Helmholtz scattering by random domains: First-order sparse boundary element approximation

Paul Escapil-Inchauspé, Carlos Jerez-Hanckes

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We consider the numerical solution of time-harmonic acoustic scattering by obstacles with uncertain geometries for Dirichlet, Neumann, impedance, and transmission boundary conditions. In particular, we aim to quantify diffracted fields originated by small stochastic perturbations of a given relatively smooth nominal shape. Using first-order shape Taylor expansions, we derive tensor deterministic first-kind boundary integral equations for the statistical moments of the scattering problems considered. These are then approximated by sparse tensor Galerkin discretizations via the combination technique [M. Griebel, M. Schneider, and C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, P. de Groen and P. Beauwens, eds., Elsevier, Amsterdam, 1992, pp. 263-281; H. Harbrecht, M. Peters, and M. Siebenmorgen, J. Comput. Phys., 252 (2013), pp. 128-141]. We supply extensive numerical experiments confirming the predicted error convergence rates with polylogarithmic growth in the number of degrees of freedom and accuracy in approximation of the moments. Moreover, we discuss implementation details such as preconditioning to finally point out further research avenues.

Original languageEnglish
Pages (from-to)A2561-A2592
JournalSIAM Journal on Scientific Computing
Volume42
Issue number5
DOIs
StatePublished - Sep 2020
Externally publishedYes

Keywords

  • Boundary element method
  • Combination technique
  • Helmholtz equation
  • Shape calculus
  • Uncertainty quantification

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