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This paper provides a rigorous convergence rate and complexity analysis for arecently introduced framework, called PDE acceleration, for solving problems inthe calculus of variations, and explores applications to obstacle problems. PDEacceleration grew out of a variational interpretation of momentum methods, suchas Nesterov's accelerated gradient method and Polyak's heavy ball method, thatviews acceleration methods as equations of motion for a generalized Lagrangianaction. Its application to convex variational problems yields equations ofmotion in the form of a damped nonlinear wave equation rather than nonlineardiffusion arising from gradient descent. These accelerated PDE's can beefficiently solved with simple explicit finite difference schemes whereacceleration is realized by an improvement in the CFL condition from $dt\simdx^2$ for diffusion equations to $dt\sim dx$ for wave equations. In this paper,we prove a linear convergence rate for PDE acceleration for strongly convexproblems, provide a complexity analysis of the discrete scheme, and show how tooptimally select the damping parameter for linear problems. We then apply PDEacceleration to solve minimal surface obstacle problems, including doubleobstacles with forcing, and stochastic homogenization problems with obstacles,obtaining state of the art computational results.

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