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<head><title>AMG preconditioners</title>
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<head><title>Examples</title>
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</head><body
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<!--l. 56--><div class="crosslinks"><p class="noindent"><span
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class="cmr-12">[</span><a
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class="cmr-12">next</span></a><span
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class="cmr-12">] [</span><a
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class="cmr-12">tail</span></a><span
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class="cmr-12">] [</span><a
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@ -24,376 +21,370 @@ class="cmr-12">] </span></p></div>
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<h4 class="subsectionHead"><span class="titlemark"><span
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class="cmr-12">4.1 </span></span> <a
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id="x14-130004.1"></a><span
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class="cmr-12">AMG preconditioners</span></h4>
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<!--l. 58--><p class="noindent" ><span
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class="cmr-12">In order to describe the AMG preconditioners available in MLD2P4, we consider a</span>
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<span
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class="cmr-12">linear system</span>
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<table
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class="equation"><tr><td>
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<center class="math-display" >
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<img
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src="userhtml2x.png" alt="Ax = b,
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" class="math-display" ><a
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id="x14-13001r2"></a></center></td><td class="equation-label"><span
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class="cmr-12">(2)</span></td></tr></table>
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<!--l. 62--><p class="nopar" >
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<span
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class="cmr-12">where </span><span
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class="cmmi-12">A </span><span
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class="cmr-12">= (</span><span
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class="cmmi-12">a</span><sub><span
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class="cmmi-8">ij</span></sub><span
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class="cmr-12">) </span><span
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class="cmsy-10x-x-120">∈ </span><span
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class="msbm-10x-x-120">ℝ</span><sup><span
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class="cmmi-8">n</span><span
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class="cmsy-8">×</span><span
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class="cmmi-8">n</span></sup> <span
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class="cmr-12">is a nonsingular sparse matrix; for ease of presentation we</span>
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<span
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class="cmr-12">assume </span><span
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class="cmmi-12">A </span><span
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class="cmr-12">has a symmetric sparsity pattern.</span>
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class="cmr-12">Examples</span></h4>
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<!--l. 109--><p class="noindent" ><span
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class="cmr-12">The code reported in Figure</span><span
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class="cmr-12"> </span><a
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href="#x14-130011"><span
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class="cmr-12">1</span><!--tex4ht:ref: fig:ex1 --></a> <span
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class="cmr-12">shows how to set and apply the default multilevel</span>
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<span
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class="cmr-12">preconditioner available in the real double precision version of AMG4PSBLAS</span>
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<span
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class="cmr-12">(see Table</span><span
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class="cmr-12"> </span><a
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href="userhtmlse4.html#x13-120151"><span
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class="cmr-12">1</span><!--tex4ht:ref: tab:precinit --></a><span
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class="cmr-12">). This preconditioner is chosen by simply specifying </span><span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">’ML’</span></span></span> <span
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class="cmr-12">as the</span>
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<span
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class="cmr-12">second argument of </span><span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">P%init</span></span></span> <span
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class="cmr-12">(a call to </span><span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">P%set</span></span></span> <span
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class="cmr-12">is not needed) and is applied</span>
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<span
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class="cmr-12">with the CG solver provided by PSBLAS (the matrix of the system to be</span>
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<span
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class="cmr-12">solved is assumed to be positive definite). As previously observed, the modules</span>
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<span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">psb_base_mod</span></span></span><span
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class="cmr-12">, </span><span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">amg_prec_mod</span></span></span> <span
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class="cmr-12">and </span><span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">psb_krylov_mod</span></span></span> <span
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class="cmr-12">must be used by the example</span>
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<span
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class="cmr-12">program.</span>
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<!--l. 119--><p class="indent" > <span
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class="cmr-12">The part of the code concerning the reading and assembling of the sparse matrix</span>
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<span
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class="cmr-12">and the right-hand side vector, performed through the PSBLAS routines for sparse</span>
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<span
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class="cmr-12">matrix and vector management, is not reported here for brevity; the statements</span>
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<span
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class="cmr-12">concerning the deallocation of the PSBLAS data structure are neglected too. The</span>
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<span
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class="cmr-12">complete code can be found in the example program file </span><span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">amg_dexample_ml.f90</span></span></span><span
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class="cmr-12">, in the</span>
