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This paper concerns the reconstruction of possibly complex-valuedcoefficients in a second-order scalar elliptic equation posed on a boundeddomain from knowledge of several solutions of that equation. We show that for asufficiently large number of solutions and for an open set of correspondingboundary conditions, all coefficients can be uniquely and stably reconstructedup to a well characterized gauge transformation. We also show that in somespecific situations, a minimum number of such available solutions equal to$I_n=\frac12n(n+3)$ is sufficient to uniquely and globally reconstruct theunknown coefficients. This theory finds applications in several coupled-physicsmedical imaging modalities including photo-acoustic tomography, transientelastography, and magnetic resonance elastography.

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