A Proximal Primal-Dual Approach to Generalized JKO Schemes for Doubly Nonlinear Parabolic Equations

TL;DR

This paper introduces a variational numerical method for solving nonlinear PDEs, including p-Laplace and relativistic heat equations, using proximal operators with general costs.

Contribution

It develops a proximal primal-dual approach to generalized JKO schemes, enabling efficient numerical solutions for a broad class of nonlinear equations.

Findings

Successfully recovers qualitative behavior of known PDE cases.

Provides explicit formulas for proximal operators with general costs.

Validates the numerical approach through simulations.

Abstract

Variational methods based on optimization strategies are proposed to numerically solve a large family of nonlinear partial differential equations. They are all particular instances of gradient flows with general costs, including the $p$-Laplace equation and flux-limited equations such as the relativistic heat equation. This is achieved by computing explicit formulas for proximal operators with general costs amenable to efficient numerical approximation. We showcase our numerical approach via validation of the results by recovering the qualitative behavior of particular known cases of this large family of steepest descent evolutions.