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

TL;DR

This paper introduces a new numerical method for reconstructing sources in the 3D acoustic wave equation using droplet-induced asymptotics, enabling stable inversion from single-point measurements even with noise.

Contribution

The paper presents a novel asymptotic expansion and numerical schemes for inverse source reconstruction in 3D wave equations, requiring only single-point data and providing error analysis.

Findings

Accurate source recovery demonstrated in 3D with noisy data

Error bounds established for droplet size and spectral modes

Method overcomes traditional ill-posedness in inverse problems

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 non-destructive testing. Key contributions include: 1. Implementation of a theoretically justified asymptotic expansion, from [33], 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 $1/a$), with error analysis for the droplet size (of the order $a$) and the number of spectral modes $N$.