Practical guidelines for fast, efficient and robust simulations of yield-stress flows without regularisation using accelerated proximal gradient or augmented Lagrangian methods Treskatis, Timm; Roustaei, Ali; Frigaard, Ian; Wachs, Anthony M., 1983-
The mathematically sound resolution of yield stress fluid flows involves nonsmooth convex optimisation problems. Traditionally, augmented Lagrangian methods developed in the 1980's have been used for this purpose. The main drawback of these algorithms is their frustratingly slow O(1/√k) worst-case convergence, where k is the iteration counter. Recently, an improved 'dual FISTA' algorithm (short: FISTA*) was introduced, which achieves the higher and provably optimal rate of O(1/k). When implementing these algorithms in two finite-element packages (FreeFem++ by Frédéric Hecht, UPMC Paris and Rheolef by Pierre Saramito, UGA Grenoble), we observed that these theoretical convergence rates are not generally attained. In this article, we present four common numerical pitfalls that adversely impact the convergence of the optimisation algorithms. By means of constructive and practical guidelines we point out how a careful implementation can not only recover the full order of convergence, but also reduce the computational cost per iteration for further efficiency gains. Furthermore, we assess the performance and accuracy of FISTA* for the practical case of flow in wavy walled channel and demonstrate significant speed-up when FISTA* is employed instead of the classical augmented Lagrangian method.
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