On the Schwarz Alternating Method with More than two Subdomains for Nonlinear Monotone Problems

Badea, Lori

doi:10.1137/0728010

Cited by 50 publications

(43 citation statements)

References 4 publications

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“…• Nonlinearity: stationary iterations can directly be transferred to nonlinear problems, in particular the ones with monotone operators (positive hardening) [32,33]. The local-global method was successfully applied in many nonlinear problems like plasticity or fracture [1,22] • Overlapping version: optimized Schwarz methods also exist with overlaps.…”

Section: Connexion With Alternate Non-overlapping Schwarz Methodsmentioning

confidence: 99%

Non-invasive global–local coupling as a Schwarz domain decomposition method: acceleration and generalization

Gosselet

Blanchard

Allix

et al. 2018

Adv. Model. and Simul. in Eng. Sci.

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Section: Connexion With Alternate Non-overlapping Schwarz Methodsmentioning

confidence: 99%

Non-invasive global–local coupling as a Schwarz domain decomposition method: acceleration and generalization

Gosselet

Blanchard

Allix

et al. 2018

Adv. Model. and Simul. in Eng. Sci.

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“…It is sufficient to carry this out for i = 1. Since 2 on Ω and both are equal to the unique solution of the PDE. On…”

Section: Scalar Equationsmentioning

confidence: 99%

On Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs

Lui¹

2001

ESAIM: M2AN

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Abstract. The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. In this paper, proofs of convergence of some Schwarz alternating methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well as some coupled nonlinear PDEs are shown to converge for finitely many subdomains. These results are applicable to several models in population biology.Mathematics Subject Classification. 65N55, 65J15.

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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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On the Schwarz Alternating Method with More than two Subdomains for Nonlinear Monotone Problems

Cited by 50 publications

References 4 publications

Non-invasive global–local coupling as a Schwarz domain decomposition method: acceleration and generalization

Non-invasive global–local coupling as a Schwarz domain decomposition method: acceleration and generalization

On Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs

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

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