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

PDE acceleration: a convergence rate analysis and applications to obstacle problems 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 PDEs can be efficiently solved with simple explicit finite difference schemes where acceleration is realized by an improvement in the CFL condition from $$\mathrm{d}t\sim \mathrm{d}x^2$$ d t ∼ d x 2 for diffusion equations to $$\mathrm{d}t\sim \mathrm{d}x$$ d t ∼ d x 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
DOI
10.1007/s40687-019-0197-x
Publisher site
See Article on Publisher Site

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 PDEs can be efficiently solved with simple explicit finite difference schemes where acceleration is realized by an improvement in the CFL condition from $$\mathrm{d}t\sim \mathrm{d}x^2$$ d t ∼ d x 2 for diffusion equations to $$\mathrm{d}t\sim \mathrm{d}x$$ d t ∼ d x 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.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Oct 31, 2019

References