Reconstruction of Coefficients in Scalar Second‐Order Elliptic Equations from Knowledge of Their Solutions

This paper concerns the reconstruction of possibly complex‐valued coefficients in a second‐order scalar elliptic equation that is 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 $I_n = {1 \over 2}n(n + 3)$ 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. © 2013 Wiley Periodicals, Inc.