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373 lines
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373 lines
16 KiB
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<!--l. 1--><div class="crosslinks"><p class="noindent"><span
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<h3 class="sectionHead"><span class="titlemark"><span
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class="cmr-12">1 </span></span> <a
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id="x4-30001"></a><span
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class="cmr-12">General Overview</span></h3>
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<!--l. 5--><p class="noindent" ><span
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class="cmr-12">The </span><span
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class="cmcsc-10x-x-120">M<span
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class="small-caps">u</span><span
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class="small-caps">l</span><span
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class="small-caps">t</span><span
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class="small-caps">i</span>L<span
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class="small-caps">e</span><span
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class="small-caps">v</span><span
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class="small-caps">e</span><span
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class="small-caps">l</span> D<span
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class="small-caps">o</span><span
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class="small-caps">m</span><span
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class="small-caps">a</span><span
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class="small-caps">i</span><span
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class="small-caps">n</span> D<span
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class="small-caps">e</span><span
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class="small-caps">c</span><span
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class="small-caps">o</span><span
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class="small-caps">m</span><span
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class="small-caps">i</span><span
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class="small-caps">t</span><span
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class="small-caps">i</span><span
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class="small-caps">o</span><span
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class="small-caps">n</span> P<span
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class="small-caps">a</span><span
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class="small-caps">r</span><span
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class="small-caps">a</span><span
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class="small-caps">l</span><span
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class="small-caps">l</span><span
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class="small-caps">e</span><span
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class="small-caps">l</span> P<span
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class="small-caps">r</span><span
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class="small-caps">e</span><span
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class="small-caps">c</span><span
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class="small-caps">o</span><span
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class="small-caps">n</span><span
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class="small-caps">d</span><span
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class="small-caps">i</span><span
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class="small-caps">t</span><span
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class="small-caps">i</span><span
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class="small-caps">o</span><span
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class="small-caps">n</span><span
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class="small-caps">e</span><span
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class="small-caps">r</span><span
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class="small-caps">s</span></span>
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<span
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class="cmcsc-10x-x-120">P<span
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class="small-caps">a</span><span
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class="small-caps">c</span><span
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class="small-caps">k</span><span
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class="small-caps">a</span><span
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class="small-caps">g</span><span
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class="small-caps">e</span> <span
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class="small-caps">b</span><span
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class="small-caps">a</span><span
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class="small-caps">s</span><span
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class="small-caps">e</span><span
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class="small-caps">d</span> <span
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class="small-caps">o</span><span
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class="small-caps">n</span> PSBLAS (MLD2P4</span><span
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class="cmr-12">) provides parallel Algebraic MultiGrid</span>
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<span
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class="cmr-12">(AMG) and Domain Decomposition preconditioners (see, e.g., </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
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class="cmr-12"> </span><a
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href="userhtmlli4.html#XStuben_01"><span
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class="cmr-12">24</span></a><span
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class="cmr-12">,</span><span
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class="cmr-12"> </span><a
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href="userhtmlli4.html#Xdd2_96"><span
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class="cmr-12">22</span></a><span
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class="cmr-12">]</span></span><span
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class="cmr-12">), to be used</span>
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<span
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class="cmr-12">in the iterative solution of linear systems,</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="userhtml0x.png" alt="Ax = b,
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" class="math-display" ><a
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id="x4-3001r1"></a></center></td><td class="equation-label"><span
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class="cmr-12">(1)</span></td></tr></table>
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<!--l. 12--><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">is a square, real or complex, sparse matrix. The name of the package comes</span>
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<span
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class="cmr-12">from its original implementation, containing multilevel additive and hybrid Schwarz</span>
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<span
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class="cmr-12">preconditioners, as well as one-level additive Schwarz preconditioners. The current</span>
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<span
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class="cmr-12">version extends the original plan by including multilevel cycles and smoothers widely</span>
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<span
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class="cmr-12">used in multigrid methods.</span>
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<!--l. 24--><p class="indent" > <span
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class="cmr-12">The multilevel preconditioners implemented in MLD2P4 are obtained by</span>
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<span
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class="cmr-12">combining AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and</span>
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<span
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class="cmr-12">K-cycles</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
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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">20</span></a><span
