Multi-parameter identification in systems of PDEs from internal data

TL;DR

This paper develops a theoretical and numerical framework for recovering elliptic parameters in PDE systems from internal solution data, providing stability estimates, null space characterization, and finite element discretization for anisotropic elastic parameters.

Contribution

It introduces a unified analysis of inverse PDE problems, including stability, null space characterization, and a finite element method for parameter recovery.

Findings

Proved a closed range property leading to $L^2$-stability estimates.

Characterized the null space using conservative third-order tensor fields.

Demonstrated the effectiveness of the finite element discretization through numerical examples.

Abstract

This article aims to present a general analysis of a class of inverse problems that consists in recovering the elliptic parameter maps in systems of PDEs, such as the linear elastic system, from the knowledge of some of their solutions. This identification problem is reformulated as a first-order linear system of the form $\nabla\bm{\mu} + \bm{B} \cdot \bm{\mu} = F$, where $F$ and $\g B$ are tensor fields constructed from the data. A closed range property is proved, which induces $L^2$-stability estimates. We then characterize the null space by introducing the concept of conservative third-order tensor field. Finally, a discretization based on the finite element method is proposed and some numerical examples show the efficiency of this approach to recover anisotropic elastic parameters from both static and dynamic solutions of the PDE system.