An inverse source problem for the stochastic wave equation

TL;DR

This paper investigates an inverse source problem for a stochastic wave equation driven by fractional Brownian motion, focusing on determining source properties from final-time data, with theoretical analysis and numerical reconstructions.

Contribution

It establishes well-posedness for the direct problem, proves uniqueness for the inverse problem under certain conditions, and introduces a regularization method for numerical reconstruction.

Findings

The direct problem has a unique mild solution.

Uniqueness of the inverse problem is proved for specific function classes.

Numerical experiments demonstrate effective source reconstruction.

Abstract

This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.