Stability for finite element discretization of some elliptic inverse parameter problems from internal data -- application to elastography

TL;DR

This paper establishes stability estimates for finite element discretizations of certain elliptic inverse problems, introduces a novel hexagonal finite element method, and demonstrates its efficiency and generality in numerical reconstructions relevant to elastography.

Contribution

It provides a new stability analysis for inverse elliptic problems, introduces a simple hexagonal finite element discretization satisfying stability conditions, and offers an efficient, non-iterative inversion method.

Findings

Stability estimates are proven for the discretized inverse problem.

A new hexagonal finite element discretization satisfies the stability condition.

Numerical reconstructions demonstrate the method's efficiency and generality.

Abstract

In this article, we provide stability estimates for the finite element discretization of a class of inverse parameter problems of the form $-\nabla\cdot(\mu S) = \g f$ in a domain $\Omega$ of $\R^d$. Here $\mu$ is the unknown parameter to recover, the matrix valued function $S$ and the vector valued distribution $\g f$ are known. As uniqueness is not guaranteed in general for this problem, we prove a Lipschitz-type stability estimate in an hyperplane of $L^2(\Omega)$. This stability is obtained through an adaptation of the so-called discrete \emph{inf-sup} constant or LBB constant to a large class of first-order differential operators. We then provide a simple and original discretization based on hexagonal finite element that satisfies the discrete stability condition and shows corresponding numerical reconstructions. The obtained algebraic inversion method is efficient as it does not require any iterative solving of the forward problem and is very general as it does not require any smoothness hypothesis for the data nor any additional information at the boundary.