# An Adaptive Sparse Grid Semi-Lagrangian Scheme for First Order Hamilton-Jacobi Bellman Equations

Jochen Garcke*

We start with an introduction on sparse grids for high-dimensional function approximation. Following that, we propose a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations. We focus in particular on front propagation models in higher dimensions which are related to control problems. We test the numerical efficiency of the method on several benchmark problems up to space dimension $d=8$, and give evidence of convergence towards the exact viscosity solution. In addition, we study how the complexity and precision scale with the dimension of the problem.

Mathematics Subject Classification: 65D05

Keywords: sparse grids; semi-lagrangian; front propagation

Minisymposion: Dynamic Programming Approach to Optimal Control Methods and Applications