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<span
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class="cmr-12">directory </span><span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">examples/fileread</span></span></span> <span
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class="cmr-12">of the AMG4PSBLAS implementation (see</span>
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<span
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class="cmr-12">Section</span><span
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class="cmr-12"> </span><a
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href="userhtmlsu5.html#x12-110003.5"><span
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class="cmr-12">3.5</span><!--tex4ht:ref: sec:ex_and_test --></a><span
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class="cmr-12">). A sample test problem along with the relevant input data is available in</span>
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<span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">examples/fileread/runs</span></span></span><span
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class="cmr-12">. For details on the use of the PSBLAS routines, see the</span>
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<span
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class="cmr-12">PSBLAS User’s Guide</span><span
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class="cmr-12"> </span><span class="cite"><span
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class="cmr-12">[</span><a
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href="userhtmlli4.html#XPSBLASGUIDE"><span
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class="cmr-12">16</span></a><span
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class="cmr-12">]</span></span><span
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class="cmr-12">.</span>
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<!--l. 131--><p class="indent" > <span
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class="cmr-12">The setup and application of the default multilevel preconditioner for the real single</span>
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<!--l. 67--><p class="indent" > <span
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class="cmr-12">Let us consider as finest index space the set of row (column) indices of </span><span
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class="cmmi-12">A</span><span
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class="cmr-12">,</span>
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<span
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class="cmr-12">i.e., Ω = </span><span
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class="cmsy-10x-x-120">{</span><span
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class="cmr-12">1</span><span
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class="cmmi-12">, </span><span
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class="cmr-12">2</span><span
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class="cmmi-12">,</span><span
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class="cmmi-12">…</span><span
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class="cmmi-12">,n</span><span
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class="cmsy-10x-x-120">}</span><span
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class="cmr-12">. Any algebraic multilevel preconditioners implemented in</span>
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<span
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class="cmr-12">MLD2P4 generates a hierarchy of index spaces and a corresponding hierarchy of</span>
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<span
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class="cmr-12">matrices,</span>
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<center class="math-display" >
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<img
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src="userhtml3x.png" alt=" 1 2 nlev 1 2 nlev
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Ω ≡ Ω ⊃ Ω ⊃ ...⊃ Ω , A ≡ A, A ,...,A ,
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" class="math-display" ></center>
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<!--l. 72--><p class="nopar" > <span
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class="cmr-12">by using the information contained in </span><span
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class="cmmi-12">A</span><span
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class="cmr-12">, without assuming any knowledge of</span>
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<span
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class="cmr-12">the geometry of the problem from which </span><span
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class="cmmi-12">A </span><span
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class="cmr-12">originates. A vector space </span><span
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class="msbm-10x-x-120">ℝ</span><sup><span
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class="cmmi-8">n</span><sub><span
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class="cmmi-6">k</span></sub></sup> <span
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class="cmr-12">is</span>
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<span
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class="cmr-12">associated with Ω</span><sup><span
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class="cmmi-8">k</span></sup><span
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class="cmr-12">, where </span><span
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class="cmmi-12">n</span><sub>
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<span
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class="cmmi-8">k</span></sub> <span
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class="cmr-12">is the size of Ω</span><sup><span
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class="cmmi-8">k</span></sup><span
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class="cmr-12">. For all </span><span
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class="cmmi-12">k < nlev</span><span
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class="cmr-12">, a restriction</span>
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<span
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class="cmr-12">operator and a prolongation one are built, which connect two levels </span><span
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class="cmmi-12">k </span><span
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class="cmr-12">and</span>
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<span
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class="cmmi-12">k </span><span
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class="cmr-12">+ 1:</span>
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<center class="math-display" >
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<img
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src="userhtml4x.png" alt="P k ∈ ℝnk×nk+1, Rk ∈ ℝnk+1×nk ;
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" class="math-display" ></center>
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<!--l. 82--><p class="nopar" > <span
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class="cmr-12">the matrix </span><span
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class="cmmi-12">A</span><sup><span
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class="cmmi-8">k</span><span
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class="cmr-8">+1</span></sup> <span
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class="cmr-12">is computed by using the previous operators according to the</span>
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<span
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class="cmr-12">Galerkin approach, i.e.,</span>
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<center class="math-display" >
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<img
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src="userhtml5x.png" alt=" k+1 k k k
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A = R A P .
