# Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction

## TL;DR

This paper establishes theoretical local stability results for determining spatial coefficients in hyperbolic equations and proposes a numerical iterative method using Tikhonov regularization and Poisson equations, demonstrated through 1D examples.

## Contribution

It provides new theoretical stability estimates for coefficient inverse problems in hyperbolic equations and introduces a practical iterative reconstruction method with numerical validation.

## Key findings

- Local Hölder stability under geometric conditions
- Effective iterative reconstruction method based on Tikhonov regularization
- Numerical examples demonstrating the method's performance

## Abstract

In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local H\"older stability with both partial boundary and interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.