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374
Graph Partitioning Algorithms With Applications To Scientific Computing
 Parallel Numerical Algorithms
, 1997
"... Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of su ..."
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Cited by 41 (0 self)
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Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of subsets such that few edges join two vertices in different subsets. Several new graph partitioning algorithms have been developed in the past few years, and we survey some of this activity. We describe the terminology associated with graph partitioning, the complexity of computing good separators, and graphs that have good separators. We then discuss early algorithms for graph partitioning, followed by three new algorithms based on geometric, algebraic, and multilevel ideas. The algebraic algorithm relies on an eigenvector of a Laplacian matrix associated with the graph to compute the partition. The algebraic algorithm is justified by formulating graph partitioning as a quadratic assignment p...
An EnergyMinimizing Interpolation For Robust Multigrid Methods
 SIAM J. SCI. COMPUT
, 1998
"... We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coe ..."
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Cited by 37 (6 self)
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We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coefficient of the underlying PDE, in addition to being independent of the mesh size. We demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem and an oscillatory coefficient problem. We also show using a onedimensional Helmholtz problem that the energy minimization principle can be applied to solving elliptic problems that are not positive definite.
Optimized Schwarz methods without overlap for the Helmholtz equation
 SIAM J. Sci. Comput
"... Abstract. The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz eq ..."
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Cited by 37 (12 self)
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Abstract. The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps. These conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the Schwarz method. This leads to an algorithm in the class of optimized Schwarz methods. We present an asymptotic analysis of the optimized Schwarz method for two types of transmission conditions, Robin conditions and transmission conditions with second order tangential derivatives. Numerical results illustrate the effectiveness of the optimized Schwarz method on a model problem and on a problem from industry.
Rate Of Convergence Of Some Space Decomposition Methods For Linear And Nonlinear Problems
 SIAM J. Numer. Anal
, 1998
"... . Convergence of a space decomposition method is proved for a class of convex programming problems. A space decomposition refers to a method that decomposes a space into a sum of subspaces, which could be a domain decomposition or a multilevel method when applied to partial di#erential e ..."
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Cited by 37 (14 self)
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.<F3.784e+05> Convergence of a space decomposition method is proved for a class of convex programming problems. A space decomposition refers to a method that decomposes a space into a sum of subspaces, which could be a domain decomposition or a multilevel method when applied to partial di#erential equations. Two algorithms are proposed. Both can be used for linear as well as nonlinear elliptic problems, and they reduce to the standard additive and multiplicative Schwarz methods for linear elliptic problems.<F4.005e+05> Key words.<F3.784e+05> parallel, domain decomposition, nonlinear, elliptic equation, space decomposition<F4.005e+05> AMS subject classifications.<F3.784e+05> 65J10, 65M55, 65Y05<F4.005e+05> PII.<F3.784e+05> S0036142996297461<F4.795e+05> 1. Introduction.<F4.397e+05> We use space decomposition methods to solve a convex programming problem. When the minimization space is suitably decomposed into subspaces, two algorithms are proposed to solve the minimization problem. The f...
Globalized NewtonKrylovSchwarz algorithms and software for parallel implicit CFD
 Int. J. High Performance Computing Applications
, 1998
"... Key words. NewtonKrylovSchwarz algorithms, parallel CFD, implicit methods Abstract. Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is e ..."
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Cited by 36 (14 self)
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Key words. NewtonKrylovSchwarz algorithms, parallel CFD, implicit methods Abstract. Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is essential. The pseudotransient matrixfree NewtonKrylovSchwarz (ΨNKS) algorithmic framework is presented as a widely applicable answer. This article shows that, for the classical problem of threedimensional transonic Euler flow about an M6 wing, ΨNKS can simultaneously deliver • globalized, asymptotically rapid convergence through adaptive pseudotransient continuation and Newton’s method; • reasonable parallelizability for an implicit method through deferred synchronization and favorable communicationtocomputation scaling in the Krylov linear solver; and • high perprocessor performance through attention to distributed memory and cache locality, especially through the Schwarz preconditioner. Two discouraging features of ΨNKS methods are their sensitivity to the coding of the underlying PDE discretization and the large number of parameters that must be selected to govern convergence. We therefore distill several recommendations from our experience and from our reading of the literature on various algorithmic components of ΨNKS, and we describe a freely available, MPIbased portable parallel software implementation of the solver employed here. 1. Introduction. Disparate
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 35 (10 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.
SupportGraph Preconditioners
 SIAM Journal on Matrix Analysis and Applications
"... . We present a littleknown preconditioning technique, called supportgraph preconditioning, and use it to analyze two classes of preconditioners. The technique was first described in a talk by Pravin Vaidya, who did not formally publish his results. Vaidya used the technique to devise and analyze a ..."
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Cited by 35 (13 self)
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. We present a littleknown preconditioning technique, called supportgraph preconditioning, and use it to analyze two classes of preconditioners. The technique was first described in a talk by Pravin Vaidya, who did not formally publish his results. Vaidya used the technique to devise and analyze a class of novel preconditioners. The technique was later extended by Gremban and Miller, who used it in the development and analysis of yet another class of new preconditioners. This paper extends the technique further and uses it to analyze two classes of existing preconditioners: modified incompleteCholesky and multilevel diagonal scaling. The paper also contains a presentation of Vaidya's preconditioners, which was previously missing from the literature. 1. Introduction. This paper presents new applications of a littleknown technique for constructing and analyzing preconditioners called supportgraph preconditioning. The technique was first proposed and used by Pravin Vaidya [11], who ...
Nonlinearly preconditioned inexact Newton algorithms
 SIAM J. Sci. Comput
, 2000
"... Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at loc ..."
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Cited by 35 (14 self)
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Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of �F �, especiallyfor problems with unbalanced nonlinearities, because the methods do not have builtin machineryto deal with the unbalanced nonlinearities. To find the same solution u ∗ , one maywant to solve instead an equivalent nonlinearlypreconditioned system F(u ∗ ) = 0 whose nonlinearities are more balanced. In this paper, we propose and studya nonlinear additive Schwarzbased parallel nonlinear preconditioner and show numericallythat the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails. Key words. nonlinear preconditioning, inexact Newton methods, Krylov subspace methods, nonlinear additive Schwarz, domain decomposition, nonlinear equations, parallel computing, incompressible
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 35 (6 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...
Distributed Schur Complement Techniques for General Sparse Linear Systems
 SIAM J. SCI. COMPUT
, 1997
"... This paper presents a few preconditioning techniques for solving general sparse linear systems on distributed memory environments. These techniques utilize the Schur complement system for deriving the preconditioning matrix in a number of ways. Two of these preconditioners consist of an approxima ..."
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Cited by 33 (13 self)
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This paper presents a few preconditioning techniques for solving general sparse linear systems on distributed memory environments. These techniques utilize the Schur complement system for deriving the preconditioning matrix in a number of ways. Two of these preconditioners consist of an approximate solution process for the global system, which exploit approximate LU factorizations for diagonal blocks of the Schur complement. Another preconditioner uses a sparse approximateinverse technique to obtain certain local approximations of the Schur complement. Comparisons are reported for systems of varying difficulty.