Abstract

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 as a constrained optimization problem with total variation regularization. The 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 property of the objective functional is established for the discrete optimization problem. We propose a novel modification to the traditional alternating direction method of multipliers, and prove the convergence of the modified algorithm. Finally, we show some numerical experiments to illustrate the efficiency of the proposed method.

中文翻译：

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全变分正则化逆涡流问题的迭代方法

摘要

本文考虑了逆涡流问题中的电导率重建。通过对域边界部分的电场测量，我们将重构问题表述为具有全变分正则化的约束优化问题。证明了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，建立了目标函数的梯度Lipschitz性质。我们对传统的乘法器交替方向方法提出了一种新颖的修改，并证明了修改算法的收敛性。最后，我们展示了一些数值实验来说明所提出方法的效率。