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" class="math-display" ></center>
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<!--l. 87--><p class="nopar" > <span
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class="cmr-12">In the current implementation of MLD2P4 we have </span><span
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class="cmmi-12">R</span><sup><span
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class="cmmi-8">k</span></sup> <span
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class="cmr-12">= (</span><span
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class="cmmi-12">P</span><sup><span
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class="cmmi-8">k</span></sup><span
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class="cmr-12">)</span><sup><span
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class="cmmi-8">T</span> </sup> <span
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class="cmr-12">A smoother with</span>
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<span
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class="cmr-12">iteration matrix </span><span
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class="cmmi-12">M</span><sup><span
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class="cmmi-8">k</span></sup> <span
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class="cmr-12">is set up at each level </span><span
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class="cmmi-12">k < nlev</span><span
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class="cmr-12">, and a solver is set up at the</span>
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<span
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class="cmr-12">coarsest level, so that they are ready for application (for example, setting up a solver</span>
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<span
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class="cmr-12">based on the </span><span
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class="cmmi-12">LU </span><span
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class="cmr-12">factorization means computing and storing the </span><span
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class="cmmi-12">L </span><span
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class="cmr-12">and </span><span
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class="cmmi-12">U </span><span
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class="cmr-12">factors). The</span>
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<span
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class="cmr-12">construction of the hierarchy of AMG components described so far corresponds to the</span>
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<span
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class="cmr-12">so-called build phase of the preconditioner.</span>
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<!--l. 95--><p class="indent" > <hr class="figure"><div class="figure"
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<span
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class="cmr-12">precision and the complex, single and double precision, versions are obtained</span>
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<span
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class="cmr-12">with straightforward modifications of the previous example (see Section</span><span
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class="cmr-12"> </span><a
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href="userhtmlse5.html#x15-140005"><span
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class="cmr-12">5</span><!--tex4ht:ref: sec:userinterface --></a> <span
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class="cmr-12">for</span>
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<span
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class="cmr-12">details). If these versions are installed, the corresponding codes are available in</span>
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<span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">examples/fileread/</span></span></span><span
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class="cmr-12">.</span>
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<!--l. 137--><p class="indent" > <hr class="figure"><div class="figure"
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>
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<a
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id="x14-130021"></a>
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id="x14-130011"></a>
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<div class="center"
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<!--l. 96--><p class="noindent" >
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<div class="fbox"> <div class="minipage"><table
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cellpadding="0" border="0" cellspacing="0"
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class="tabbing"><tr
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style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing"></td></tr></table>
|
|
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|
<!--l. 115--><p class="noindent" ><table
|
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|
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|
cellpadding="0" border="0" cellspacing="0"
|
|
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|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td
|
|
|
|
|
class="tabbing">procedure V-cycle<img
|
|
|
|
|
src="userhtml6x.png" alt="( k k k)
|
|
|
|
|
k,A ,b ,u" class="left" align="middle">
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</td></tr></table>
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|
