Linearized Boundary Control Method for Damping Reconstruction in an Acoustic Inverse Boundary Value Problem

TL;DR

This paper introduces a linearized boundary control method to reconstruct damping coefficients in a damped wave equation, providing stability estimates and numerical validation, advancing inverse boundary value problem solutions.

Contribution

It develops a novel linearized boundary control algorithm for damping reconstruction with stability analysis, applicable in multiple dimensions and validated numerically in one dimension.

Findings

Reconstructive algorithm with stability estimates derived for constant background damping.

Numerical implementation successfully validated in one dimension.

Established increasing stability estimate in higher dimensions.

Abstract

We develop a linearized boundary control method for the inverse boundary value problem of determining the damping coefficient in the damped wave equation. The objective is to reconstruct an unknown perturbation in a known background damping from the linearized Neumann-to-Dirichlet map. When the linearization is at a constant background damping, we derive a reconstructive algorithm with stability estimates based on the boundary control method in dimension $n\geq 1$. The reconstruction algorithm is implemented in one dimension to validate its numerical feasibility. When the linearization is at a non-constant background damping, we establish an increasing stability estimate in the time domain in dimension $n\geq 3$.