Deterministic-Statistical Approach for an Inverse Acoustic Source Problem using Multiple Frequency Limited Aperture Data

TL;DR

This paper introduces a deterministic-statistical approach combining direct sampling, eigenfunction expansion, and Bayesian inference to solve inverse acoustic source problems with limited aperture data across multiple frequencies.

Contribution

It presents a novel integrated method that leverages direct sampling and Bayesian inversion to improve source reconstruction stability and accuracy with limited data.

Findings

Effective for smooth and non-smooth sources

Stable with respect to measurement data

Utilizes Bayesian inference with MCMC sampling

Abstract

We propose a deterministic-statistical method for an inverse source problem using multiple frequency limited aperture far field data. The direct sampling method is used to obtain a disc such that it contains the compact support of the source. The Dirichlet eigenfunctions of the disc are used to expand the source function. Then the inverse problem is recast as a statistical inference problem for the expansion coefficients and the Bayesian inversion is employed to reconstruct the coefficients. The stability of the statistical inverse problem with respect to the measured data is justified in the sense of Hellinger distance. A preconditioned Crank-Nicolson (pCN) Metropolis-Hastings (MH) algorithm is implemented to explore the posterior density function of the unknowns. Numerical examples show that the proposed method is effective for both smooth and non-smooth sources given limited-aperture data.