Bayesian identification of sound sources with the Helmholtz equation

TL;DR

This paper presents a Bayesian method for identifying sound sources using the Helmholtz equation, incorporating a problem-specific prior and finite element discretization, with proven convergence and numerical validation.

Contribution

It introduces a novel Bayesian framework with a tailored prior for source parameters and analyzes the convergence of discretized posteriors for Helmholtz-based acoustics.

Findings

Proposed a specific prior distribution for source parameters.

Proved convergence rates of the discretized posterior.

Numerical experiments confirm the sharpness of the theoretical rates.

Abstract

In this work we discuss the problem of identifying sound sources from pressure measurements with a Bayesian approach. The acoustics are modelled by the Helmholtz equation and the goal is to get information about the number, strength and position of the sound sources, under the assumption that measurements of the acoustic pressure are noisy. We propose a problem specific prior distribution of the number, the amplitudes and positions of the sound sources and algorithms to compute an approximation of the associated posterior. We also discuss a finite element discretization of the Helmholtz equation for the practical computation and prove convergence rates of the resulting discretized posterior to the true posterior. The theoretical results are illustrated by numerical experiments, which indicate that the proven rates are sharp.