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class="cmr-12">]</span></span> <span
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class="cmr-12">are available, which allow to define almost all the preconditioners in</span>
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<span
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class="cmr-12">the package, including the multilevel hybrid Schwarz ones; a specific cycle is</span>
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<span
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class="cmr-12">implemented to obtain multilevel additive Schwarz preconditioners. The Jacobi,</span>
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<span
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class="cmr-12">hybrid forward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz</span>
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<span
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class="cmr-12">methods are available as smoothers. An algebraic approach is used to generate a</span>
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<span
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class="cmr-12">hierarchy of coarse-level matrices and operators, without explicitly using any</span>
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<span
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class="cmr-12">information on the geometry of the original problem, e.g., the discretization of a</span>
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<span
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class="cmr-12">PDE. To this end, the smoothed aggregation technique</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#XBREZINA_VANEK"><span
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class="cmr-12">2</span></a><span
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class="cmr-12">,</span><span
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class="cmr-12"> </span><a
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href="userhtmlli4.html#XVANEK_MANDEL_BREZINA"><span
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class="cmr-12">26</span></a><span
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class="cmr-12">]</span></span> <span
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class="cmr-12">is applied.</span>
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<span
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class="cmr-12">Either exact or approximate solvers can be used on the coarsest-level system.</span>
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<span
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class="cmr-12">Specifically, different sparse LU factorizations from external packages, and</span>
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<span
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class="cmr-12">native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel, and</span>
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<span
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class="cmr-12">block-Jacobi solvers are available. All smoothers can be also exploited as one-level</span>
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<span
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class="cmr-12">preconditioners.</span>
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<!--l. 42--><p class="indent" > <span
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class="cmr-12">MLD2P4 is written in Fortran</span><span
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class="cmr-12"> 2003, following an object-oriented design</span>
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<span
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class="cmr-12">through the exploitation of features such as abstract data type creation, type</span>
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<span
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class="cmr-12">extension, functional overloading, and dynamic memory management. The</span>
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<span
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class="cmr-12">parallel implementation is based on a Single Program Multiple Data (SPMD)</span>
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<span
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class="cmr-12">paradigm. Single and double precision implementations of MLD2P4 are available</span>
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<span
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class="cmr-12">for both the real and the complex case, which can be used through a single</span>
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<span
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class="cmr-12">interface.</span>
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<!--l. 53--><p class="indent" > <span
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class="cmr-12">MLD2P4 has been designed to implement scalable and easy-to-use multilevel</span>
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<span
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class="cmr-12">preconditioners in the context of the PSBLAS (Parallel Sparse BLAS) computational</span>
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<span
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class="cmr-12">framework</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#Xpsblas_00"><span
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class="cmr-12">15</span></a><span
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class="cmr-12">,</span><span
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class="cmr-12"> </span><a
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href="userhtmlli4.html#XPSBLAS3"><span
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class="cmr-12">14</span></a><span
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class="cmr-12">]</span></span><span
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class="cmr-12">. PSBLAS provides basic linear algebra operators and data</span>
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<span
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class="cmr-12">management facilities for distributed sparse matrices, as well as parallel Krylov solvers</span>
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<span
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class="cmr-12">which can be used with the MLD2P4 preconditioners. The choice of PSBLAS has been</span>
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<span
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class="cmr-12">mainly motivated by the need of having a portable and efficient software infrastructure</span>
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<span
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class="cmr-12">implementing “de facto” standard parallel sparse linear algebra kernels, to pursue goals</span>
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<span
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class="cmr-12">such as performance, portability, modularity ed extensibility in the development of the</span>
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<span
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class="cmr-12">preconditioner package. On the other hand, the implementation of MLD2P4 has led to</span>
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<span
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class="cmr-12">some revisions and extentions of the original PSBLAS kernels. The inter-process</span>
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<span
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class="cmr-12">comunication required by MLD2P4 is encapsulated in the PSBLAS routines; therefore,</span>
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<span
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class="cmr-12">MLD2P4 can be run on any parallel machine where PSBLAS implementations are</span>
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<span
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class="cmr-12">available.</span>
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<!--l. 70--><p class="indent" > <span
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class="cmr-12">MLD2P4 has a layered and modular software architecture where three main layers</span>
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<span
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class="cmr-12">can be identified. The lower layer consists of the PSBLAS kernels, the middle one</span>
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<span
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class="cmr-12">implements the construction and application phases of the preconditioners, and the</span>
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<span
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class="cmr-12">upper one provides a uniform interface to all the preconditioners. This architecture</span>
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<span
