Inverse point source location with the Helmholtz equation on a bounded domain

TL;DR

This paper develops a method to recover acoustic monopole sources from limited frequency measurements using sparse optimization and the Helmholtz equation, with theoretical recovery guarantees and a finite element implementation.

Contribution

It introduces a novel sparse measure-based optimization framework with weighted norms for source recovery, including optimality conditions and a numerical algorithm.

Findings

Effective source recovery demonstrated in simulations

Weighted norms improve localization accuracy near observation points

The method provides theoretical guarantees under small noise

Abstract

The problem of recovering acoustic sources, more specifically monopoles, from point-wise measurements of the corresponding acoustic pressure at a limited number of frequencies is addressed. To this purpose, a family of sparse optimization problems in measure space in combination with the Helmholtz equation on a bounded domain is considered. A weighted norm with unbounded weight near the observation points is incorporated into the formulation. Optimality conditions and conditions for recovery in the small noise case are discussed, which motivates concrete choices of the weight. The numerical realization is based on an accelerated conditional gradient method in measure space and a finite element discretization.