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We consider a class of nonlinear saddle point problems with various applications in PDEs and optimal control problems and propose an algorithmic framework based on some inexact Uzawa methods in the...

The alternating direction method of multipliers (ADMM) has been extensively used in a wide variety of different applications. When large datasets with high-dimensional variables are considered, subproblems arising from the ADMM must be inexactly solved even though they may theoretically have closed-form solutions. Such a scenario immediately poses mathematical ambiguities such as how these subproblems should be accurately solved and whether the convergence can still be guaranteed. Although the ADMM is well known, it seems that these topics should be deeply investigated. In this paper, we study the mathematics of how to implement the ADMM for a large dataset scenarios. More specifically, we attempt to focus on the convex programming case where there is a quadratic function with extremely high-dimensional variables in the objective function of the model; thereby there is a huge-scale system for linear equations needing to be solved at each iteration of the ADMM. It is revealed that there is no need, indeed ...

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.