Convergence analysis of an alternating direction method of multipliers for the identification of nonsmooth diffusion parameters with total variation

Abstract

The paper presents a numerical method for identifying discontinuous conductivities in elliptic equations from boundary observations. The solutions to this inverse problem are obtained through a constrained optimization problem, where the cost functional is a combination of the Kohn–Vogelius and Total Variation functionals. Instead of regularizing the Total Variation stabilization functional, which is commonly used in the literature, we introduce an Alternating Direction Method of Multipliers to preserve the favorable properties of non-smoothness and convexity. The discretization is carried out using a mixed finite element/volume method, while the numerical solutions are iteratively computed using a variant of the Uzawa algorithm. We show the surjectivity of the derivatives of the constraints related to the discrete optimization problem and derive a source condition for the discrete inverse problem. We then investigate the convergence analysis and establish the convergence rate. Finally, we conclude with some numerical experiments to illustrate the efficiency of the proposed method.