Theoretical stability and numerical reconstruction for an inverse source problem for hyperbolic equations

TL;DR

This paper establishes theoretical stability results for an inverse source problem in hyperbolic equations and proposes an efficient numerical reconstruction method using Tikhonov regularization and iterative thresholding.

Contribution

It introduces a new Carleman estimate for hyperbolic operators and develops a computationally simple algorithm for source reconstruction.

Findings

Proved local Hölder stability for the inverse problem.

Developed an explicit solution-based iterative algorithm.

Numerical experiments confirm accuracy and efficiency.

Abstract

In this paper, we investigate the inverse problem on determining the spatial component of the source term in a hyperbolic equation with time-dependent principal part. Based on a newly established Carleman estimate for general hyperbolic operators, we prove a local stability result of H\"older type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to transform the inverse problem into an output least-squares minimization, which can be solved by the iterative thresholding algorithm. The proposed algorithm is computationally easy and efficient: the minimizer at each step has explicit solution. Abundant amounts of numerical experiments are presented to demonstrate the accuracy and efficiency of the algorithm.