# A Splitting Method For Overcoming the Curse of Dimensionality in Hamilton-Jacobi Equations Arising from Nonlinear Optimal Control and Differential Games with Applications to Trajectory Generation

## TL;DR

This paper introduces a novel splitting method based on the Chambolle-Pock algorithm for solving high-dimensional Hamilton-Jacobi equations from nonlinear optimal control and differential games, enabling direct solution computation without spatial grids and improving scalability.

## Contribution

The paper develops a new splitting algorithm leveraging recent formulas for Hamilton-Jacobi equations, allowing direct pointwise solutions and trajectories in high dimensions, reducing the curse of dimensionality.

## Key findings

- Computes solutions at points directly without spatial grids.
- Scales polynomially with time in numerical experiments.
- Algorithm is embarrassingly parallelizable.

## Abstract

Recent observations have been made that bridge splitting methods arising from optimization, to the Hopf and Lax formulas for Hamilton-Jacobi Equations with Hamiltonians $H(p)$. This has produced extremely fast algorithms in computing solutions of these PDEs. More recent observations were made in generalizing the Hopf and Lax formulas to state-and-time-dependent cases $H(x,p,t)$. In this article, we apply a new splitting method based on the Primal Dual Hybrid Gradient algorithm (a.k.a. Chambolle-Pock) to nonlinear optimal control and differential games problems, based on techniques from the derivation of the new Hopf and Lax formulas, which allow us to compute solutions at points $(x,t)$ directly, i.e. without the use of grids in space. This algorithm also allows us to create trajectories directly. Thus we are able to lift the curse of dimensionality a bit, and therefore compute solutions in much higher dimensions than before. And in our numerical experiments, we actually observe that our computations scale polynomially in time. Furthermore, this new algorithm is embarrassingly parallelizable.