Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  author = {R.~Hochmuth and S.~Knapek and G.~Zumbusch},
  title = {Tensor products of {S}obolev spaces and applications},
  journal = {submitted},
  year = {2000},
  annote = {refereed,article},
  note = {also as Technical Report 685, SFB 256, Univ.~Bonn},
  abstract = {In many cases the approximation of solutions to
		  variational problems involving isotropic Sobolev spaces has
		  a complexity which depends exponentially on the dimension.
		  However, if the solutions possess dominating mixed
		  derivatives one can find discretizations to the
		  corresponding variational problems with a lower complexity
		  -- sometimes even independent of the dimension. In order to
		  analyse these effects, we relate tensor products of Sobolev
		  spaces with spaces with dominating mixed derivatives. Based
		  on these considerations we construct families of finite
		  dimensional anisotropic approximation spaces which
		  generalize in particular sparse grids. The obtained
		  estimates demonstrate, in which cases a complexity
		  independent or nearly independent of the dimension can be
		  expected. Finally numerical experiments demonstrate the
		  usefulness of the suggested approximation spaces.},
  ps = { 1},
  pdf = { 1}