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.

对于有限元离散化的迭代解，考虑了非光滑最小化问题，采用了乘数交变方向法（ADMM），这是解决一类凸最小化问题的灵活方法。特殊的功能是它在涉及的步长及其直接适用性方面的无条件收敛。特别是，本文讨论了具有可变步长的ADMM，并根据残差的单调性为步长设计了调整规则，并讨论了适当的停止准则。拟议的方案适用于障碍问题和Rudin-Osher-Fatemi图像降噪问题的有限元公式化，数值实验表明，相对于已建立的ADMM变体而言，有了显着的改进。