We consider an adaptive sparse grid approach for the numerical solution of Hamilton-Jacobi-Bellman (HJB) equations in high dimensions. HJB equations arise for example for the solution of optimal control (OC) problems, or in the field of reinforcement learning that is closely related to optimal control. In this talk, we concentrate on a specific kind of HJB equations that describe optimal control problems in continuous time and state space with finite time horizon.
We use spatial adaptive sparse grids for the discretization of the continuous state spaces and apply a semi lagrangian scheme based on the dynamic programming approach in order to approximate the HJB-solution. We show numerical results where sparse grids enable us to treat problems in up to 9 dimensions.