An Efficient Primal-Dual Method for the Obstacle Problem

Zosso, Dominique; Osting, Braxton; Xia, Mandy; Osher, Stanley

doi:10.1007/s10915-017-0420-0

Cited by 11 publications

(48 citation statements)

References 23 publications

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“…Here, we need k = 4 π 2 dx 2 log(Cε −1 ) iterations for convergence. Table 1 gives a comparison of the performance of PDE acceleration, gradient descent, and the primal dual algorithm from [48] for solving the Dirichlet problem on various grid sizes. We used the boundary condition g(x 1 , x 2 ) = sin(2πx 2 1 )+sin(2πx 2 2 ) and ran each algorithm until the finite difference scheme was satisfied with an error of less than dx 2 .…”

Section: Dirichlet Problemmentioning

confidence: 99%

“…The initial conditions for both algorithms were u(x, 0) = g(x). For the primal dual algorithm [48] we set r 1 = 4π 2 r 2 , which is provably optimal using similar methods as in Section 2.3. We see that PDE acceleration is more than twice as fast as primal dual, while both significantly outperform standard gradient descent.…”

Section: Dirichlet Problemmentioning

confidence: 99%

“…In the context of the Dirichlet problem, there is a close connection between primal dual methods [48], and PDE acceleration. This was explored briefly in [48], where it was observed that their primal dual algorithm for solving the Dirichlet problem can be interpreted as a numerical scheme for a damped wave equation. We go further here, and give a PDE interpretation of primal dual methods and show exactly how they are related to PDE acceleration for the Dirichlet problem.…”

Section: 3mentioning

confidence: 99%

“…Recently in [48], a primal dual algorithm was proposed for obstacle problems, and it was shown to be several orders of magnitude faster than existing state of the art methods. We compare PDE acceleration against an improved version of the primal dual algorithm from [48], which is described below.…”

Section: Primal Dual Algorithmsmentioning

confidence: 99%

“…Finally, we apply the PDE acceleration method to efficiently solve minimal surface obstacle problems [12,38,48]. Solving obstacle problems requires resolving a free boundary, which makes efficient solutions challenging to obtain.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Calder

Yezzi

2019

Res Math Sci

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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.

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Section: Dirichlet Problemmentioning

confidence: 99%

Section: Dirichlet Problemmentioning

confidence: 99%