An adaptive POD approximation method for the control of evolutive equations
Alessandro Alla, Maurizio Falcone · 2012
equation, choosing in particular there advection‐ diusion equation. The basic ingredient of the method is the coupling between an adaptive reduced basis representation of the solution and a Dynamic Programming scheme for the evolutive Hamilton-Jacobi equation which gives the characterization of the value function. Although the approximation schemes available for the HJB are shown to be convergent for any dimension, in practice we need to restrict the dimension to rather low number (typically 4) and this limitation aects the accuracy of the POD approximation. In fact with only few basis functions the POD method does not have enough informations to follow the solution of the advectiondiusion problem. A way to circumvent this problem is to update our POD basis splitting the problem into subproblems. Every sub-problem is set in an interval [tj;tj+1] and in that interval we recompute the POD basis. Several strategies can be applied to determine an optimal way to generate the partition of the time interval so to adapt the POD basis choice. Since the solution based on the HJB equation allows to compute the value function and the optimal feedback for general nonlinear problems, this technique can be applied also to nonlinear cost functions. We will show some numerical tests to explain our problem and to show the eectiveness of the method.