Large deformation shape uncertainty quantification in acoustic scattering

• Published in: Advances in computational mathematics Vol. 44; no. 5; pp. 1475 - 1518
• Main Authors: Hiptmair, R., Scarabosio, L., Schillings, C., Schwab, Ch
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2018; Springer Nature B.V
• Subjects: Acoustic scattering; Adaptive algorithms; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Configurations; Deformation; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Parameterization; Uncertainty; Visualization; 35R60; 65N12; Stochastic parametrization; Uncertainty quantification; High-dimensional approximation; Helmholtz equation; Dimension-adaptive Smolyak quadrature; Random interface; 60H35; 65N75
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-018-9594-8

We address shape uncertainty quantification for the two-dimensional Helmholtz transmission problem, where the shape of the scatterer is the only source of uncertainty. In the framework of the so-called deterministic approach, we provide a high-dimensional parametrization for the interface. Each domain configuration is mapped to a nominal configuration, obtaining a problem on a fixed domain with stochastic coefficients. To compute surrogate models and statistics of quantities of interest, we apply an adaptive, anisotropic Smolyak algorithm, which allows to attain high convergence rates that are independent of the number of dimensions activated in the parameter space. We also develop a regularity theory with respect to the spatial variable, with norm bounds that are independent of the parametric dimension. The techniques and theory presented in this paper can be easily generalized to any elliptic problem on a stochastic domain.