PDE acceleration: a convergence rate analysis and applications to obstacle problems

Abstract

This paper provides a rigorous convergence rate and complexity analysis for a recently introduced framework, called PDE acceleration, for solving problems in the calculus of variations, and explores applications to obstacle problems. PDE acceleration grew out of a variational interpretation of momentum methods, such as Nesterov's accelerated gradient method and Polyak's heavy ball method, that views acceleration methods as equations of motion for a generalized Lagrangian action. Its application to convex variational problems yields equations of motion in the form of a damped nonlinear wave equation rather than nonlinear diffusion arising from gradient descent. These accelerated PDE's can be efficiently solved with simple explicit finite difference schemes where acceleration is realized by an improvement in the CFL condition from dt ∼ dx 2 for diffusion equations to dt ∼ dx for wave equations. In this paper, we prove a linear convergence rate for PDE acceleration for strongly convex problems, provide a complexity analysis of the discrete scheme, and show how to optimally select the damping parameter for linear problems. We then apply PDE acceleration to solve minimal surface obstacle problems, including double obstacles with forcing, and stochastic homogenization problems with obstacles, obtaining state of the art computational results.

Physics-inspired pseudo-transient method and its application in modelling focused fluid flow with geological complexity

“…Recent studies with matrix-free methods have demonstrated the ability to solve up to 10 12 degrees of freedom with a parallel efficiency of >90 per cent (Bauer et al 2019), which is far beyond what matrix-based methods can reach. At the same time, the iterative procedure of Pseudo-transient continuation, inspired by physical processes, has been proven to solve multiphysics problems efficiently (Frankel 1950;Calder & Yezzi 2019;Räss et al 2019;Benyamin et al 2020). With matrix-free implementation, Rass et al 2019 applied a pseudo-transient solver to simulate the long-term evolution of large-scale 3-D problems (1000 3 ) and showed its ability to handle the non-linearity of Kozeny-Carman permeability and decompaction weakening.…”

“…This method is also widely known as the 2nd Richardson or relaxation method (Frankel 1950), dynamic relaxation method (Otter et al 1966;Zhang & Yu 1989) and inertial method (Poliakov et al 1993). Recently, a general framework for this type of numerical methods has been developed, known as 'PDE acceleration', which shows great potentials both for the forward and inverse modeling problems (Calder & Yezzi 2018;Benyamin et al 2020). As demonstrated by previous studies, fast PT solvers usually involve 2nd time-derivatives or 'damping factors' (Frankel 1950;Zhang & Yu 1989;Chen 2009;Benyamin et al 2020) that enable the iteration number to be linearly scaled with grid number in one direction.…”

Flow of the Bingham-Papanastasiou Regularized Material in a Channel in the Presence of Obstacles: Correlation between Hydrodynamic Forces and Spacing of Obstacles

“…Recently, Koblitz et al [33] investigated the hydrodynamic forces for very high values of Bn. Calder and Yezzi [34] considered a rate of convergence and application for obstacle problems. In some studies mentioned therein [35][36][37][38], authors have numerically investigated the flows of incompressible non-Newtonian fluids in various computational domains.…”