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It is well known that time dependent Hamilton-Jacobi-Isaacs partialdifferential equations (HJ PDE), play an important role in analyzing continuousdynamic games and control theory problems. An important tool for such problemswhen they involve geometric motion is the level set method. This was first usedfor reachability problems. The cost of these algorithms, and, in fact, all PDEnumerical approximations is exponential in the space dimension and time.  In this work we propose and test methods for solving a large class of the HJPDE relevant to optimal control problems without the use of grids or numericalapproximations. Rather we use the classical Hopf formulas for solving initialvalue problems for HJ PDE. We have noticed that if the Hamiltonian is convexand positively homogeneous of degree one that very fast methods exist to solvethe resulting optimization problem. This is very much related to fast methodsfor solving problems in compressive sensing, based on $\ell_1$ optimization. Weseem to obtain methods which are polynomial in the dimension. Our algorithm isvery fast, requires very low memory and is totally parallelizable. We canevaluate 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 thenumerical solution of the associated initial valued HJ-PDE for a class ofoptimal control problems. We show that our algorithms compute all thequantities we need to obtain easily the controller.  The term curse of dimensionality, was coined by Richard Bellman in 1957 whenconsidering problems in dynamic optimization.

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