We consider the sparse grid approximation of the Riccati operator P arising from closed loop parabolic control problems. In particular, we concentrate on the linear quadratic regulator (LQR) problems, i.e. we are looking for an optimal control uopt in the linear state feedback form uopt(t, •) = P x(t, •), where x(t, •) is the solution of the controlled partial differential equation (PDE) for a time point t. Under sufficient regularity assumptions, the Riccati operator P fulfills the algebraic Riccati equation (ARE)where A, B, and Q are linear operators associated to the LQR problem. By expressing P in terms of an integral kernel p, the weak form of the ARE leads to a non-linear partial integro-differential equation for the kernel p -the Riccati-IDE. We represent the kernel function as an element of a sparse grid space, which considerably reduces the cost to solve the Riccati IDE. Numerical results are given to validate the approach.

We consider a semi-Lagrangian approach for the computation of the value function of a Hamilton-Jacobi-Bellman equation. This problem arises when one solves optimal feedback control problems for evaluationary partial differential equations. A time discretization with Runge-Kutta methods leads in general to a complexity of the optimization problem for the control which is exponential in the number of stages of the time scheme. Motivated by this, we introduce a time discretization based on Runge-Kutta composition methods, which achieves higher order approximation with respect to time, but where the overall optimization costs increase only linearly with respect to the number of stages of the employed Runge-Kutta method. In numerical tests we can empirically confirm an approximately linear complexity with respect to the number of stages. The presented algorithm is in particular of interest for those optimal control problems which do involve a costly minimization over the control set.