An ADMM-Newton-CNN numerical approach to a TV model for identifying discontinuous diffusion coefficients in elliptic equations: convex case with gradient observations

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.

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

“…However, the non-smoothness property of the original term can make algorithmic design challenging. To address this issue, recent work in [40] applied an ADMM, which allowed the non-smoothing properties of the coefficient to be identified everywhere, while maintaining the convexity of the data fidelity function and proposing an algorithm that is easy to implement. To the best of our knowledge, there is no study of stability and convergence for such an approach for the parameter identification in partial differential equations in the literature, neither in the continuous nor in the discrete setting.…”

Convergence rate analysis of the coefficient identification problems in a Kuramoto–Sivashinsky equation

“…The TV regularization can be utilized to reserve the piecewise-constant property, and has advantageous to recover non-smooth or discontinuous solutions, such as image denoising or reconstruction and identification of discontinuous coefficients in partial differential equations, see [12][13][14][15][16][17] and the references therein. In addition, the L p regularization also preserves some special feature to the unknown coefficient, such as the sparsity or localized oscillating profiles by L 1 regularization, see [18][19][20] for instance.…”