An inverse Cauchy problem of a stochastic hyperbolic equation

TL;DR

This paper addresses an inverse problem for a stochastic hyperbolic equation, proposing a regularization method with theoretical guarantees and validating it through numerical experiments.

Contribution

It introduces a novel regularization approach for the inverse Cauchy problem in stochastic hyperbolic equations, combining Carleman estimates and kernel-based learning.

Findings

Established a Lipschitz type observability estimate.

Developed a Tikhonov-type regularization method.

Validated algorithms with numerical experiments.

Abstract

In this paper, we investigate an inverse Cauchy problem for a stochastic hyperbolic equation. A Lipschitz type observability estimate is established using a pointwise Carleman identity. By minimizing the constructed Tikhonov-type functional, we obtain a regularized approximation to the problem. The properties of the approximation are studied by means of the Carleman estimate and Riesz representation theorem. Leveraging kernel-based learning theory, we simulate numerical algorithms based on the proposed regularization method. These reconstruction algorithms are implemented and validated through several numerical experiments, demonstrating their feasibility and accuracy.