Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation

TL;DR

This paper introduces a discrete regularization method for solving the inverse boundary value problem of the 1+1 dimensional wave equation, enabling the recovery of wave speed from boundary data with quantifiable error bounds.

Contribution

It proposes a novel discrete regularization strategy for the inverse wave problem and establishes convergence with explicit error estimates.

Findings

Achieved a Hölder-type stability estimate for wave speed reconstruction.

Developed a regularization method applicable to noisy boundary data.

Provided theoretical guarantees for the convergence of the approximation.

Abstract

An inverse boundary value problem for the 1+1 dimensional wave equation $(\partial_t^2 - c(x)^2 \partial_x^2)u(x,t)=0,\quad x\in\mathbb{R}_+$ is considered. We give a discrete regularization strategy to recover wave speed $c(x)$ when we are given the boundary value of the wave, $u(0,t)$, that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed $\widetilde c$, satisfying a H\"older type estimate $\| \widetilde c-c\|\leq C \epsilon^{\gamma}$, where $\epsilon$ is the noise level.