Algorithms for overcoming the curse of dimensionality for certain Hamilton–Jacobi equations arising in control theory and elsewhere

It is well known that time-dependent Hamilton–Jacobi–Isaacs partial differential equations (HJ PDEs) play an important role in analyzing continuous dynamic games and control theory problems. An important tool for such problems when they involve geometric motion is the level set method (Osher and Sethian in J Comput Phys 79(1):12–49, 1988). This was first used for reachability problems in Mitchell et al. (IEEE Trans Autom Control 50(171):947–957, 2005) and Mitchell and Tomlin (J Sci Comput 19(1–3):323–346, 2003). The cost of these algorithms and, in fact, all PDE numerical approximations is exponential in the space dimension and time. In Darbon (SIAM J Imaging Sci 8(4):2268–2293, 2015), some connections between HJ PDE and convex optimization in many dimensions are presented. In this work, we propose and test methods for solving a large class of the HJ PDE relevant to optimal control problems without the use of grids or numerical approximations. Rather we use the classical Hopf formulas for solving initial value problems for HJ PDE (Hopf in J Math Mech 14:951–973, 1965). We have noticed that if the Hamiltonian is convex and positively homogeneous of degree one (which the latter is for all geometrically based level set motion and control and differential game problems) that very fast methods exist to solve the resulting optimization problem. This is very much related to fast methods for solving problems in compressive sensing, based on $$\ell _1$$ℓ1 optimization (Goldstein and Osher in SIAM J Imaging Sci 2(2):323–343, 2009; Yin et al. in SIAM J Imaging Sci 1(1):143–168, 2008). We seem to obtain methods which are polynomial in the dimension. Our algorithm is very fast, requires very low memory and is totally parallelizable. We can evaluate the solution and its gradient in very high dimensions at $$10^{-4}$$10-4–$$10^{-8}$$10-8 s per evaluation on a laptop. We carefully explain how to compute numerically the optimal control from the numerical solution of the associated initial valued HJ PDE for a class of optimal control problems. We show that our algorithms compute all the quantities we need to obtain easily the controller. In addition, as a step often needed in this procedure, we have developed a new and equally fast way to find, in very high dimensions, the closest point y lying in the union of a finite number of compact convex sets $$\Omega $$Ω to any point x exterior to the $$\Omega $$Ω. We can also compute the distance to these sets much faster than Dijkstra type “fast methods,” e.g., Dijkstra (Numer Math 1:269–271, 1959). The term “curse of dimensionality” was coined by Bellman (Adaptive control processes, a guided tour. Princeton University Press, Princeton, 1961; Dynamic programming. Princeton University Press, Princeton, 1957), when considering problems in dynamic optimization.