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130
Schwarz Analysis Of Iterative Substructuring Algorithms For Elliptic Problems In Three Dimensions
 SIAM J. Numer. Anal
, 1993
"... . Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate sol ..."
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Cited by 144 (32 self)
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. Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equation restricted to a set of subregions into which the given region has been divided. In addition, the preconditioner is often augmented by a coarse, second level approximation, that provides additional, global exchange of information, and which can enhance the rate of convergence considerably. The iterative substructuring methods, based on decompositions of the region into nonoverlapping subregions, form one of the main families of such algorithms. Many domain decomposition algorithms can conveniently be described and analyzed as Schwarz methods. These algorithms are fully defined in terms of a set of subspaces and auxiliary bilinear forms. A gener...
Domain Decomposition Algorithms With Small Overlap
, 1994
"... Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear ..."
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Cited by 101 (12 self)
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Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables interior to the subregions have been eliminated. In this paper, a supporting theory is developed.
Some Nonoverlapping Domain Decomposition Methods
, 1998
"... . The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the Neumann ..."
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Cited by 57 (11 self)
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. The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the NeumannNeumanntype methods. The basic framework used for analysis is the parallel subspace correction method or additive Schwarz method, and other technical tools include localglobal and globallocal techniques. The analyses for both two and threedimensional cases are carried out simultaneously. Some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived. Key words. nonoverlapping domain decomposition, Schur complement, localglobal and globallocal techniques, jumps in coe#cients, substructuring, NeumannNeumann, balancing methods AMS subject classifications. 65N30, 65N55, 65F10 PII. S0036144596306800 1. Introduction. T...
A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions
 SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 48 (16 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
Some Schwarz Methods For Symmetric And Nonsymmetric Elliptic Problems
 Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations
, 1992
"... . This paper begins with an introduction to additive and multiplicative Schwarz methods. A twolevel method is then reviewed and a new result on its rate of convergence is established for the case when the overlap is small. Recent results by Xuejun Zhang, on multilevel Schwarz methods, are formulat ..."
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Cited by 38 (2 self)
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. This paper begins with an introduction to additive and multiplicative Schwarz methods. A twolevel method is then reviewed and a new result on its rate of convergence is established for the case when the overlap is small. Recent results by Xuejun Zhang, on multilevel Schwarz methods, are formulated and discussed. The paper is concluded with a discussion of recent joint results with XiaoChuan Cai on nonsymmetric and indefinite problems. Key Words. domain decomposition, Schwarz methods, finite elements, nonsymmetric and indefinite elliptic problems AMS(MOS) subject classifications. 65F10, 65N30 1. Introduction. Over the last few years, a general theory has been developed for the study of additive and multiplicative Schwarz methods. Many domain decomposition and certain multigrid methods can now be successfully analyzed inside this framework. Early work by P.L. Lions [23], [24] gave an important impetus to this effort. The additive Schwarz methods were then developed by Dryja and ...
On The Spectra Of Sums Of Orthogonal Projections With Applications To Parallel Computing
, 1991
"... Many parallel iterative algorithms for solving symmetric, positive definite problems proceed by solving in each iteration, a number of independent systems on subspaces. The convergence of such methods is determined by the spectrum of the sums of orthogonal projections on those subspaces, while the c ..."
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Cited by 31 (3 self)
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Many parallel iterative algorithms for solving symmetric, positive definite problems proceed by solving in each iteration, a number of independent systems on subspaces. The convergence of such methods is determined by the spectrum of the sums of orthogonal projections on those subspaces, while the convergence of a related sequential method is determined by the spectrum of the product of complementary projections. We study spectral properties of sums of orthogonal projections and in the case of two projections, characterize the spectrum of the sum completely in terms of the spectrum of the product.
Width and finite extinction time of Ricci flow
, 2007
"... This is an expository article with complete proofs intended for a general nonspecialist audience. The results are twofold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2spheres. For instance, when M i ..."
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Cited by 26 (1 self)
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This is an expository article with complete proofs intended for a general nonspecialist audience. The results are twofold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2spheres. For instance, when M is a homotopy 3sphere, the width is loosely speaking the
Prioritization methods for accelerating MDP solvers
 Journal of Machine Learning Research
, 2005
"... The performance of value and policy iteration can be dramatically improved by eliminating redundant or useless backups, and by backing up states in the right order. We study several methods designed to accelerate these iterative solvers, including prioritization, partitioning, and variable reorderin ..."
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Cited by 26 (2 self)
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The performance of value and policy iteration can be dramatically improved by eliminating redundant or useless backups, and by backing up states in the right order. We study several methods designed to accelerate these iterative solvers, including prioritization, partitioning, and variable reordering. We generate a family of algorithms by combining several of the methods discussed, and present extensive empirical evidence demonstrating that performance can improve by several orders of magnitude for many problems, while preserving accuracy and convergence guarantees.
Asynchronous Weighted Additive Schwarz Methods
, 1997
"... A class of asynchronous Schwarz methods for the parallel solution of nonsingular linear systems of the formAx=fis investigated. This class includes, in particular, an asynchronous algebraic Schwarz method as well as asynchronous multisplitting. Theorems are obtained demonstrating convergence for the ..."
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Cited by 25 (14 self)
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A class of asynchronous Schwarz methods for the parallel solution of nonsingular linear systems of the formAx=fis investigated. This class includes, in particular, an asynchronous algebraic Schwarz method as well as asynchronous multisplitting. Theorems are obtained demonstrating convergence for the cases whenA?1 is nonnegative and when A is an Hmatrix. The results shown are for both the situations with or without overlap between the domains in which an underlying mesh is divided, if such a mesh exists. Numerical experiments on systems of up to over ten million variables on up to 256 processors are presented. They illustrate the convergence properties of the method, as well as the fact that when the domains are not all of the same size, the asynchronous method can be up to 50 % faster than the corresponding synchronous one.