A Domain Decomposition Method for the Helmholtz Equation and Related Optimal Control Problems

Benamou, Jean-David; Després, Bruno

doi:10.1006/jcph.1997.5742

Cited by 195 publications

(181 citation statements)

References 24 publications

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“…For optimal control problems governed by the deterministic Helmholtz equation, see, for example, [9] and the references therein. The optimal control of the stochastic Helmholtz equation is studied in [16] in the context of controlling radiated engine noise.…”

Section: Optimal Control Of Stochastic Helmholtz Equationmentioning

confidence: 99%

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2013

SIAM J. Sci. Comput.

108

116

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Abstract. The numerical solution of optimization problems governed by partial differential equations (PDEs) with random coefficients is computationally challenging because of the large number of deterministic PDE solves required at each optimization iteration. This paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse-grid collocation for the discretization of the PDE in the stochastic space and a trust-region framework for optimization and fidelity management of the stochastic discretization. The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem. It frequently uses few collocation points initially and increases the number of collocation points only as necessary, thereby keeping the number of deterministic PDE solves low while guaranteeing convergence. Currently an error indicator is used to estimate gradient errors due to adaptive stochastic collocation. The algorithm is applied to three examples, and the numerical results demonstrate a significant reduction in the total number of PDE solves required to obtain an optimal solution when compared with a Newton conjugate gradient algorithm applied to a fixed high-fidelity discretization of the optimization problem.

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Section: Optimal Control Of Stochastic Helmholtz Equationmentioning

confidence: 99%

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2013

SIAM J. Sci. Comput.

108

116

View full text Add to dashboard Cite

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“…The pattern on a photolithography mask is transfered onto Email addresses: schaedle@zib.de, lin.zschiedrich@jcmwave.com, sven.burger@jcmwave.com, klose@zib.de, frank.schmidt@zib.de (Frank Schmidt). 1 Supported by the DFG Research Center Matheon "Mathematics for key technologies" in Berlin. the chip by optical projection.…”

Section: Introductionmentioning

confidence: 99%

“…Després and Shaidurov proposed to balance the energy fluxes across domain interfaces for Helmholtz problems, [9,26]. This idea was further expanded, [1,8,7,13,6,12]. Toselli used the PML method at the interfaces of the sub-domains, [27].…”

Section: Introductionmentioning

confidence: 99%

Domain decomposition method for Maxwell’s equations: Scattering off periodic structures

Schädle

Zschiedrich

Burger

et al. 2007

Journal of Computational Physics

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We present a domain decomposition approach for the computation of the electromagnetic field within periodic structures. We use a Schwarz method with transparent boundary conditions at the interfaces of the domains. Transparent boundary conditions are approximated by the perfectly matched layer method (PML). To cope with Wood anomalies appearing in periodic structures an adaptive strategy to determine optimal PML parameters is developed.We focus on the application to typical EUV lithography line masks. Light propagation within the multi-layer stack of the EUV mask is treated analytically. This results in a drastic reduction of the computational costs and allows for the simulation of next generation lithography masks on a standard personal computer.

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“…Interface boundary conditions are the key ingredient to design efficient domain decomposition methods, see Chevalier and Nataf [1998], Benamou and Després [1997], Gander et al [2002]. However, convergence cannot be obtained for any method in a number of iterations less than the number of subdomains minus one in the case of a one-way splitting.…”

Section: Introductionmentioning

confidence: 99%

Approximation of Optimal Interface Boundary Conditions for Two-Lagrange Multiplier FETI Method

Roux

Magoulès

Séries

et al.

Lecture Notes in Computational Science and Engineering

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Summary. Interface boundary conditions are the key ingredient to design efficient domain decomposition methods. However, convergence cannot be obtained for any method in a number of iterations less than the number of subdomains minus one in the case of a one-way splitting. This optimal convergence can be obtained with generalized Robin type boundary conditions associated with an operator equal to the Schur complement of the outer domain. Since the Schur complement is too expensive to compute exactly, a new approach based on the computation of the exact Schur complement for a small patch around each interface node is presented for the two-Lagrange multiplier FETI method.

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A Domain Decomposition Method for the Helmholtz Equation and Related Optimal Control Problems

Cited by 195 publications

References 24 publications

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Domain decomposition method for Maxwell’s equations: Scattering off periodic structures

Approximation of Optimal Interface Boundary Conditions for Two-Lagrange Multiplier FETI Method

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