<!--l. 115--><p class="noindent" >
|
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<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing">if <img
|
|
|
|
|
src="userhtml7x.png" alt="(k ⁄= nlev)" class="left" align="middle"> then</td></tr></table>
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|
<!--l. 115--><p class="noindent" >
|
|
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|
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<table
|
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|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing"><span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup> = <span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup> + <span
|
|
|
|
|
class="cmmi-10x-x-109">M</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup><img
|
|
|
|
|
src="userhtml8x.png" alt="( )
|
|
|
|
|
bk - Akuk" class="left" align="middle"></td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" >
|
|
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|
|
<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing"><span
|
|
|
|
|
class="cmmi-10x-x-109">b</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span><span
|
|
|
|
|
class="cmr-8">+1</span></sup> = <span
|
|
|
|
|
class="cmmi-10x-x-109">R</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span><span
|
|
|
|
|
class="cmr-8">+1</span></sup><img
|
|
|
|
|
src="userhtml9x.png" alt="(bk - Akuk )" class="left" align="middle"></td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" >
|
|
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|
|
<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing"><span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span><span
|
|
|
|
|
class="cmr-8">+1</span></sup> = V-cycle<img
|
|
|
|
|
src="userhtml10x.png" alt="( )
|
|
|
|
|
k + 1,Ak+1,bk+1,0" class="left" align="middle"></td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" >
|
|
|
|
|
<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing"><span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup> = <span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup> + <span
|
|
|
|
|
class="cmmi-10x-x-109">P</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span><span
|
|
|
|
|
class="cmr-8">+1</span></sup><span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span><span
|
|
|
|
|
class="cmr-8">+1</span></sup></td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" >
|
|
|
|
|
<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing"><span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup> = <span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup> + <span
|
|
|
|
|
class="cmmi-10x-x-109">M</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup><img
|
|
|
|
|
src="userhtml11x.png" alt="( )
|
|
|
|
|
bk - Akuk" class="left" align="middle"></td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" >
|
|
|
|
|
<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing">else</td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" >
|
|
|
|
|
<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing"><span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup> = <img
|
|
|
|
|
src="userhtml12x.png" alt="( )
|
|
|
|
|
Ak" class="left" align="middle"><sup><span
|
|
|
|
|
class="cmsy-8">-</span><span
|
|
|
|
|
class="cmr-8">1</span></sup><span
|
|
|
|
|
class="cmmi-10x-x-109">b</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup></td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" >
|
|
|
|
|
<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing">endif</td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" >
|
|
|
|
|
<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td style="width:16;"
|
|
|
|
|
class="tabbing"> </td><td
|
|
|
|
|
class="tabbing">return <span
|
|
|
|
|
class="cmmi-10x-x-109">u</span><sup><span
|
|
|
|
|
class="cmmi-8">k</span></sup></td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" >
|
|
|
|
|
<table
|
|
|
|
|
cellpadding="0" border="0" cellspacing="0"
|
|
|
|
|
class="tabbing"><tr
|
|
|
|
|
style="vertical-align:baseline;" class="tabbing"><td
|
|
|
|
|
class="tabbing">end</td></tr></table>
|
|
|
|
|
<!--l. 115--><p class="noindent" > </div> </div>
|
|
|
|
|
<!--l. 138--><p class="noindent" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<div class="minipage"><pre class="verbatim" id="verbatim-6">
|
|
|
|
|
  use psb_base_mod
|
|
|
|
|
  use amg_prec_mod
|
|
|
|
|
  use psb_krylov_mod
|
|
|
|
|
... ...
|
|
|
|
|
!
|
|
|
|
|
! sparse matrix
|
|
|
|
|
  type(psb_dspmat_type) :: A
|
|
|
|
|
! sparse matrix descriptor
|
|
|
|
|
  type(psb_desc_type)   :: desc_A
|
|
|
|
|
! preconditioner
|
|
|
|
|
  type(amg_dprec_type)  :: P
|
|
|
|
|
! right-hand side and solution vectors
|
|
|
|
|
  type(psb_d_vect_type) :: b, x
|
|
|
|
|
... ...
|
|
|
|
|
!
|
|
|
|
|
! initialize the parallel environment
|
|
|
|
|
  call psb_init(ictxt)
|
|
|
|
|
  call psb_info(ictxt,iam,np)
|
|
|
|
|
... ...
|
|
|
|
|
!
|
|
|
|
|
! read and assemble the spd matrix A and the right-hand side b
|
|
|
|
|
! using PSBLAS routines for sparse matrix / vector management
|
|
|
|
|
... ...
|
|
|
|
|
!
|
|
|
|
|
! initialize the default multilevel preconditioner, i.e. V-cycle
|
|
|
|
|
! with basic smoothed aggregation, 1 hybrid forward/backward
|
|
|
|
|
! GS sweep as pre/post-smoother and UMFPACK as coarsest-level
|
|
|
|
|
! solver
|
|
|
|
|
  call P%init(’ML’,info)
|
|
|
|
|
!
|
|
|
|
|
! build the preconditioner
|
|
|
|
|
  call P%hierarchy_build(A,desc_A,info)
|
|
|
|
|
  call P%smoothers_build(A,desc_A,info)
|
|
|
|
|
!
|
|
|
|
|
! set the solver parameters and the initial guess
|
|
|
|
|
  ... ...
|
|
|
|
|
!
|
|
|
|
|
! solve Ax=b with preconditioned CG
|
|
|
|
|
  call psb_krylov(’CG’,A,P,b,x,tol,desc_A,info)
|
|
|
|
|
  ... ...
|
|
|
|
|
!
|
|
|
|
|
! deallocate the preconditioner
|
|
|
|
|
  call P%free(info)
|
|
|
|
|
!
|
|
|
|
|
! deallocate other data structures
|
|
|
|
|
  ... ...
|
|
|
|
|
!
