Inverse source problems for the stochastic wave equations

TL;DR

This paper develops a computational framework for solving inverse source problems in stochastic wave equations, addressing ill-posedness and providing theoretical and numerical solutions for equations driven by jump processes.

Contribution

It introduces a novel approach for reconstructing sources in stochastic wave equations with jump processes, including theoretical analysis and numerical schemes.

Findings

Established existence of mild solutions with stability estimates.

Proposed a method to reconstruct unknown sources from wave data.

Demonstrated applicability to wave equations with non-Gaussian stochastic properties.

Abstract

To address the ill-posedness of the inverse source problem for the one-dimensional stochastic Helmholtz equations without attenuation, this study develops a novel computational framework designed to mitigate this inherent challenge at the numerical implementation level. For the stochastic wave equation driven by a finite-jump L\'evy process (assuming that its jump amplitude obeys a Gaussian distribution and the jump time interval obeys a Poisson distribution), this paper firstly establish the existence of a mild solution to its direct problem satisfying a particular stability estimate. Building upon these theoretical foundations, we further investigate the well-posedness of the inverse problem and develop a methodology to reconstruct the unknown source terms $f$ and $g$ using the data of the wave field at the final time point $u(x,T)$. This work not only provides rigorous theoretical analysis and effective numerical schemes for solving inverse source problems in these two specific classes of stochastic wave equations, but also offers new perspectives and methodological approaches for addressing a broader range of wave propagation inverse problems characterized by non-Gaussian stochastic properties. The proposed framework demonstrates significant relevance for characterizing physical phenomena influenced by jump-type stochastic perturbations, offering promising applications in diverse domains including but not limited to seismic wave propagation analysis and financial market volatility modeling.