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Recent observations have been made that bridge splitting methods arising fromoptimization, to the Hopf and Lax formulas for Hamilton-Jacobi Equations withHamiltonians $H(p)$. This has produced extremely fast algorithms in computingsolutions of these PDEs. More recent observations were made in generalizing theHopf and Lax formulas to state-and-time-dependent cases $H(x,p,t)$. In thisarticle, we apply a new splitting method based on the Primal Dual HybridGradient algorithm (a.k.a. Chambolle-Pock) to nonlinear optimal control anddifferential games problems, based on techniques from the derivation of the newHopf 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 usto create trajectories directly. Thus we are able to lift the curse ofdimensionality a bit, and therefore compute solutions in much higher dimensionsthan before. And in our numerical experiments, we actually observe that ourcomputations scale polynomially in time. Furthermore, this new algorithm isembarrassingly parallelizable.

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