Parallelizing preconditioned conjugate gradient algorithms

Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for positive definite, symmetric problems are extended to certain nonsymmetric problems, which can also have some eigenvalues in the left half plane.We first consider an additive Schwarz method applied to linear, second order, symmetric or nonsymmetric, indefinite elliptic boundary value problems in two and three dimensions. An alternative linear system, which has the same solution as the original problem, is derived and this system is then solved by using GMRES, an iterative method of conjugate gradient type. In each iteration step, a coarse mesh finite element problem and a number of local problems are solved on small, overlapping subregions into which the original region is subdivided. We show that the rate of convergence is independent of the number of degrees of freedom and the number of local problems if the coarse mesh is fine enough. The performance of the method in two dimensions is illustrated by results of several numerical experiments.We also consider two other iterative method for solving the same class of elliptic problems in two dimensions. Using an observation of Dryja and Widlund, we show that the rate of convergence of certain iterative substructuring methods deteriorates only quite slowly when the local problems increase in size. A similar result is established for Yserentant's hierarchical basis method.

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“…Numerical results for positive definite problems, both symmetric and nonsymmetric, have previously been given in [3], [4], [12].…”

Section: Numerical Resultsmentioning

confidence: 99%

Domain Decomposition Algorithms for Indefinite Elliptic Problems

Cai¹,

Widlund²

1992

SIAM J. Sci. and Stat. Comput.

166

137

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“…Moreover, the computational work per step is O(h -1 even for highly nonuniform and refined meshes. For numerical experiments on parallel computers, see [1], [16]. -1 for problems with discontinuous coefficients.…”

Section: Hierarchical Basis Preconditioner (Hb)mentioning

confidence: 99%

“…Unfortunately, previous works [12], [16] have shown that for many of the classical preconditioners, there is a fundamental trade-off in the ease of parallelization and the rate of convergence. A principal obstacle to parallelization is the sequential manner in which many preconditioners traverse the computational gridthe data dependence implicitly prescribed by the method fundamentally limits the amount of parallelism available.…”

mentioning

confidence: 99%

Multilevel Filtering Elliptic Preconditioners

Kuo¹,

Chan²,

Tong³

1990

SIAM J. Matrix Anal. & Appl.

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Abstract. A class of preconditioners for elliptic problems built on ideas borrowed from the digital filtering theory and implemented on a multilevel grid structure is presented. These preconditioners are designed to be both rapidly convergent and highly parallelizable. The digital filtering viewpoint allows for the use of filter design techniques for constructing elliptic preconditioners and also provides an alternative framework for understanding several other recently proposed multilevel preconditioners. Numerical results are presented to assess the convergence behavior of the new methods and to compare them with other preconditioners of multilevel type, including the usual multigrid method as preconditioner, the hierarchical basis method, and a recent method proposed by Bramble-Pasciak-Xu.

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“…Since the closed set J consists of accumulation points (at which the inequality (11) holds because the left side is zero) and a countable number of isolated points, we may integrate over (0, t) to obtain…”

Section: Lemmamentioning

confidence: 99%

Finite element approximation of diffusion equations with convolution terms

Peszyńska

1996

Math. Comp.

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Abstract. Approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered. Such problems arise from modeling of flows in fissured media. Convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented. The advantages of implementation of the algorithm in a multiprocessing environment are discussed.

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Parallelizing preconditioned conjugate gradient algorithms

Cited by 25 publications

References 5 publications

Domain Decomposition Algorithms for Indefinite Elliptic Problems

Domain Decomposition Algorithms for Indefinite Elliptic Problems

Multilevel Filtering Elliptic Preconditioners

Finite element approximation of diffusion equations with convolution terms

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