Inverse boundary value problem for the Helmholtz equation: quantitative conditional Lipschitz stability estimates

TL;DR

This paper investigates the inverse boundary value problem for the Helmholtz equation, providing a conditional Lipschitz stability estimate for reconstructing wavespeeds that are piecewise constant, with implications for convergence and computational validation.

Contribution

It introduces a stability estimate for the inverse Helmholtz problem with piecewise constant wavespeeds and analyzes the stability constant's growth and bounds.

Findings

Stability constant grows exponentially with the number of subdomains.

Established an order optimal upper bound for the stability constant.

Performed computational experiments demonstrating stability constant evolution.

Abstract

We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequencies as the data. A conditional Lipschitz stability estimate for the inverse problem holds in the case of wavespeeds that are a linear combination of piecewise constant functions (following a domain partition) and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partition increases. We establish an order optimal upper bound for the stability constant. We eventually realize computational experiments to demonstrate the stability constant evolution for three dimensional wavespeed reconstruction.