An ADMM-Newton-CNN Numerical Approach to a TV Model for Identifying Discontinuous Diffusion Coefficients in Elliptic Equations: Convex Case with Gradient Observations

TL;DR

This paper introduces an efficient numerical method combining ADMM, Newton, and CNN techniques to solve a convex TV-regularized inverse problem for identifying discontinuous diffusion coefficients in elliptic equations, preserving key mathematical properties.

Contribution

The paper develops a novel ADMM-Newton-CNN approach that maintains the convexity and nonsmoothness of TV models while improving computational efficiency for inverse diffusion problems.

Findings

The method effectively preserves TV model properties.

It demonstrates high efficiency in higher-dimensional problems.

The CNN component accelerates the solution of subproblems.

Abstract

Identifying the discontinuous diffusion coefficient in an elliptic equation with observation data of the gradient of the solution is an important nonlinear and ill-posed inverse problem. Models with total variational (TV) regularization have been widely studied for this problem, while the theoretically required nonsmoothness property of the TV regularization and the hidden convexity of the models are usually sacrificed when numerical schemes are considered in the literature. In this paper, we show that the favorable nonsmoothness and convexity properties can be entirely kept if the well-known alternating direction method of multipliers (ADMM) is applied to the TV-regularized models, hence it is meaningful to consider designing numerical schemes based on the ADMM. Moreover, we show that one of the ADMM subproblems can be well solved by the active-set Newton method along with the Schur complement reduction method, and the other one can be efficiently solved by the deep convolutional neural network (CNN). The resulting ADMM-Newton-CNN approach is demonstrated to be easily implementable and very efficient even for higher-dimensional spaces with fine mesh discretization.