@Article{	  Griebel.Oeltz.Vassilevski:2005,
  author	= {Michael Griebel and Daniel Oeltz and Panayot Vassilevski},
  title		= {Space-time approximation with Sparse Grids},
  institution	= {Lawrence Livermore National Laboratory},
  year		= {2005},
  journal	= {SIAM J. Sci. Comput.},
  volume	= {28},
  number	= {2},
  pages		= {701-727},
  pdf		= {http://wissrech.ins.uni-bonn.de/research/pub/oeltz/IM-319022-3-preprint.pdf}
		  ,
  annote	= {article,C2},
  abstract	= {In this report we introduce approximation spaces for
		  parabolic problems which are based on the tensor product
		  construction of a multiscale basis in space and a
		  multiscale basis in time. Proper truncation then leads to
		  so-called space-time sparse grid spaces. For a uniform
		  discretization of the spatial space of dimension d with
		  O(N^d) degrees of freedom, these spaces involve for d > 1
		  also only O(N^d) degrees of freedom for the discretization
		  of the whole space-time problem. But they provide the same
		  approximation rate as classical space-time Finite Element
		  spaces which need O(N^(d+1)) degrees of freedoms. This
		  makes these approximation spaces well suited for
		  conventional parabolic and for time-dependent optimization
		  problems.
		  
		  We analyze the approximation properties and the dimension
		  of these sparse grid space-time spaces for general stable
		  multiscale bases. We then restrict ourselves to an
		  interpolatory multiscale basis, i.e. a hierarchical basis.
		  Here, to be able to handle also complicated spatial domains
		  Omega, we construct the hierarchical basis from a given
		  spatial Finite Element basis as follows: First we determine
		  coarse grid points recursively over the levels by the
		  coarsening step of the algebraic multigrid method. Then, we
		  derive interpolatory prolongation operators between the
		  respective coarse and fine grid points by a least square
		  approach. This way we obtain an algebraic hierarchical
		  basis for the spatial domain which we then use in our
		  space-time sparse grid approach.
		  
		  We give numerical results on the convergence rate of the
		  interpolation error of these spaces for various space-time
		  problems with two spatial dimensions. Also implementational
		  issues, data structures and questions of adaptivity are
		  addressed to some extent.},
  note		= {Also as Preprint No.~222, SFB 611, University of Bonn}
}