PDE Acceleration: A convergence rate analysis and applications to obstacle problems

TL;DR

This paper analyzes the convergence and complexity of PDE acceleration, a method inspired by momentum techniques, applied to obstacle problems, demonstrating improved efficiency and state-of-the-art results in various applications.

Contribution

It provides the first rigorous convergence rate and complexity analysis for PDE acceleration and demonstrates its effectiveness on obstacle problems and related applications.

Findings

Proves linear convergence rate for strongly convex problems

Provides optimal damping parameter selection for linear problems

Achieves state-of-the-art results in obstacle problem computations

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 $dt\sim dx^2$ for diffusion equations to $dt\sim dx$ 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.