Analysis of the domain mapping method for elliptic diffusion problems on random domains

Harbrecht, Helmut; Peters, Michael; Siebenmorgen, Markus

doi:10.1007/s00211-016-0791-4

Cited by 70 publications

(138 citation statements)

References 35 publications

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“…Representation of random domains. In this section, we introduce a description of random obstacles by means of random vector fields, as they have originally been considered in [11] in the context of the domain mapping method. To that end, let (Ω, A, P) denote a complete and separable probability space with σ-algebra A and probability measure P. Here, complete means that A contains all P-null sets.…”

Section: Random Obstaclesmentioning

confidence: 99%

Rapid computation of far-field statistics for random obstacle scattering

Harbrecht

Ilić

Multerer

2019

Engineering Analysis with Boundary Elements

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In this article, we consider the numerical approximation of far-field statistics for acoustic scattering problems in the case of random obstacles. In particular, we consider the computation of the expected far-field pattern and the expected scattered wave away from the scatterer as well as the computation of the corresponding variances. To that end, we introduce an artificial interface, which almost surely contains all realizations of the random scatterer. At this interface, we directly approximate the second order statistics, i.e., the expectation and the variance, of the Cauchy data by means of boundary integral equations. From these quantities, we are able to rapidly evaluate statistics of the scattered wave everywhere in the exterior domain, including the expectation and the variance of the far-field. By employing a low-rank approximation of the Cauchy data's twopoint correlation function, we drastically reduce the cost of the computation of the scattered wave's variance. Numerical results are provided in order to demonstrate the feasibility of the proposed approach.

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Section: Random Obstaclesmentioning

confidence: 99%

Rapid computation of far-field statistics for random obstacle scattering

Harbrecht

Ilić

Multerer

2019

Engineering Analysis with Boundary Elements

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“…Besides the fictitious domain approach considered in [7], one might essentially distinguish two approaches: the domain mapping method, cf. [8,19,16,28,27], and the perturbation method, cf. [14,17], which is based on shape derivatives.…”

Section: Motivation and Backgroundmentioning

confidence: 99%

Numerical Solution of the Poisson Equation on Domains with a Thin Layer of Random Thickness

Dambrine¹,

Greff²,

Harbrecht³

et al. 2016

SIAM J. Numer. Anal.

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Abstract. The present article is dedicated to the numerical solution of the Poisson equation on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on a random domain is transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Robin boundary condition which yields a third order accurate solution in the scale parameter ε of the layer's thickness. With the help of the Karhunen-Loève expansion, we transform this random boundary value problem into a deterministic parametric one with a possibly high-dimensional parameter y. Based on the decay of the random fluctuations of the layer's thickness, we prove rates of decay of the derivatives of the random solution with respect to this parameter y which are robust in the scale parameter ε. Numerical results validate our theoretical findings.

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“…[6,21,27,38,41], and the perturbation method. They result from a description of the random domain either in Lagrangian coordinates or in Eulerian coordinates, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains

Harbrecht

Peters

2016

Lecture Notes in Computational Science and Engineering

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In this article, we propose the sparse grid combination technique for the second moment analysis of elliptic partial differential equations on random domains. By employing shape sensitivity analysis, we linearize the influence of the random domain perturbation on the solution. We derive deterministic partial differential equations to approximate the random solution's mean and its covariance with leading order in the amplitude of the random domain perturbation. The partial differential equation for the covariance is a tensor product Dirichlet problem which can efficiently be determined by Galerkin's method in the sparse tensor product space. We show that this Galerkin approximation coincides with the solution derived from the combination technique provided that the detail spaces in the related multiscale hierarchy are constructed with respect to Galerkin projections. This means that the combination technique does not impose an additional error in our construction. Numerical experiments quantify and qualify the proposed method.

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Analysis of the domain mapping method for elliptic diffusion problems on random domains

Cited by 70 publications

References 35 publications

Rapid computation of far-field statistics for random obstacle scattering

Rapid computation of far-field statistics for random obstacle scattering

Numerical Solution of the Poisson Equation on Domains with a Thin Layer of Random Thickness

Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains

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