  Recent observations have been made that bridge splitting methods arising from
optimization, to the Hopf and Lax formulas for Hamilton-Jacobi Equations with
Hamiltonians $H(p)$. This has produced extremely fast algorithms in computing
solutions of these PDEs. More recent observations were made in generalizing the
Hopf and Lax formulas to state-and-time-dependent cases $H(x,p,t)$. In this
article, we apply a new splitting method based on the Primal Dual Hybrid
Gradient algorithm (a.k.a. Chambolle-Pock) to nonlinear optimal control and
differential games problems, based on techniques from the derivation of the new
Hopf 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 us
to create trajectories directly. Thus we are able to lift the curse of
dimensionality a bit, and therefore compute solutions in much higher dimensions
than before. And in our numerical experiments, we actually observe that our
computations scale polynomially in time. Furthermore, this new algorithm is
embarrassingly parallelizable.
