Primal-dual hybrid gradient algorithms for computing time-implicit Hamilton-Jacobi equations

TL;DR

This paper introduces a primal-dual hybrid gradient method for solving Hamilton-Jacobi equations by formulating them as saddle point problems, enabling efficient computation for complex, non-smooth, and spatially dependent cases.

Contribution

It develops a novel optimization-based framework using saddle point formulations and PDHG to solve time-implicit Hamilton-Jacobi equations, extending applicability to broader Hamiltonian classes.

Findings

Effective in one-dimensional and two-dimensional examples

Handles non-smooth and spatiotemporally dependent Hamiltonians

Provides fast computational updates

Abstract

Hamilton-Jacobi (HJ) partial differential equations (PDEs) have diverse applications spanning physics, optimal control, game theory, and imaging sciences. This research introduces a first-order optimization-based technique for HJ PDEs, which formulates the time-implicit update of HJ PDEs as saddle point problems. We remark that the saddle point formulation for HJ equations is aligned with the primal-dual formulation of optimal transport and potential mean-field games (MFGs). This connection enables us to extend MFG techniques and design numerical schemes for solving HJ PDEs. We employ the primal-dual hybrid gradient (PDHG) method to solve the saddle point problems, benefiting from the simple structures that enable fast computations in updates. Remarkably, the method caters to a broader range of Hamiltonians, encompassing non-smooth and spatiotemporally dependent cases. The approach's effectiveness is verified through various numerical examples in both one-dimensional and two-dimensional examples, such as quadratic and $L^1$ Hamiltonians with spatial and time dependence.