Efficient iterative solution of finite element discretized nonsmooth minimization problems

Abstract

For the iterative solution of finite element discretized, nonsmooth minimization problems the alternating direction method of multipliers (ADMM) is considered, which is a flexible method to solve a large class of convex minimization problems. Particular features are its unconditional convergence with respect to the involved step size and its direct applicability. In particular, this article deals with the ADMM with variable step sizes and devises an adjustment rule for the step size relying on the monotonicity of the residual and discusses proper stopping criteria. The proposed scheme is applied to finite element formulations of the obstacle problem and the Rudin–Osher–Fatemi image denoising problem, and the numerical experiments show significant improvements over established variants of the ADMM.