Stability for the Calder\'on's problem for a class of anisotropic conductivities via an ad-hoc misfit functional

TL;DR

This paper establishes stability estimates for a specific class of anisotropic conductivities in Calderón's inverse problem, using a novel misfit functional that relates data to the unknown conductivity parameters.

Contribution

It introduces an ad-hoc misfit functional and proves stability estimates for anisotropic conductivities of a particular form, advancing understanding of inverse problems with anisotropic media.

Findings

Stability estimates are derived in terms of the misfit functional.

Stability estimates are also established using the local Dirichlet-to-Neumann map.

The results apply to conductivities with a known matrix part and an unknown piecewise affine scalar.

Abstract

We address the stability issue in Calder\'on's problem for a special class of anisotropic conductivities of the form $\sigma=\gamma A$ in a Lipschitz domain $\Omega\subset\mathbb{R}^n$, $n\geq 3$, where $A$ is a known Lipschitz continuous matrix-valued function and $\gamma$ is the unknown piecewise affine scalar function on a given partition of $\Omega$. We define an ad-hoc misfit functional encoding our data and establish stability estimates for this class of anisotropic conductivity in terms of both the misfit functional and the more commonly used local Dirichlet-to-Neumann map.