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 = 1/2 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. (C) 2013 Wiley Periodicals, Inc.