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 \(\mathrm{d}t\sim \mathrm{d}x^2\) for diffusion equations to \(\mathrm{d}t\sim \mathrm{d}x\) 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.

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PDE加速：收敛速度分析及其在障碍问题中的应用

本文为最近引入的称为PDE加速的框架提供了严格的收敛速度和复杂性分析，以解决变异微积分中的问题，并探讨了对障碍问题的应用。PDE加速源自对动量方法的变异解释，例如Nesterov的加速梯度方法和Polyak的重球方法，后者将加速方法视为广义拉格朗日动作的运动方程。它在凸变分问题上的应用产生了阻尼非线性波方程形式的运动方程，而不是由梯度下降引起的非线性扩散。这些加速的PDE可以通过简单的显式有限差分方案有效地解决，其中通过将CFL条件从用于扩散方程的\（\ mathrm {d} t \ sim \ mathrm {d} x ^ 2 \）改进到\（\ mathrm {d} t \ sim \ mathrm {d} x \）用于波动方程。在本文中，我们证明了针对强凸问题的PDE加速的线性收敛速度，提供了离散方案的复杂度分析，并展示了如何为线性问题最佳选择阻尼参数。然后，我们应用PDE加速来解决最小的表面障碍问题，包括带有强迫的双重障碍，以及带有障碍的随机均质化问题，从而获得最新的计算结果。