A note on the domain mapping method with rough diffusion coefficients

Abstract

In this article, we consider elliptic diffusion problems on random domains with non-smooth diffusion coefficients. We start by illustrating the problems that arise from a nonsmooth diffusion coefficient by recapitulating the corresponding regularity analysis. Then, we propose an alternative approach to address this problem by means of a perturbation method. Based on the assumption that the diffusion coefficient can be decomposed in a possibly deterministic, analytic part and a rough random perturbation, we derive approximation results in terms of the perturbations amplitude for the approximation of quantities of interest of the solution. Numerical examples are given in order to validate and quantify the theoretical results.

Rapid computation of far-field statistics for random obstacle scattering

“…An efficient way to compute the Karhunen-Loève expansion if the mean and the covariance function of the random (vector) field under consideration are known is given by the pivoted Cholesky decomposition. This accounts particularly for random vector fields, see [9,10,18].…”

Isogeometric multilevel quadrature for forward and inverse random acoustic scattering

“…Although perturbation approaches have the potential to deal with rough domain deformations, the obtained results are usually only reliable when the domain deformations are small. A combination of the two approaches was presented in [11,35], but relies on an a-prioriliy available splitting of the domain deformations into small and large amplitudes of the deformations. Summarizing, state-of-the-art methods rely on different kinds of smoothness and can either work with large and rather global or with small and rough randomness in the shape of the domain.…”