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arXiv · 2601.15787

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

Shutong Hou·Mourad Sini·Haibing Wang
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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 $N$.

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Shutong Hou, Mourad Sini, Haibing Wang. 2026-01-22. Efficient Numerical Reconstruction of Wave Equation Sources via Droplet-Induced Asymptotics. https://arxiv.org/abs/2601.15787

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