  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.

Numerical Analysis;Numerical Analysis