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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.
USING SYMMETRIES AND ANTISYMMETRIES TO ANALYZE A PARALLEL MULTIGRID ALGORITHM: THE ELLIPTIC BOUNDARY VALUE PROBLEM CASE
, 1988
"... Symmetry and antisymmetry properties of a class of elliptic partial di erential equations are exploited to prove when a particular parallel multilevel algorithm is a direct method rather than the usual iterative method. No smoothing is required for this result. Examples are presented, including vari ..."
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Cited by 18 (8 self)
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Symmetry and antisymmetry properties of a class of elliptic partial di erential equations are exploited to prove when a particular parallel multilevel algorithm is a direct method rather than the usual iterative method. No smoothing is required for this result. Examples are presented, including variable coefficient ones. A connection between the algorithm in this article and domain decomposition is established, even though this algorithm is more general and different. The parallel algorithm is also analyzed when it is iterative and it is shown how to increase processor utilization. Hackbusch's robust multigrid algorithm is analyzed for some model problems and it is shown that the parallel algorithm in this article uses much less computer time with at most the same amount of storage.
A Parallel Domain Reduction Method
, 1989
"... : We relate a particular version of a parallel multigrid method analyzed by C. Douglas, W. L. Miranker, and B. F. Smith ([4, 6]) to the domain decomposition method of D. Funaro, L. D. Marini, A. Quarteroni, and P. Zanolli ([7, 8]). We show that the parallel multigrid method is reducing computation t ..."
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Cited by 10 (7 self)
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: We relate a particular version of a parallel multigrid method analyzed by C. Douglas, W. L. Miranker, and B. F. Smith ([4, 6]) to the domain decomposition method of D. Funaro, L. D. Marini, A. Quarteroni, and P. Zanolli ([7, 8]). We show that the parallel multigrid method is reducing computation to a small portion of the domain and then extending the solution to the entire domain using the correct reflections to get the exact solution. We extend a particular example to double the parallelism in a nonobvious manner. While the techniques of this paper are applied to two dimensional problems, they can be applied to higher dimensional problems in an obvious manner. Keywords: partial differential equations, parallel multigrid, domain decomposition, direct method AMS(MOS) subject classification: 35, 65 To appear in Numerical Methods for Partial Differential Equations. In [4], a particular version of a parallel multigrid method is analyzed in a very abstract manner, but does not offer m...
A TUPLEWARE APPROACH TO DOMAIN DECOMPOSITION METHODS
"... Domain decomposition methods are highly parallel methods for solving elliptic partial differential equations. In many domain decomposition variants, the domain is partitioned into a number of (possibly overlapping) subdomains before computation begins. In contrast, the domain reduction method folds ..."
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Cited by 10 (6 self)
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Domain decomposition methods are highly parallel methods for solving elliptic partial differential equations. In many domain decomposition variants, the domain is partitioned into a number of (possibly overlapping) subdomains before computation begins. In contrast, the domain reduction method folds the domain into a number of smaller domains covering only a small portion of the entire domain. The solution over the entire domain is recovered by unfolding the solutions on the subdomains and summing them. The cost of the folding and unfolding is negligible. Results are derived measuring how much time and space can be reduced in the problem discretization phase by using these methods. Storage can be reduced by a substantial factor (a factor of eight is constructed for an example in this paper), and discretization time can be reduced by a constant factor or more, depending on the time complexity formula for a given problem. The amount of storage required is substantially less than for traditional domain decomposition implementations, or ones based on standard iterative methods or multigrid (standard or nontelescoping). A highly portable implementation using the Linda system for parallel or distributed programming is discussed. Linda adds a tuple data abstraction to a language, which is used to develop numerical software.
The Domain Reduction Method: High Way Reduction In Three Dimensions And Convergence With Inexact Solvers
 in Fourth Copper Mountain conference on multigrid methods
, 1989
"... . We study a method for parallel solution of elliptic partial differential equations which decomposes the problem into a number of independent subproblems on subspaces of the underlying solution space. Using symmetries of the domain, we obtain up to 64 such subproblems for a 3 dimensional cube and t ..."
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Cited by 9 (7 self)
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. We study a method for parallel solution of elliptic partial differential equations which decomposes the problem into a number of independent subproblems on subspaces of the underlying solution space. Using symmetries of the domain, we obtain up to 64 such subproblems for a 3 dimensional cube and the method reduces to a direct solver. In the general case, or when the subproblems are solved only approximately, the method becomes an iterative method or can be used as a preconditioner. Bounds on the resulting convergence factors and condition numbers are given. 1. Introduction. In this paper, we approximate the solution to the elliptic boundary value problem 8 ? ? ? ? ? ? ! ? ? ? ? ? ? : Lu = f in\Omega ae IR d ; d ? 0 u = 0 on @\Omega D ; where @\Omega = @\Omega D [ @\Omega N ; and @\Omega D " @\Omega N = OE; @ @n u = 0 on @\Omega N (1.1) using a combination of domain decomposition, multigrid, and projection method techniques. We assume that (1.1) is well posed a...
The Parallel UCycle Multigrid Method
 in Proceedings of the 8th Copper Mountain Conference on Multigrid Methods
, 1997
"... . A simple way to avoid idle processors in implementing the multigrid method on a parallel computer is to select a proper finer grid as the new coarsest grid. For clarity, the variant of the Vcycle generated by this approach is called the Ucycle in this paper. It is proved that the Ucycle with a ..."
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Cited by 8 (2 self)
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. A simple way to avoid idle processors in implementing the multigrid method on a parallel computer is to select a proper finer grid as the new coarsest grid. For clarity, the variant of the Vcycle generated by this approach is called the Ucycle in this paper. It is proved that the Ucycle with a finer coarsest grid can have a faster convergence rate, and the coarsest grid equations of the Ucycle can be solved approximately without increasing the total number of Ucycle iterations over what would be required using exact coarsest grid solutions. Then, a parallel Ucycle is defined by using domain partitioning techniques, which can be implemented on a MIMD multiprocessor computer without any idle processors. An analysis of the time complexity of the parallel Ucycle shows that the parallel Ucycle is fully scalable, and can have superlinear speedup in comparison to the original Vcycle. Further, the scaled efficiency of the parallel Ucycle in the memoryconstrained case is discussed...
Madpak: A family of abstract multigrid or multilevel solutions
 Computation and Applied Mathematics
, 1995
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A review of numerous parallel multigrid methods
 Applications on Advanced Architecture Computers
, 1996
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C.C.: Sparse matrix multiplication package (SMMP
 Advances in computational mathematics
, 1993
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A VARIATION OF THE SCHWARZ ALTERNATING METHOD: THE DOMAIN REDUCTION METHOD
"... Domain decomposition methods are highly parallel methods for solving elliptic partial differential equations. In many domain decomposition variants, the domain is partitioned into a number of (possibly overlapping) subdomains before computation begins. In contrast, the domain reduction method folds ..."
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Cited by 1 (0 self)
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Domain decomposition methods are highly parallel methods for solving elliptic partial differential equations. In many domain decomposition variants, the domain is partitioned into a number of (possibly overlapping) subdomains before computation begins. In contrast, the domain reduction method folds the domain into a number of smaller domains covering only a small portion of the entire domain. The solution over the entire domain is recovered by unfolding the solutions on the subdomains and summing them. The cost of the folding and unfolding is negligible. All steps of the algorithm are embarrassingly parallel.