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class="cmr-12">allows for different levels of use of the package: few black-box routines at the upper</span>
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<span
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class="cmr-12">layer allow all users to easily build and apply any preconditioner available in</span>
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<span
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class="cmr-12">MLD2P4; facilities are also available allowing expert users to extend the set of</span>
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<span
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class="cmr-12">smoothers and solvers for building new versions of the preconditioners (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="userhtmlse7.html#x29-330007"><span
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class="cmr-12">7</span><!--tex4ht:ref: sec:adding --></a><span
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class="cmr-12">).</span>
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<!--l. 81--><p class="indent" > <span
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class="cmr-12">We note that the user interface of MLD2P4 2.1 has been extended with respect to</span>
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<span
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class="cmr-12">the previous versions in order to separate the construction of the multilevel</span>
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<span
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class="cmr-12">hierarchy from the construction of the smoothers and solvers, and to allow for</span>
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<span
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class="cmr-12">more flexibility at each level. The software architecture described 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#XMLD2P4_TOMS"><span
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class="cmr-12">8</span></a><span
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class="cmr-12">]</span></span> <span
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class="cmr-12">has</span>
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<span
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class="cmr-12">significantly evolved too, in order to fully exploit the Fortran</span><span
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class="cmr-12"> 2003 features</span>
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<span
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class="cmr-12">implemented in PSBLAS 3. However, compatibility with previous versions has been</span>
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<span
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class="cmr-12">preserved.</span>
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<!--l. 88--><p class="indent" > <span
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class="cmr-12">This guide is organized as follows. General information on the distribution of the</span>
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<span
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class="cmr-12">source code is reported in Section</span><span
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class="cmr-12"> </span><a
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href="userhtmlse2.html#x5-40002"><span
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class="cmr-12">2</span><!--tex4ht:ref: sec:distribution --></a><span
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class="cmr-12">, while details on the configuration and installation</span>
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<span
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class="cmr-12">of the package are given in Section</span><span
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class="cmr-12"> </span><a
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href="userhtmlse3.html#x7-60003"><span
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class="cmr-12">3</span><!--tex4ht:ref: sec:building --></a><span
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class="cmr-12">. A short description of the preconditioners</span>
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<span
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class="cmr-12">implemented in MLD2P4 is provided in Section</span><span
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class="cmr-12"> </span><a
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href="userhtmlse4.html#x13-120004"><span
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class="cmr-12">4</span><!--tex4ht:ref: sec:background --></a><span
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class="cmr-12">, to help the users in choosing among</span>
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<span
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class="cmr-12">them. The basics for building and applying the preconditioners with the Krylov solvers</span>
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<span
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class="cmr-12">implemented in PSBLAS are reported in</span><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="userhtmlse5.html#x17-160005"><span
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class="cmr-12">5</span><!--tex4ht:ref: sec:started --></a><span
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class="cmr-12">, where the Fortran codes of a few</span>
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<span
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class="cmr-12">sample programs are also shown. A reference guide for the user interface routines is</span>
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<span
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class="cmr-12">provided in Section</span><span
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class="cmr-12"> </span><a
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href="userhtmlse6.html#x19-180006"><span
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class="cmr-12">6</span><!--tex4ht:ref: sec:userinterface --></a><span
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class="cmr-12">. Information on the extension of the package through the</span>
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<span
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class="cmr-12">addition of new smoothers and solvers is reported in Section</span><span
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class="cmr-12"> </span><a
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href="userhtmlse7.html#x29-330007"><span
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class="cmr-12">7</span><!--tex4ht:ref: sec:adding --></a><span
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class="cmr-12">. The error handling</span>
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<span
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class="cmr-12">mechanism used by the package is briefly described in Section</span><span
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class="cmr-12"> </span><a
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href="userhtmlse8.html#x30-340008"><span
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class="cmr-12">8</span><!--tex4ht:ref: sec:errors --></a><span
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class="cmr-12">. The copyright</span>
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<span
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class="cmr-12">terms concerning the distribution and modification of MLD2P4 are reported in</span>
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<span
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class="cmr-12">Appendix</span><span
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class="cmr-12"> </span><a
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href="userhtmlse9.html#x31-35000A"><span
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class="cmr-12">A</span><!--tex4ht:ref: sec:license --></a><span
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class="cmr-12">.</span>
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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="userhtmlse2.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="userhtmlli2.html" ><span
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class="cmr-12">prev</span></a><span
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class="cmr-12">] [</span><a
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href="userhtmlli2.html#tailuserhtmlli2.html" ><span
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class="cmr-12">prev-tail</span></a><span
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class="cmr-12">] [</span><a
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href="userhtmlse1.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="userhtml.html#userhtmlse1.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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