Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions

TL;DR

This paper develops a method to uniquely and stably reconstruct complex coefficients in second-order elliptic equations from multiple solutions, with applications in medical imaging modalities like photo-acoustic tomography and elastography.

Contribution

It establishes conditions for unique and stable reconstruction of coefficients from multiple solutions, including minimal solution counts in specific cases, advancing inverse problem theory.

Findings

Unique and stable reconstruction of coefficients up to gauge transformations.

Minimal number of solutions needed for global reconstruction in certain cases.

Applications demonstrated in medical imaging techniques.

Abstract

This paper concerns the reconstruction of possibly complex-valued coefficients in a second-order scalar elliptic equation posed on a bounded domain from knowledge of several solutions of that equation. We show that for a sufficiently large number of solutions and for an open set of corresponding boundary conditions, all coefficients can be uniquely and stably reconstructed up to a well characterized gauge transformation. We also show that in some specific situations, a minimum number of such available solutions equal to $I_n=\frac12n(n+3)$ is sufficient to uniquely and globally reconstruct the unknown coefficients. This theory finds applications in several coupled-physics medical imaging modalities including photo-acoustic tomography, transient elastography, and magnetic resonance elastography.