  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.