|
|
|
|
|
! exit the parallel environment
|
|
|
|
|
  call psb_exit(ictxt)
|
|
|
|
|
  stop
|
|
|
|
|
</pre>
|
|
|
|
|
<!--l. 193--><p class="nopar" >
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
<br /> <div class="caption"
|
|
|
|
|
><span class="id">Figure 1: </span><span
|
|
|
|
|
class="content">Application phase of a V-cycle preconditioner.</span></div><!--tex4ht:label?: x14-130021 -->
|
|
|
|
|
class="content">setup and application of the default multilevel preconditioner (example 1). </span></div><!--tex4ht:label?: x14-130011 -->
|
|
|
|
|
</div>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<!--l. 118--><p class="indent" > </div><hr class="endfigure">
|
|
|
|
|
<!--l. 120--><p class="indent" > <span
|
|
|
|
|
class="cmr-12">The components produced in the build phase may be combined in several ways to</span>
|
|
|
|
|
<!--l. 199--><p class="indent" > </div><hr class="endfigure">
|
|
|
|
|
<!--l. 201--><p class="indent" > <span
|
|
|
|
|
class="cmr-12">Different versions of the multilevel preconditioner can be obtained by changing the</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">obtain different multilevel preconditioners; this is done in the application phase, i.e., in</span>
|
|
|
|
|
class="cmr-12">default values of the preconditioner parameters. The code reported in Figure</span><span
|
|
|
|
|
class="cmr-12"> </span><a
|
|
|
|
|
href="#x14-130022"><span
|
|
|
|
|
class="cmr-12">2</span><!--tex4ht:ref: fig:ex2 --></a> <span
|
|
|
|
|
class="cmr-12">shows</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">the computation of a vector of type </span><span
|
|
|
|
|
class="cmmi-12">w </span><span
|
|
|
|
|
class="cmr-12">= </span><span
|
|
|
|
|
class="cmmi-12">B</span><sup><span
|
|
|
|
|
class="cmsy-8">-</span><span
|
|
|
|
|
class="cmr-8">1</span></sup><span
|
|
|
|
|
class="cmmi-12">v</span><span
|
|
|
|
|
class="cmr-12">, where </span><span
|
|
|
|
|
class="cmmi-12">B </span><span
|
|
|
|
|
class="cmr-12">denotes the preconditioner,</span>
|
|
|
|
|
class="cmr-12">how to set a V-cycle preconditioner which applies 1 block-Jacobi sweep as pre- and</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">usually within an iteration of a Krylov solver </span><span class="cite"><span
|
|
|
|
|
class="cmr-12">[</span><a
|
|
|
|
|
href="userhtmlli4.html#XSaad_book"><span
|
|
|
|
|
class="cmr-12">24</span></a><span
|
|
|
|
|
class="cmr-12">]</span></span><span
|
|
|
|
|
class="cmr-12">. An example of such a combination,</span>
|
|
|
|
|
class="cmr-12">post-smoother, and solves the coarsest-level system with 8 block-Jacobi sweeps. Note</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">that the ILU(0) factorization (plus triangular solve) is used as local solver for the</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">block-Jacobi sweeps, since this is the default associated with block-Jacobi</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">and set by</span><span
|
|
|
|
|
class="cmr-12"> </span><span class="obeylines-h"><span class="verb"><span
|
|
|
|
|
class="cmtt-12">P%init</span></span></span><span
|
|
|
|
|
class="cmr-12">. Furthermore, specifying block-Jacobi as coarsest-level</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">solver implies that the coarsest-level matrix is distributed among the processes.</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">known as V-cycle, is given in Figure</span><span
|
|
|
|
|
class="cmr-12">Figure</span><span
|
|
|
|
|
class="cmr-12"> </span><a
|
|
|
|
|
href="#x14-130021"><span
|
|
|
|
|
class="cmr-12">1</span><!--tex4ht:ref: fig:application_alg --></a><span
|
|
|
|
|
class="cmr-12">. In this case, a single iteration of the same</span>
|
|
|
|
|
href="#x14-130033"><span
|
|
|
|
|
class="cmr-12">3</span><!--tex4ht:ref: fig:ex3 --></a> <span
|
|
|
|
|
class="cmr-12">shows how to set a W-cycle preconditioner using the Coarsening</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">smoother is used before and after the the recursive call to the V-cycle (i.e., in the</span>
|
|
|
|
|
class="cmr-12">based on Compatible Weighted Matching. It applies 2 hybrid Gauss-Seidel</span>
|
|
|
|
|
<span
|
|
|
|
|
class="cmr-12">pre-smoothing and post-smoothing phases); however, different choices can be</span>
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class="cmr-12">sweeps as pre- and post-smoother, and solves the coarsest-level system with the</span>
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<span
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class="cmr-12">performed. Other cycles can be defined; in MLD2P4, we implemented the</span>
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class="cmr-12">multifrontal LU factorization implemented in MUMPS. It is specified that the</span>
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<span
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class="cmr-12">standard V-cycle and W-cycle</span><span
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class="cmr-12"> </span><span class="cite"><span
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class="cmr-12">[</span><a
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href="userhtmlli4.html#XBriggs2000"><span
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class="cmr-12">3</span></a><span
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class="cmr-12">]</span></span><span
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class="cmr-12">, and a version of the K-cycle described</span>
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class="cmr-12">coarsest-level matrix is distributed, since MUMPS can be used on both replicated and</span>
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<span
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class="cmr-12">in</span><span
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class="cmr-12"> </span><span class="cite"><span
