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Identifying the discontinuous diffusion coefficient in an elliptic equationwith observation data of the gradient of the solution is an important nonlinearand ill-posed inverse problem. Models with total variational (TV)regularization have been widely studied for this problem, while thetheoretically required nonsmoothness property of the TV regularization and thehidden convexity of the models are usually sacrificed when numerical schemesare considered in the literature. In this paper, we show that the favorablenonsmoothness and convexity properties can be entirely kept if the well-knownalternating direction method of multipliers (ADMM) is applied to theTV-regularized models, hence it is meaningful to consider designing numericalschemes based on the ADMM. Moreover, we show that one of the ADMM subproblemscan be well solved by the active-set Newton method along with the Schurcomplement reduction method, and the other one can be efficiently solved by thedeep convolutional neural network (CNN). The resulting ADMM-Newton-CNN approachis demonstrated to be easily implementable and very efficient even forhigher-dimensional spaces with fine mesh discretization.

inverse problems

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