Splitting Enables Overcoming the Curse of Dimensionality

Abstract

In this chapter we briefly outline a new and remarkably fast algorithm for solving a large class of high dimensional Hamilton-Jacobi (H-J) initial value problems arising in optimal control and elsewhere [1]. This is done without the use of grids or numerical approximations. Moreover, by using the level set method [8] we can rapidly compute projections of a point in $\mathbb{R}^{n}$, n large to a fairly arbitrary compact set [2]. The method seems to generalize widely beyond what will we present here to some nonconvex Hamiltonians, new linear programming algorithms, differential games, and perhaps state dependent Hamiltonians.