Algorithm for overcoming the curse of dimensionality for state-dependent Hamilton-Jacobi equations

Abstract

In this paper, we develop algorithms to overcome the curse of dimensionality in non-convex state-dependent Hamilton-Jacobi partial differential equations (HJ PDEs) arising from optimal control and differential game problems. The subproblems are independent and they can be implemented in an embarrassingly parallel fashion. This is ideal for perfect scaling in parallel computing. The algorithm is proposed to overcome the curse of dimensionality [1,2] when solving HJ PDE. The major contribution of the paper is to change either the solving of a PDE problem or an optimization problem over a space of curves to an optimization problem of a single vector, which goes beyond the work of [40]. We extend the method in [7,9,15], and conjecture a (Lax-type) minimization principle to solve state-dependent HJ PDE when the Hamiltonian is convex, as well as a (Hopf-type) maximization principle to solve state-dependent HJ PDE when the Hamiltonian is non-convex, as a generalization of the well-known Hopf formula in [18,25,50]. We showed the validity of the formula under restricted assumption for the sake of completeness, and would like to bring our readers to [62] which validates our conjectures in a more general setting. We conjectured the weakest assumption of our formula to hold is a pseudoconvexity assumption similar to one stated in [50]. Our method is expected to have application in control theory, differential game problems and elsewhere.