title:
Solving optimal feedback control problems for partial differential equations using adaptive sparse grids 
name:
Garcke 
first name:
Jochen

location/conference:
SPPJT14

PRESENTATIONlink:
http://www.dfgspp1324.de/nuhagtools/event_NEW/dateien/SPPJT14/slides/garcke_fc14.pdf 
abstract:
An approach to solve finite time horizon optimal feedback control problems for partial differential equations using adaptive sparse grids is presented. A semidiscrete optimal control problem is introduced and
the feedback control is derived from the corresponding value function.
The value function can be characterized as the solution of an evolutionary HamiltonJacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semidiscrete system. Besides a low dimensional
semidiscretization it is important to solve the HJB equation efficiently to address the curse of dimensionality.
We apply a semiLagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the high(er) dimensional value functions arising in the numerical scheme since the
curse of dimensionality of full grid methods arises to a much smaller extent. For additional efficiency an adaptive grid refinement procedure is explored.
We present several numerical examples studying the effect of the parameters characterizing the sparse grid on the accuracy of the value function and optimal trajectories. Furthermore we analyze the behaviour
of the trajectories in case of noise. 