Isogeometric multilevel quadrature for forward and inverse random acoustic scattering

TL;DR

This paper introduces a fast isogeometric boundary element method combined with multilevel quadrature for efficiently solving forward and inverse 3D acoustic scattering problems involving randomly shaped obstacles, enabling uncertainty quantification and shape inference.

Contribution

The paper develops a novel isogeometric boundary element framework that efficiently handles random obstacle shapes and integrates multilevel quadrature for uncertainty quantification in acoustic scattering.

Findings

Efficient computation of scattered waves' expectation and variance.

Successful inference of obstacle shape and uncertainty from noisy measurements.

Numerical validation demonstrating the method's feasibility for forward and inverse problems.

Abstract

We study the numerical solution of forward and inverse acoustic scattering problems by randomly shaped obstacles in three-dimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the scatterer. This way, we end up with a random deformation field. In particular, we show that the knowledge of the deformation field's expectation and covariance at the surface of the scatterer are already sufficient to compute the surface Karhunen-Lo\`eve expansion. Leveraging on the isogeometric framework, we utilize multilevel quadrature methods for the efficient approximation of quantities of interest, such as the scattered wave's expectation and variance. Computing the wave's Cauchy data at an artificial, fixed interface enclosing the random obstacle, we can also directly infer quantities of interest in free space. Adopting the Bayesian paradigm, we finally compute the expected shape and the variance of the scatterer from noisy measurements of the scattered wave at the artificial interface. Numerical results for the forward and inverse problem are given to demonstrate the feasibility of the proposed approach.