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问题。基于部分边界上的电场测量，我们将重建问题转化为带有全变分正则化的约束优化问题。证明了该优化问题解的存在性和稳定性。采用有限元方法对优化问题进行离散化。针对离散优化问题，建立了目标泛函的梯度Lipschitz性质。我们提出采用交替方向乘子法求解离散问题。基于梯度Lipschitz性质，通过将可行集扩展到整个有限元空间，证明了该方法的收敛性。最后，通过数值实验验证了所提方法的有效性。