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class="cmr-12">[</span><a
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href="userhtmlli4.html#XNotay2008"><span
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class="cmr-12">23</span></a><span
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class="cmr-12">]</span></span><span
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class="cmr-12">distributed matrices, and by default it is used on replicated ones. The code</span>
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<span
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class="cmr-12">fragments shown in Figures</span><span
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class="cmr-12"> </span><a
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href="#x14-130022"><span
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class="cmr-12">2</span><!--tex4ht:ref: fig:ex2 --></a> <span
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class="cmr-12">and </span><a
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href="#x14-130033"><span
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class="cmr-12">3</span><!--tex4ht:ref: fig:ex3 --></a> <span
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class="cmr-12">are included in the example program file</span>
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<span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">amg_dexample_ml.f90</span></span></span> <span
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class="cmr-12">too. </span><span
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class="cmbx-12">DA CORREGGERE NEL CODICE ESEMPIO</span>
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<span
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class="cmbx-12">3</span>
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<!--l. 225--><p class="indent" > <span
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class="cmr-12">Finally, Figure</span><span
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class="cmr-12"> </span><a
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href="#x14-130044"><span
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class="cmr-12">4</span><!--tex4ht:ref: fig:ex4 --></a> <span
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class="cmr-12">shows the setup of a one-level additive Schwarz preconditioner,</span>
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<span
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class="cmr-12">i.e., RAS with overlap 2. Note also that a Krylov method different from CG</span>
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<span
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class="cmr-12">must be used to solve the preconditioned system, since the preconditione in</span>
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<span
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class="cmr-12">nonsymmetric. The corresponding example program is available in the file</span>
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<span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">amg_dexample_1lev.f90</span></span></span><span
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class="cmr-12">.</span>
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<!--l. 232--><p class="indent" > <span
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class="cmr-12">For all the previous preconditioners, example programs where the sparse matrix and</span>
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<span
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class="cmr-12">the right-hand side are generated by discretizing a PDE with Dirichlet boundary</span>
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<span
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class="cmr-12">conditions are also available in the directory </span><span class="obeylines-h"><span class="verb"><span
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class="cmtt-12">examples/pdegen</span></span></span><span
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class="cmr-12">.</span>
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<!--l. 236--><p class="indent" > <hr class="figure"><div class="figure"
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>
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<a
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id="x14-130022"></a>
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<div class="center"
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>
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<!--l. 237--><p class="noindent" >
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<div class="minipage"><pre class="verbatim" id="verbatim-7">
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... ...
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! build a V-cycle preconditioner with 1 block-Jacobi sweep (with
|
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! ILU(0) on the blocks) as pre- and post-smoother, and 8  block-Jacobi
|
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|
! sweeps (with ILU(0) on the blocks) as coarsest-level solver
|
|
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|
|
  call P%init(’ML’,info)
|
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|
  call_P%set(’SMOOTHER_TYPE’,’BJAC’,info)
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  call P%set(’COARSE_SOLVE’,’BJAC’,info)
|
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  call P%set(’COARSE_SWEEPS’,8,info)
|
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  call P%hierarchy_build(A,desc_A,info)
|
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  call P%smoothers_build(A,desc_A,info)
|
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|
... ...
