Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere

Abstract

It is well known that time dependent Hamilton-Jacobi-Isaacs partial differential equations (HJ PDE), play an important role in analyzing continuous dynamic games and control theory problems. An important tool for such problems when they involve geometric motion is the level set method. This was first used for reachability problems. The cost of these algorithms, and, in fact, all PDE numerical approximations is exponential in the space dimension and time. In this work we propose and test methods for solving a large class of the HJ PDE relevant to optimal control problems without the use of grids or numerical approximations. Rather we use the classical Hopf formulas for solving initial value problems for HJ PDE. We have noticed that if the Hamiltonian is convex and positively homogeneous of degree one that very fast methods exist to solve the resulting optimization problem. This is very much related to fast methods for solving problems in compressive sensing, based on $\ell_1$ optimization. We seem to obtain methods which are polynomial in the dimension. Our algorithm is very fast, requires very low memory and is totally parallelizable. We can evaluate the solution and its gradient in very high dimensions at $10^{-4}$ to $10^{-8}$ seconds per evaluation on a laptop. We carefully explain how to compute numerically the optimal control from the numerical solution of the associated initial valued HJ-PDE for a class of optimal control problems. We show that our algorithms compute all the quantities we need to obtain easily the controller. The term curse of dimensionality, was coined by Richard Bellman in 1957 when considering problems in dynamic optimization.