We present a very simple and fast algorithm for the numerical solution of viscoplastic flow problems without prior regularisation. Compared to the widespread alternating direction method of multipliers (ADMM / ALG2), the new method features three key advantages: firstly, it accelerates the worst-case convergence rate from $O(1/\sqrt{k})$ to $O(1/k)$, where $k$ is the iteration counter. Secondly, even for nonlinear constitutive models like those of Casson or Herschel-Bulkley, no nonlinear systems of equations have to be solved in the subproblems of the algorithm. Thirdly, there is no need to augment the Lagrangian, which eliminates the difficulty of choosing a penalty parameter heuristically.
In this paper, we transform the usual velocity-based formulation of viscoplastic flow problems to a dual formulation in terms of the stress. For the numerical solution of this dual problem we apply FISTA, an accelerated first-order optimisation algorithm from the class of so-called proximal gradient methods. Finally, we conduct a series of numerical experiments, focussing on stationary flow in two-dimensional square cavities.
Our results confirm that Algorithm FISTA*, the new dual-based FISTA, outperforms state-of-the-art algorithms such as ADMM / ALG2 by several orders of magnitude. We demonstrate how this speedup can be exploited to identify the free boundary between yielded and unyielded regions with previously unknown accuracy. Since the accelerated algorithm relies solely on Stokes-type subproblems and nonlinear function evaluations, existing code based on augmented Lagrangians would require only few minor adaptations to obtain an implementation of FISTA*.
Accelerated Primal-Dual Proximal Gradient Splitting Methods for Convex-Concave Saddle-Point Problems
National Center for Applied Mathematics in Chongqing, Chongqing Normal University（重庆师范大学应用数学中心）；Chongqing Research Institute of Big Data, Peking University（北京大学重庆大数据研究院）
Inexact accelerated proximal gradient method with line search and reduced complexity for affine-constrained and bilinear saddle-point structured convex problems
Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling
University of Edinburgh（爱丁堡大学）；Heriot-Watt University（赫瑞-瓦特大学）；Maxwell Institute for Mathematical Sciences（麦克斯韦数学科学研究所）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
Accelerated Implicit GDA Schemes: Theoretical Guarantees and Application to Proximal Augmented Lagrangian Methods
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究所）；Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学与交叉学科中心）
Linear convergence of forward-backward accelerated algorithms without knowledge of the modulus of strong convexity
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
