Hierarchical matrix approximation for the uncertainty quantification of potentials on random domains

Abstract

Computing statistical quantities of interest of the solution of PDE on random domains is an important and challenging task in engineering. We consider the computation of these quantities by the perturbation approach. Especially, we discuss how third order accurate expansions of the mean and the correlation can numerically be computed. These expansions become even fourth order accurate for certain types of boundary variations. The correction terms are given by the solution of correlation equations in the tensor product domain, which can efficiently be computed by means of H-matrices. They have recently been shown to be an efficient tool to solve correlation equations with rough data correlations, that is, with low Sobolev smoothness or small correlation length, in almost linear time. Numerical experiments in three dimensions for higher order ansatz spaces show the feasibility of the proposed algorithm. The application to a non-smooth domain is also included.

Shape differentiability of Helmholtz scattering problems via explicit shape calculus

“…and consequently, the desired result for first element of ( 20) since λ = γ 0 U, and for (21) by using (18). For Neumann traces, as…”

“…Besides the obvious applications of this analysis in optimization, one sees it in inverse problems (cf. [9,Chapter 4 and 5] and [26,27,39,40]), first-order second statistical moment approximation for uncertainty quantification [18,20], and even shape holomorphy analysis [31]. Appealing to bi-Lipschitz diffeomorphisms between nominal and perturbed domains, one carries out shape calculus-see [1,25,42] for an exhaustive overview of these techniques.…”

Isogeometric multilevel quadrature for forward and inverse random acoustic scattering

“…To reduce the number of required samples, quasi-Monte Carlo (QMC) methods [8] and sparse grids [6] exploit parametric smoothness and may be combined with multilevel sampling techniques [22,26]. Perturbation approaches compute the statistical moments by approximating them with truncated Taylor expansions in the Fréchet sense [5,14,15,28]. The additional correction terms can often be computed without the need for high-dimensional quadrature methods.…”