  We study an inverse problem associated with an eddy current model. We first
address the ill-posedness of the inverse problem by proving the compactness of
the forward map with respect to the conductivity and the non-uniqueness of the
recovery process. Then by virtue of non-radiating source conceptions, we
establish a regularity result for the tangential trace of the true solution on
the boundary, which is necessary to justify our subsequent mathematical
formulation. After that, we formulate the inverse problem as a constrained
optimization problem with an appropriate regularization and prove the existence
and stability of the regularized minimizers. To facilitate the numerical
solution of the nonlinear non-convex constrained optimization, we introduce a
feasible Lagrangian and its discrete variant. Then the gradient of the
objective functional is derived using the adjoint technique. By means of the
gradient, a nonlinear conjugate gradient method is formulated for solving the
optimization system, and a Sobolev gradient is incorporated to accelerate the
iterative process. Numerical examples are provided to demonstrate the
feasibility of the proposed algorithm.
