Regularization strategy for inverse problem for 1+1 dimensional wave equation

TL;DR

This paper proposes a regularization method for reconstructing the wave speed in a 1+1 dimensional wave equation from noisy boundary measurements, providing stability estimates and a new reconstruction formula.

Contribution

It introduces a novel regularization strategy and a new formula for stable inversion of the wave speed from perturbed Neumann-to-Dirichlet maps.

Findings

Reconstruction error is bounded by the measurement error to the power of 1/18.

The method can handle measurements outside the range of the forward map.

A new explicit formula for wave speed reconstruction is developed.

Abstract

An inverse boundary value problem for a 1+1 dimensional wave equation with wave speed $c(x)$ is considered. We give a regularisation strategy for inverting the map $\mathcal A:c\mapsto \Lambda,$ where $\Lambda$ is the hyperbolic Neumann-to-Dirichlet map corresponding to the wave speed $c$. More precisely, we consider the case when we are given a perturbation of the Neumann-to-Dirichlet map $\tilde \Lambda=\Lambda +\mathcal E $, where $\mathcal E$ corresponds to the measurement errors, and reconstruct an approximate wave speed $\tilde c$. We emphasize that $\tilde \Lambda$ may not not be in the range of the map $\mathcal A$. We show that the reconstructed wave speed $\tilde c$ satisfies $\| \tilde c-c\|_{L^\infty}<C \|E\|^{1/18}$. Our regularization strategy is based on a new formula to compute $c$ from $\Lambda$.