Linearized Boundary Control Method for Density Reconstruction in Acoustic Wave Equations

TL;DR

This paper introduces a linearized boundary control method to reconstruct unknown density perturbations in acoustic wave equations, providing algorithms with stability estimates and numerical validation for inverse boundary value problems.

Contribution

It develops a novel linearized boundary control approach for density reconstruction, including algorithms and stability analysis, applicable to both constant and non-constant background densities.

Findings

Algorithms successfully reconstruct density perturbations.

Stability estimates are established for different background densities.

Numerical experiments confirm the method's feasibility.

Abstract

We develop a linearized boundary control method for the inverse boundary value problem of determining a density in the acoustic wave equation. The objective is to reconstruct an unknown perturbation in a known background density from the linearized Neumann-to-Dirichlet map. A key ingredient in the derivation is a linearized Blagovescenskii's identity with a free parameter. When the linearization is at a constant background density, we derive two reconstructive algorithms with stability estimates based on the boundary control method. When the linearization is at a non-constant background density, we establish an increasing stability estimate for the recovery of the density perturbation. The proposed reconstruction algorithms are implemented and validated with several numerical experiments to demonstrate the feasibility.