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original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION00060000000000000000"></A><A NAME="sec:background"></A>
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<BR>
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Multi-level Domain Decomposition Background
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</H1>
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<P>
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<I>Domain Decomposition</I> (DD) preconditioners, coupled with Krylov iterative
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solvers, are widely used in the parallel solution of large and sparse linear systems.
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These preconditioners are based on the divide and conquer technique: the matrix
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to be preconditioned is divided into submatrices, a ``local'' linear system
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involving each submatrix is (approximately) solved, and the local solutions are used
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to build a preconditioner for the whole original matrix. This process
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often corresponds to dividing a physical domain associated to the original matrix
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into subdomains, e.g. in a PDE discretization, to (approximately) solving the
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subproblems corresponding to the subdomains and to building an approximate
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solution of the original problem from the local solutions
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[<A
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HREF="node25.html#Cai_Widlund_92">6</A>,<A
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HREF="node25.html#dd1_94">7</A>,<A
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HREF="node25.html#dd2_96">22</A>].
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<P>
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<I>Additive Schwarz</I> preconditioners are DD preconditioners using overlapping
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submatrices, i.e. with some common rows, to couple the local information
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related to the submatrices (see, e.g., [<A
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HREF="node25.html#dd2_96">22</A>]).
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The main motivation for choosing Additive Schwarz preconditioners is their
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intrinsic parallelism. A drawback of these
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preconditioners is that the number of iterations of the preconditioned solvers
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generally grows with the number of submatrices. This may be a serious limitation
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on parallel computers, since the number of submatrices usually matches the number
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of available processors. Optimal convergence rates, i.e. iteration numbers
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independent of the number of submatrices, can be obtained by correcting the
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preconditioner through a suitable approximation of the original linear system
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in a coarse space, which globally couples the information related to the single
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submatrices.
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<P>
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<I>Two-level Schwarz</I> preconditioners are obtained
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by combining basic (one-level) Schwarz preconditioners with a coarse-level
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correction. In this context, the one-level preconditioner is often
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called `smoother'. Different two-level preconditioners are obtained by varying the
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choice of the smoother and of the coarse-level correction, and the
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way they are combined [<A
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HREF="node25.html#dd2_96">22</A>]. The same reasoning can be applied starting
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from the coarse-level system, i.e. a coarse-space correction can be built
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from this system, thus obtaining <I>multi-level</I> preconditioners.
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<P>
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It is worth noting that optimal preconditioners do not necessarily correspond
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to minimum execution times. Indeed, to obtain effective multi-level preconditioners
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a tradeoff between optimality of convergence and the cost of building and applying
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the coarse-space corrections must be achieved. The choice of the number of levels,
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i.e. of the coarse-space corrections, also affects the effectiveness of the
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preconditioners. One more goal is to get convergence rates as less sensitive
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as possible to variations in the matrix coefficients.
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<P>
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Two main approaches can be used to build coarse-space corrections. The geometric approach
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applies coarsening strategies based on the knowledge of some physical grid associated
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to the matrix and requires the user to define grid transfer operators from the fine
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to the coarse levels and vice versa. This may result difficult for complex geometries;
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furthermore, suitable one-level preconditioners may be required to get efficient
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interplay between fine and coarse levels, e.g. when matrices with highly varying coefficients
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are considered. The algebraic approach builds coarse-space corrections using only matrix
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information. It performs a fully automatic coarsening and enforces the interplay between
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the fine and coarse levels by suitably choosing the coarse space and the coarse-to-fine
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interpolation [<A
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HREF="node25.html#StubenGMD69_99">24</A>].
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<P>
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MLD2P4 uses a pure algebraic approach for building the sequence of coarse matrices
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starting from the original matrix. The algebraic approach is based on the <I>smoothed
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aggregation</I> algorithm [<A
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HREF="node25.html#BREZINA_VANEK">1</A>,<A
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HREF="node25.html#VANEK_MANDEL_BREZINA">26</A>]. A decoupled version
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of this algorithm is implemented, where the smoothed aggregation is applied locally
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to each submatrix [<A
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HREF="node25.html#TUMINARO_TONG">25</A>]. In the next two subsections we provide
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a brief description of the multi-level Schwarz preconditioners and of the smoothed
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aggregation technique as implemented in MLD2P4. For further details the reader
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is referred to [<A
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HREF="node25.html#para_04">2</A>,<A
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HREF="node25.html#aaecc_07">3</A>,<A
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HREF="node25.html#apnum_07">4</A>,<A
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HREF="node25.html#MLD2P4_TOMS">8</A>,<A
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HREF="node25.html#dd2_96">22</A>].
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html197"
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HREF="node12.html">Multi-level Schwarz Preconditioners</A>
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<LI><A NAME="tex2html198"
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HREF="node13.html">Smoothed Aggregation</A>
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</UL>
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HREF="node10.html">Example and test programs</A>
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