We consider inverse problem for a non-linear wave equation with a time-depending metric tensor on manifolds. In addition, we study the question, do the observation of the solutions of coupled Einstein equations and matter field equations in an open subset U of the space-time M corresponding to sources supported in U determine the properties of the metric in a maximal domain where waves can propagate from U and return back to U. To study these problems we define the concept of light observation sets and show that these sets determine the conformal class of the metric. The results have been done in collaboration with Yaroslav Kurylev and Gunther Uhlmann.

In this talk I will describe a multiple frequency approach to the boundary control of Helmholtz and Maxwell equations. We give boundary conditions and a finite number of frequencies such that the corresponding solutions satisfy certain non-zero constraints inside the domain. The suitable boundary conditions and frequencies are explicitly constructed and do not depend on the coefficients, in contrast to the illuminations given as traces of complex geometric optics solutions. This theory finds applications in several hybrid imaging modalities: these constraints are needed to prove stability and to apply explicit reconstruction formulae. Similarly, multiple frequencies can be used to prove uniqueness and stability for the linearized inverse problem in acousto-electromagnetic tomography, thereby obtaining the convergence of a Landweber iteration scheme.

Magneto-acoustic tomography with magnetic induction (MAT-MI) is a coupled-physics medical imaging modality for determining conductivity distribution in biological tissue. MAT-MI involves two steps. The first step is a well-posed inverse source problem for acoustic wave equation, which has been well studied in the literature. This work concerns mathematical analysis of the second step, a quantitative reconstruction of the conductivity from knowledge of the internal data recovered in the first step, using techniques such as time reversal. The problem is modeled by a system derived from Maxwell's equations. We show that a single internal data determines the conductivity and a global Lipschitz type stability estimate is obtained. A numerical approach for recovering the conductivity is proposed. A linear convergent rate is proved for the proposed scheme and observed in the computational experiments.