In this talk I will present a recent result showing that linear combinations of tensor products of $k$ gradients of harmonic functions, with $k$ at least three, are dense in $C(\overline{\Omega})$, for any bounded domain $\Omega$ in dimension 3 or higher. This kind of density result has applications to inverse problems for elliptic quasilinear equations/systems in divergence form, where the nonlinear part of the "conductivity'' is anisotropic. The talk will be based on two papers written in collaboration with A. Feizmohammadi.

We study the problem of recovering an electrical anisotropic conductivity from interior power density measurements on a two-dimensional Riemannian manifold. This problem arises in Acousto-Electric Tomography and is motivated by the geometric Calderón problem of recovering the metric from the Dirichlet-to-Neumann map. In contrast to the geometric Calderón problem, we consider a conductive Riemannian manifold and treat the conductivity and metric separately. Assuming that the metric is known, for two-dimensional Riemannian manifolds with genus zero, we highlight in this talk that under certain assumptions on the power density data it is possible to recover the conductivity uniquely and constructively from the data. We illustrate our findings with a numerical experiment and comment on how added noise on the manifold affects the reconstructed conductivity.

In this talk, we discuss a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. The approach is based on a regularized output least-squares formulation with the standard $L^2$ penalty, which is then discretized by the standard Galerkin finite element method. We discuss the analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of $H$-convergence, when the discretization is uniform or adaptive. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.

In this talk we investigate the issues of stability and reconstruction in inverse problems in the presence of anisotropy. As is well-known, there is a fundamental obstruction to the unique determination of the anisotropic conductivity of materials. Such obstruction is based on the observation that any deffeomorphism of a domain $\Omega$ that keeps its boundary $\partial\Omega$ fixed, changes the conductivity in $\Omega$ by keeping the boundary measurements unchanged. In this talk we will investigate how to circumvent this obstruction and restore well-posedness in the problem.