Conductivity reconstruction in an inverse eddy current problem is considered in the present paper. With the electric field measurement on part of domain boundary, we formulate the reconstruction problem to a constrained optimization problem with total variation regularization. Existence and stability are proved for the solution to the optimization problem. The finite element method is employed to discretize the optimization problem. The gradient Lipschitz properties of the objective functional are established for the the discrete optimization problems. We propose the alternating direction method of multipliers to solve the discrete problem. Based on the the gradient Lipschitz property, we prove the convergence by extending the admissible set to the whole finite element space. Finally, we show some numerical experiments to illustrate the efficiency of the proposed methods.

翻译：本文研究了逆涡流问题中的电导率重建。基于域部分边界上的电场测量，我们将重建问题表述为带全变差正则化的约束优化问题。证明了该优化问题解的存在性与稳定性。采用有限元方法离散优化问题，并建立了离散优化问题目标泛函的梯度Lipschitz性质。提出应用交替方向乘子法求解离散问题，基于梯度Lipschitz性质，通过将允许集扩展至整个有限元空间证明了收敛性。最后通过数值实验验证了所提方法的有效性。