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<!--l. 133--><div class="crosslinks"><p class="noindent"><span
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<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>
<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"> </span><a
href="userhtmlli4.html#XVANEK_MANDEL_BREZINA"><span
class="cmr-12">26</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 Ω</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 Ω</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 Ω</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">to obtain Ω</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"> </span><span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli4.html#XVANEK_MANDEL_BREZINA"><span
class="cmr-12">26</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">∈ </span><span
class="cmr-12">Ω</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 Ω</span><sub><span
class="cmmi-8">j</span></sub><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">of Ω</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">∈ </span><span
class="cmr-12">Ω</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=" { ∘ -------}
Ωkj ⊂ N ki (θ) = r ∈ Ωk : |akir| > θ |akiiakrr| ∪ {i} ,
" class="math-display" ><a
id="x15-14007r3"></a></center></td><td class="equation-label"><span
class="cmr-12">(3)</span></td></tr></table>
<!--l. 160--><p class="nopar" >
<span
class="cmr-12">for a given threshold </span><span
class="cmmi-12">θ </span><span
class="cmsy-10x-x-120">∈ </span><span
class="cmr-12">[0</span><span
class="cmmi-12">, </span><span
class="cmr-12">1] (see</span><span
class="cmr-12"> </span><span class="cite"><span
class="cmr-12">[</span><a
href="userhtmlli4.html#XVANEK_MANDEL_BREZINA"><span
class="cmr-12">26</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"> </span><a
href="userhtmlli4.html#Xapnum_07"><span
class="cmr-12">7</span></a><span
class="cmr-12">,</span><span
class="cmr-12"> </span><a
href="userhtmlli4.html#XTUMINARO_TONG"><span
class="cmr-12">25</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">∈ </span><span
class="msbm-10x-x-120">ℝ</span><sup><span
class="cmmi-8">n</span><sub><span
class="cmmi-6">k</span></sub><span
class="cmsy-8">×</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 ∈ Ω 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 Ω</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 Ω</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">∈ </span><span
class="cmr-12">Ω</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">∈ </span><span
class="msbm-10x-x-120">ℝ</span><sup><span
class="cmmi-8">n</span><sub><span
class="cmmi-6">k</span></sub><span
class="cmsy-8">×</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"> </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"> </span><a
href="userhtmlli4.html#XStuben_01"><span
class="cmr-12">24</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 - ω (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 ∑
a�kij = aij if j ∈ N i (θ), (j ⁄= i), �akii = akii - (akij - �akij),
0 otherwise, j⁄=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">ω</span><sup><span
class="cmmi-8">k</span></sup> <span
class="cmr-12">is an approximation of 4</span><span
class="cmmi-12">∕</span><span
class="cmr-12">(3</span><span
class="cmmi-12">ρ</span><sup><span
class="cmmi-8">k</span></sup><span
class="cmr-12">), where </span><span
class="cmmi-12">ρ</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">∥</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">∥</span><sub>
<span
class="cmsy-8">∞</span></sub> <span
class="cmr-12">as an estimate of </span><span
class="cmmi-12">ρ</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>
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