Institute for Numerical Simulation
Rheinische Friedrich-Wilhelms-Universität Bonn

  abstract = {An approach to solve finite time horizon suboptimal
		  feedback control problems for partial differential
		  equations is proposed by solving dynamic programming
		  equations on adaptive sparse grids. A semi-discrete 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 Hamilton--Jacobi Bellman (HJB) equation which
		  is defined over a state space whose dimension is equal to
		  the dimension of the underlying semi-discrete system.
		  Besides a low dimensional semi-discretization it is
		  important to solve the HJB equation efficiently to address
		  the curse of dimensionality. We propose to apply a
		  semi-Lagrangian scheme using spatially adaptive sparse
		  grids. Sparse grids allow the discretization of the value
		  functions in (higher) space dimensions 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. The approach is
		  illustrated for the wave equation and an extension to
		  equations of Schr{\{}{\"{o}}{\}}dinger type is indicated.
		  We present several numerical examples studying the effect
		  the parameters characterizing the sparse grid have on the
		  accuracy of the value function and the optimal trajectory.},
  author = {Garcke, Jochen and Kr{\"{o}}ner, Axel},
  doi = {10.1007/s10915-016-0240-7},
  issn = {1573-7691},
  note = {also available as INS Preprint No. 1518},
  annote = {journal},
  inspreprintnum = {1518},
  pdf = { 1},
  journal = {Journal of Scientific Computing},
  number = {1},
  pages = {1--28},
  title = {{Suboptimal Feedback Control of PDEs by Solving HJB
		  Equations on Adaptive Sparse Grids}},
  volume = {70},
  http = {},
  year = {2017}