Stability analysis of an inverse coefficients problem in a system of partial differential equations

TL;DR

This paper investigates the stability of recovering mechanical properties like density and Lamé parameters in a medium from boundary measurements, providing Lipschitz stability estimates under certain conditions.

Contribution

It offers new Lipschitz stability estimates for inverse problems of determining density and elastic parameters from boundary data, including simultaneous recovery.

Findings

Lipschitz stability for recovering density with known Lamé parameters.

Lipschitz stability for simultaneous recovery of all parameters.

Applicability to piecewise constant and bounded parameters.

Abstract

In this study, we address the inverse problem of recovering the Lam\'e parameters ($\lambda, \mu$) and the density $\rho$ of a medium from the Neumann-to-Dirichlet map for any dimension $d\geq 2$. This inverse problem finds its motivation in the reconstruction of mechanical properties of tissues in medical diagnostics. We first assume that the Lam\'e parameters ($\lambda, \mu$) are know and we look for the inverse problem of recovering the density $\rho$. In this context, we derive a constrcutive Lipschitz stability estimate in terms of the Neumann to Dirichlet map in the case of piecewise constant parameters. Then, we look for the inverse problem of recovering $\lambda$, $\mu$ and $\rho$ simultameousely. We establish Lipschitz stability estimate, provided that the parameters $\lambda$, $\mu$ and $\rho$ have upper and lower bounds and belong to a known finite-dimensional subspace. The proofs hinge on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, coupled with the techniques of localized potentials.