High-order approximation of the finite horizon control problem via a tree structure algorithm

Solving optimal control problems via Dynamic Programming is a difficult task that suffers for the”curse of dimensionality”. This limitation has reduced its practical impact in real world applications since the construction of numerical methods for nonlinear PDEs in very high dimension is practically unfeasible. Recently, we proposed a new numerical method to compute the value function avoiding the construction of a space grid and the need for interpolation techniques. The method is based on a tree structure that mimics the continuous dynamics and allows to solve optimal control problems in high-dimension. This property is particularly useful to attack control problems with PDE constraints. We present a new high-order approximation scheme based on the tree structure and show some numerical results.