A fast implicit method for time-dependent Hamilton-Jacobi PDEs

TL;DR

This paper introduces an efficient implicit numerical method for solving time-dependent Hamilton-Jacobi PDEs, offering unconditional stability and improved efficiency over explicit methods in certain scenarios.

Contribution

The authors develop a novel implicit discretization approach that reinterprets the problem as a static PDE, enabling fast, non-iterative solutions and a hybrid method combining explicit and implicit advantages.

Findings

Implicit method is unconditionally stable and efficient for larger time steps.

Reinterpreting as a static PDE allows for fast, non-iterative solutions.

Hybrid approach balances explicit and implicit method benefits.

Abstract

We present a new efficient computational approach for time-dependent first-order Hamilton-Jacobi-Bellman PDEs. Since our method is based on a time-implicit Eulerian discretization, the numerical scheme is unconditionally stable, but discretized equations for each time-slice are coupled and non-linear. We show that the same system can be re-interpreted as a discretization of a static Hamilton-Jacobi-Bellman PDE on the same physical domain. The latter was shown to be ``causal' in [Vladimirsky 2006], making fast (non-iterative)methods applicable. The implicit discretization results in higher computational cost per time slice compared to the explicit time marching. However, the latter is subject to a CFL-stability condition, and the implicit approach becomes significantly more efficient whenever the accuracy demands on the time-step are less restrictive than the stability. We also present a hybrid method, which aims to combine the advantages of both the explicit and implicit discretizations. We demonstrate the efficiency of our approach using several examples in optimal control of isotropic fixed-horizon processes.