Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming

TL;DR

This paper introduces a novel approach linking inverse elliptic coefficient problems to convex non-linear semidefinite programming, enabling better measurement estimates, error analysis, and overcoming local minima in solutions.

Contribution

It establishes a new connection between inverse elliptic problems and semidefinite programming, providing a convex reformulation for improved solvability and analysis.

Findings

Reformulation as a convex non-linear semidefinite problem

Explicit measurement estimates for desired resolution

Error bounds for noisy data and avoidance of local minima

Abstract

Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.