Efficient Numerical Reconstruction of Wave Equation Sources via Droplet-Induced Asymptotics

Abstract

SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 1018-1042, June 2026.
Abstract. In this paper, we develop and numerically implement a novel approach for solving the inverse source problem of the acoustic wave equation in three dimensions. By injecting a small high-contrast droplet into the medium, we exploit the resulting wave field perturbation measured at a single external point over time. The method enables stable source reconstructions where conventional approaches fail due to ill-posedness, with potential applications in medical imaging and nondestructive testing. Key contributions include: (1) Implementation of a theoretically justified asymptotic expansion, from [], using the eigensystem of the Newtonian operator, with error analysis for the spectral truncation. (2) Novel numerical schemes for solving the time-domain Lippmann–Schwinger equation and reconstructing the source via Riesz basis expansions and mollification-based numerical differentiations. (3) Reconstruction requiring only single-point measurements, overcoming traditional spatial data limitations. (4) 3D numerical experiments demonstrating accurate source recovery under noise (SNR of the order [math]), with error analysis for the droplet size (of the order [math]) and the number of spectral modes [math].