A RBF approach to the control of PDEs using Dynamic Programming equations

Semi-Lagrangian schemes for discretization of the dynamic programming principle are based on a time discretization projected on a state-space grid.
The use of a structured grid makes this approach not feasible for high-dimensional problems due to the curse of dimensionality.
Here, we present a new approach for infinite horizon optimal control problems where the value function is computed using Radial Basis Functions (RBF) by the Shepard’s moving least squares approximation method on scattered grids.
We propose a new method to generate a scattered mesh driven by the dynamics and an optimal routine to select the shape parameter in the RBF. This mesh will help to localize the problem and approximate the dynamic programming principle in high dimension. Error estimates for the value function are also provided. Numerical tests for high dimensional problems will show the effectiveness of the proposed method. In addition to the optimal control of classical PDEs, we show how the method can also be applied to the control of non-local equations.