Conditional stability for an inverse problem of a fully-discrete stochastic hyperbolic equation

TL;DR

This paper establishes a Lipschitz stability result for a fully discrete inverse problem involving stochastic hyperbolic equations, using a new Carleman estimate and discrete data at boundary and final time.

Contribution

It introduces a novel Carleman estimate for fully discrete stochastic hyperbolic equations and proves stability of the inverse problem with boundary and final time data.

Findings

Proved a new Carleman estimate for the discrete stochastic hyperbolic equation.

Established Lipschitz stability for the inverse problem with boundary and final data.

Identified an additional term related to mesh size in the stability estimate.

Abstract

In this paper, we investigate a discrete inverse problem of determining three unknowns, i.e. initial displacement, initial velocity and random source term, in a fully discrete approximation of one-dimensional stochastic hyperbolic equation. We firstly prove a new Carleman estimate for the fully-discrete stochastic hyperbolic equation. Based on this Carleman estimate, we then establish a Lipschitz stability for this discrete inverse problem by the discrete spatial derivative data at the left endpoint and the measurements of the solution and its time derivative at the final time. Owing to the discrete setting, an extra term with respect to mesh size arises in the right-hand side of the stability estimate.