|
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|
</pre>
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<!--l. 252--><p class="nopar" ></div>
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<br /><div class="caption"
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><span class="id">Figure 2: </span><span
|
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|
class="content">setup of a multilevel preconditioner based on the default decoupled coarsening</span></div><!--tex4ht:label?: x14-130022 -->
|
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</div>
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<!--l. 133--><div class="crosslinks"><p class="noindent"><span
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<!--l. 258--><p class="indent" > </div><hr class="endfigure">
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<!--l. 260--><p class="indent" > <hr class="figure"><div class="figure"
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>
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<a
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id="x14-130033"></a>
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<div class="center"
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>
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<!--l. 261--><p class="noindent" >
|
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|
|
<div class="minipage"><pre class="verbatim" id="verbatim-8">
|
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|
|
... ...
|
|
|
|
|
! build a W-cycle preconditioner with 2 hybrid Gauss-Seidel sweeps
|
|
|
|
|
! as pre- and post-smoother, a distributed coarsest
|
|
|
|
|
! matrix, and MUMPS as coarsest-level solver
|
|
|
|
|
  call P%init(’ML’,info)
|
|
|
|
|
  call P%set(’PAR_AGGR_ALG’,’COUPLED’,info)
|
|
|
|
|
  call P%set(’ML_CYCLE’,’WCYCLE’,info)
|
|
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|
|
  call P%set(’SMOOTHER_TYPE’,’FBGS’,info)
|
|
|
|
|
  call P%set(’SMOOTHER_SWEEPS’,2,info)
|
|
|
|
|
  call P%set(’COARSE_SOLVE’,’MUMPS’,info)
|
|
|
|
|
  call P%set(’COARSE_MAT’,’DIST’,info)
|
|
|
|
|
  call P%hierarchy_build(A,desc_A,info)
|
|
|
|
|
  call P%smoothers_build(A,desc_A,info)
|
|
|
|
|
... ...
|
|
|
|
|
</pre>
|
|
|
|
|
<!--l. 279--><p class="nopar" ></div>
|
|
|
|
|
<br /> <div class="caption"
|
|
|
|
|
><span class="id">Figure 3: </span><span
|
|
|
|
|
class="content">setup of a multilevel preconditioner based on the coupled coarsening based
|
|
|
|
|
on weighted matching</span></div><!--tex4ht:label?: x14-130033 -->
|
|
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|
|
</div>
|
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<!--l. 284--><p class="indent" > </div><hr class="endfigure">
|
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|
<!--l. 286--><p class="indent" > <hr class="figure"><div class="figure"
|
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>
|
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<a
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id="x14-130044"></a>
|
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<div class="center"
|
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>
|
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|
<!--l. 287--><p class="noindent" >
|
|
|
|
|
<div class="minipage"><pre class="verbatim" id="verbatim-9">
|
|
|
|
|
... ...
|
|
|
|
|
! set RAS with overlap 2 and ILU(0) on the local blocks
|
|
|
|
|
  call P%init(’AS’,info)
|
|
|
|
|
  call P%set(’SUB_OVR’,2,info)
|
|
|
|
|
  call P%bld(A,desc_A,info)
|
|
|
|
|
... ...
|
|
|
|
|
! solve Ax=b with preconditioned BiCGSTAB
|
|
|
|
|
  call psb_krylov(’BICGSTAB’,A,P,b,x,tol,desc_A,info)
|
|
|
|
|
</pre>
|
|
|
|
|
<!--l. 299--><p class="nopar" ></div>
|
|
|
|
|
<br /> <div class="caption"
|
|
|
|
|
><span class="id">Figure 4: </span><span
|
|
|
|
|
class="content">setup of a one-level Schwarz preconditioner.</span></div><!--tex4ht:label?: x14-130044 -->
|
|
|
|
|
</div>
|
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|
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|
<!--l. 304--><p class="indent" > </div><hr class="endfigure">
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<!--l. 1--><div class="crosslinks"><p class="noindent"><span
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class="cmr-12">[</span><a
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href="userhtmlsu7.html" ><span
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class="cmr-12">next</span></a><span
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class="cmr-12">] [</span><a
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href="userhtmlsu6.html" ><span
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class="cmr-12">front</span></a><span
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class="cmr-12">] [</span><a
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href="userhtmlse4.html#userhtmlsu6.html" ><span
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class="cmr-12">up</span></a><span
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class="cmr-12">] </span></p></div>
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<!--l. 133--><p class="indent" > <a
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<!--l. 1--><p class="indent" > <a
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id="tailuserhtmlsu6.html"></a>
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</body></html>
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Salvatore Filippone
4 years ago