TY - JOUR

T1 - Helmholtz scattering by random domains

T2 - First-order sparse boundary element approximation

AU - Escapil-Inchauspé, Paul

AU - Jerez-Hanckes, Carlos

N1 - Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics

PY - 2020/9

Y1 - 2020/9

N2 - 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.

AB - 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.

KW - Boundary element method

KW - Combination technique

KW - Helmholtz equation

KW - Shape calculus

KW - Uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=85091853797&partnerID=8YFLogxK

U2 - 10.1137/19M1279277

DO - 10.1137/19M1279277

M3 - Article

AN - SCOPUS:85091853797

SN - 1064-8275

VL - 42

SP - A2561-A2592

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

IS - 5

ER -