Mathematical and numerical study of a three-dimensional inverse eddy current problem

TL;DR

This paper investigates the mathematical and numerical aspects of an inverse eddy current problem in three dimensions, addressing ill-posedness, regularization, and proposing an efficient algorithm with numerical validation.

Contribution

It introduces a new regularization framework for the inverse eddy current problem and develops a conjugate gradient method with Sobolev gradient acceleration for numerical solutions.

Findings

Proved the compactness of the forward map.

Established existence and stability of regularized solutions.

Demonstrated the effectiveness of the proposed algorithm through numerical examples.

Abstract

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.