Perspectives on Characteristics Based Curse-of-Dimensionality-Free Numerical Approaches for Solving Hamilton–Jacobi Equations

Abstract

This paper extends the considerations of the works [1,2] regarding curse-of-dimensionality-free numerical approaches to solve certain types of Hamilton-Jacobi equations arising in optimal control problems, differential games and elsewhere. A rigorous formulation and justification for the extended Hopf-Lax formula of [2] is provided together with novel theoretical and practical discussions including useful recommendations. By using the method of characteristics, the solutions of some problem classes under convexity/concavity conditions on Hamiltonians (in particular, the solutions of Hamilton-Jacobi-Bellman equations in optimal control problems) are evaluated separately at different initial positions. This allows for the avoidance of the curse of dimensionality, as well as for choosing arbitrary computational regions. The corresponding feedback control strategies are obtained at selected positions without approximating the partial derivatives of the solutions. The results of numerical simulations demonstrate the high potential of the proposed techniques. It is also pointed out that, despite the indicated advantages, the related approaches still have a limited range of applicability, and their extensions to Hamilton-Jacobi-Isaacs equations in zero-sum two-player differential games are currently developed only for sufficiently narrow classes of control systems. That is why further extensions are worth investigating.

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

“…In particular the conjectured (Hopf-type) maximization principle is a generalization of the well-known Hopf formula in [17,24,49]. We showed the validity of the formula under restricted assumption for the sake of completeness, and would like to bring our readers to [61] which validates that our conjectures hold in a more general setting after a previous version of our paper is on arXiv. We conjectured the weakest assumption of our formula to hold is a psuedoconvexity assumption similar to one stated in [49].The optimization problems are of the same dimension as the dimension of the HJ PDE.…”

“…We would like to bring the readers to notice that after a previous version of our paper in arXiv is out, in [61] showed that our conjectures hold in a more general setting than we did. However for the sake of completeness, we keep the proof with the restricted assumptions, and we refer our readers to the more general case shown in [61] 2 Review of Hamilton-Jacobi Equations and Differential Games…”

“…In particular the conjectured (Hopf-type) maximization principle is a generalization of the wellknown Hopf formula in [17,24,49]. We validated our conjectures under restricted assumptions and refers to [61] for the validation of our formula under a less restricted setting. The optimization problems are of the same dimension as the dimensions of the HJ PDE.…”

Algorithm for Hamilton–Jacobi Equations in Density Space Via a Generalized Hopf Formula

“…The continuous analog of (3.4) is discussed and proposed in Section 2.2. The minimal assumptions of validity for (3.4) to hold may be an interesting direction to explore, and some possibilities are discussed in, e.g., [12,22,28].…”

Fourier approximation methods for first-order nonlocal mean-field games

“…Additionally, our approach yields a mesh-free numerical approximation of u and m. More precisely, we directly recover the optimal trajectories of the agents rather than the values of u and m on a given mesh. In particular, our methods may blend well with recently developed ideas for fast and curse-of-the-dimensionality-resistant solution approach for first-order Hamilton-Jacobi equations [21,33,36]. Hence, our techniques may lead to numerical schemes for nonlocal MFG that are efficient in high dimensions.…”