img
src="userhtml14x.png" alt=" { k
P¯k = (¯pkij), p¯kij = 1 if i &#x2208; &#x03A9; j,
0 otherwise,
" class="math-display" ><a
id="x15-14008r4"></a></center></td><td class="equation-label"><span
class="cmr-12">(4)</span></td></tr></table>
<!--l. 181--><p class="nopar" >
<span
class="cmr-12">where &#x03A9;</span><sub><span
class="cmmi-8">j</span></sub><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">is the aggregate of &#x03A9;</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">corresponding to the index </span><span
class="cmmi-12">j </span><span
class="cmsy-10x-x-120">&#x2208; </span><span
class="cmr-12">&#x03A9;</span><sup><span
class="cmmi-8">k</span><span
class="cmr-8">+1</span></sup><span
class="cmr-12">. </span><span
class="cmmi-12">P</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">is obtained</span>
<span
class="cmr-12">by applying to</span> <span class="bar-css"><span
class="cmmi-12">P</span></span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">a smoother </span><span
class="cmmi-12">S</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmsy-10x-x-120">&#x2208; </span><span
class="msbm-10x-x-120">&#x211D;</span><sup><span
class="cmmi-8">n</span><sub><span
class="cmmi-6">k</span></sub><span
class="cmsy-8">&#x00D7;</span><span
class="cmmi-8">n</span><sub><span
class="cmmi-6">k</span></sub></sup><span
class="cmr-12">:</span>
<center class="math-display" >
<img
src="userhtml15x.png" alt="Pk = Sk ¯Pk, " class="math-display" ></center> <span
class="cmr-12">in</span>
<span
class="cmr-12">order to remove nonsmooth components from the range of the prolongator, and hence</span>
<span
class="cmr-12">to improve the convergence properties of the multilevel method</span><span
class="cmr-12">&#x00A0;</span><span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli4.html#XBREZINA_VANEK"><span
class="cmr-12">2</span></a><span
class="cmr-12">,</span><span
class="cmr-12">&#x00A0;</span><a
href="userhtmlli4.html#XStuben_01"><span
class="cmr-12">27</span></a><span
class="cmr-12">]</span></span><span
class="cmr-12">. A simple</span>
<span
class="cmr-12">choice for </span><span
class="cmmi-12">S</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">is the damped Jacobi smoother:</span>
<center class="math-display" >
<img
src="userhtml16x.png" alt=" k k k -1 k
S = I - &#x03C9; (D ) A F,
" class="math-display" ></center>
<!--l. 195--><p class="nopar" > <span
class="cmr-12">where </span><span
class="cmmi-12">D</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">is the diagonal matrix with the same diagonal entries as </span><span
class="cmmi-12">A</span><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">, </span><span
class="cmmi-12">A</span><sub>
<span
class="cmmi-8">F</span> </sub><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">= (</span><span class="bar-css"><span
class="cmmi-12">a</span></span><sub>
<span
class="cmmi-8">ij</span></sub><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">) is</span>
<span
class="cmr-12">the filtered matrix defined as</span>
<table
class="equation"><tr><td>
<center class="math-display" >
<img
src="userhtml17x.png" alt=" { k k &#x2211;
a¯kij = aij if j &#x2208; N i (&#x03B8;), (j &#x2044;= i), ¯akii = akii - (akij - ¯akij),
0 otherwise, j&#x2044;=i
" class="math-display" ><a
id="x15-14009r5"></a></center></td><td class="equation-label"><span
class="cmr-12">(5)</span></td></tr></table>
<!--l. 208--><p class="nopar" >
<span
class="cmr-12">and </span><span
class="cmmi-12">&#x03C9;</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">is an approximation of 4</span><span
class="cmmi-12">&#x2215;</span><span
class="cmr-12">(3</span><span
class="cmmi-12">&#x03C1;</span><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">), where </span><span
class="cmmi-12">&#x03C1;</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">is the spectral radius of (</span><span
class="cmmi-12">D</span><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">)</span><sup><span
class="cmsy-8">-</span><span
class="cmr-8">1</span></sup><span
class="cmmi-12">A</span><sub>
<span
class="cmmi-8">F</span> </sub><sup><span
class="cmmi-8">k</span></sup>
<span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli4.html#XBREZINA_VANEK"><span
class="cmr-12">2</span></a><span
class="cmr-12">]</span></span><span
class="cmr-12">. In MLD2P4 this approximation is obtained by using </span><span
class="cmsy-10x-x-120">&#x2225;</span><span
class="cmmi-12">A</span><sub><span
class="cmmi-8">F</span> </sub><sup><span
class="cmmi-8">k</span></sup><span
class="cmsy-10x-x-120">&#x2225;</span><sub>
<span
class="cmsy-8">&#x221E;</span></sub> <span
class="cmr-12">as an estimate of </span><span
class="cmmi-12">&#x03C1;</span><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">.</span>
<span
class="cmr-12">Note that for systems coming from uniformly elliptic problems, filtering the matrix </span><span
class="cmmi-12">A</span><sup><span
class="cmmi-8">k</span></sup>
<span
class="cmr-12">has little or no effect, and </span><span
class="cmmi-12">A</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">can be used instead of </span><span
class="cmmi-12">A</span><sub>
<span
class="cmmi-8">F</span> </sub><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">. The latter choice is the</span>
<span
class="cmr-12">default in MLD2P4.</span>
<!--l. 216--><div class="crosslinks"><p class="noindent"><span
<!--l. 73--><div class="crosslinks"><p class="noindent"><span
class="cmr-12">[</span><a
href="userhtmlsu8.html" ><span
class="cmr-12">next</span></a><span
class="cmr-12">] [</span><a
href="userhtmlsu6.html" ><span
class="cmr-12">prev</span></a><span
class="cmr-12">] [</span><a
href="userhtmlsu6.html#tailuserhtmlsu6.html" ><span
class="cmr-12">prev-tail</span></a><span
class="cmr-12">] [</span><a
href="userhtmlsu7.html" ><span
class="cmr-12">front</span></a><span
class="cmr-12">] [</span><a
href="userhtmlse4.html#userhtmlsu7.html" ><span
href="userhtmlse5.html#userhtmlsu7.html" ><span
class="cmr-12">up</span></a><span
class="cmr-12">] </span></p></div>
<!--l. 216--><p class="indent" > <a
<!--l. 73--><p class="indent" > <a
id="tailuserhtmlsu7.html"></a>
</body></html>