An additive Schwarz method  for variational inequalities

Badea, Lori; Wang, Junping

doi:10.1090/s0025-5718-99-01164-3

Cited by 41 publications

(33 citation statements)

References 12 publications

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“…The linear complementarity problem (5.2) can be solved by the projected SOR method [26], multilevel methods [38,57], domain decomposition methods [6], and interior point methods [8]; see [35] and references therein. We used the projected SOR method in our simulations.…”

Section: Implementation Detailsmentioning

confidence: 99%

A posteriori error analysis for a class of integral equations and variational inequalities

2010

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We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order 2s ∈ (0, 2]. Our main motivation is the pricing of European or American options under Lévy processes, in particular pure jump processes or jump diffusion processes with tempered stable processes. The problem is discretized using piecewise linear finite elements in space and the implicit Euler method in time. We construct a residual-type a posteriori error estimator which gives a computable upper bound for the actual error in H s -norm. The estimator is localized in the sense that the residuals are restricted to the discrete non-contact region. Numerical experiments illustrate the accuracy of the space and time estimators, and show that they can be used to measure local errors and drive adaptive algorithms.

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Section: Implementation Detailsmentioning

confidence: 99%

A posteriori error analysis for a class of integral equations and variational inequalities

2010

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“…. , π m ) may be viewed as the algorithmic mapping associated with the block Jacobi method for solving (1). Consider an asynchronous version of the block Jacobi method, parameterized by a stepsize γ ∈ (0, 1], which for simplicity we assume to be fixed, that generates a sequence of iterates (u 1 (t), .…”

Section: An Asynchronous Space Decomposition Methodsmentioning

confidence: 99%

“…. , K m are not all subspaces, there are various convergence studies for synchronous methods (see [1,12,23,25,28,29,30,31,34,35,36,47,50] and references therein) but, again, none for asynchronous methods.…”

Section: Introductionmentioning

confidence: 99%

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Convergence rate analysis of an asynchronous space decomposition method for convex Minimization

Tai

Tseng

2001

Math. Comp.

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Abstract. We analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space. This method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations. In particular, the method generalizes the additive Schwarz domain decomposition methods to allow for asynchronous updates. It also generalizes the BPX multigrid method to allow for use as solvers instead of as preconditioners, possibly with asynchronous updates, and is applicable to nonlinear problems. Applications to an overlapping domain decomposition for obstacle problems are also studied. The method of this work is also closely related to relaxation methods for nonlinear network flow. Accordingly, we specialize our convergence rate results to the above methods. The asynchronous method is implementable in a multiprocessor system, allowing for communication and computation delays among the processors.

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“…Badea and Tai et al [15][16][17][18][19][20][21][22] have previously applied domain decomposition techniques to a constrained convex minimization problem coming from variational inequalities using space decomposition method. It has been shown that multigrid can be viewed as a special case of such space decomposition.…”

Section: 14mentioning

confidence: 99%

Two-level domain decomposition algorithm for a nonlinear inverse DOT problem

2005

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Diffuse optical tomography is to find the value of optical coefficients in a tissue using near infra-red lights, which is usually modelled by an optimization problem being composed of two steps: the forward solver to compute the photon density function and the inverse solver to update the coefficients based on the forward solver. Since the resulting problem is mathematically nonlinear ill-posed inverse problem and numerically large-scale computational problem demanding high quality image, it is highly desirable to reduce the amount of computation needed. In this paper, domain decomposition method is adopted to decrease the computation complexity of the problem. Among many methods of domain decomposition techniques, two level multiplicative overlapping domain decomposition method and two level space decomposition method are used to the forward and inverse solver, respectively. The convergence and computational cost of each method are described. And the efficiency of using combined two methods is verified by the implementation of reconstructing the absorption coefficient on square domain and thin domain.

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An additive Schwarz method for variational inequalities

Cited by 41 publications

References 12 publications

A posteriori error analysis for a class of integral equations and variational inequalities

A posteriori error analysis for a class of integral equations and variational inequalities

Convergence rate analysis of an asynchronous space decomposition method for convex Minimization

Two-level domain decomposition algorithm for a nonlinear inverse DOT problem

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