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

  author = {Jan Hamaekers},
  title = {{Tensor Product Multiscale Many--Particle Spaces with
		  Finite--Order Weights for the Electronic Sch\"{o}dinger
  school = {Institut f\"ur Numerische Simulation, Universit\"{a}t Bonn},
  year = {2009},
  annote = {INSdiss,thesis},
  type = {Dissertation},
  month = jul,
  url = {},
  http_pdf = {},
  abstract = { We study tensor product multiscale many-particle spaces
		  with finite-order weights and their application for the
		  electronic Schr{\"o}dinger equation. Any numerical solution
		  of the electronic Schr{\"o}dinger equation using
		  conventional discretization schemes is impossible due to
		  its high dimensionality. Therefore, typically Monte Carlo
		  methods (VMC/DMC) or nonlinear model approximations like
		  Hartree-Fock (HF), coupled cluster (CC) or density
		  functional theory (DFT) are used. In this work we develop
		  and implement in parallel a numerical method based on
		  adaptive sparse grids and a particle-wise subspace
		  splitting with respect to one-particle functions which stem
		  from a nonlinear rank-1 approximation. Sparse grids allow
		  to overcome the exponential complexity exhibited by
		  conventional discretization procedures and deliver a
		  convergent numerical approach with guaranteed convergence
		  rates. In particular, the introduced weighted many-particle
		  tensor product multiscale approximation spaces include the
		  common configuration interaction (CI) spaces as a special
		  To realize our new approach, we first introduce general
		  many-particle Sobolev spaces, which particularly include
		  the standard Sobolev spaces as well as Sobolev spaces of
		  dominated mixed smoothness. For this novel variant of
		  sparse grid spaces we show estimates for the approximation
		  and complexity orders with respect to the smoothness and
		  decay parameters. With known regularity properties of the
		  electronic wave function it follows that, up to logarithmic
		  terms, the convergence rate is independent of the number of
		  electrons and almost the same as in the two-electron case.
		  However, besides the rate, also the dependence of the
		  complexity constants on the number of electrons plays an
		  important role for a truly practical method. Based on a
		  splitting of the one-particle space we construct a subspace
		  splitting of the many-particle space, which particularly
		  includes the known ANOVA decomposition, the HDMR
		  decomposition and the CI decomposition as special cases.
		  Additionally, we introduce weights for a restriction of
		  this subspace splitting. In this way weights of finite
		  order q lead to many-particle spaces in which the problem
		  of an approximation of an N-particle function reduces to
		  the problem of the approximation of q-particle functions.
		  To obtain as small as possible constants with respect to
		  the cost complexity, we introduce a heuristic adaptive
		  scheme to build a sequence of finite-dimensional subspaces
		  of a weighted tensor product multiscale many-particle
		  approximation space. Furthermore, we construct a multiscale
		  Gaussian frame and apply Gaussians and modulated Gaussians
		  for the nonlinear rank-1 approximation. In this way, all
		  matrix entries of the corresponding discrete eigenvalue
		  problem can be computed in terms of analytic formulae for
		  the one and two particle operator integrals.
		  Finally, we apply our novel approach to small atomic and
		  diatomic systems with up to 6 electrons (18 space
		  dimensions). The numerical results demonstrate that our new
		  method indeed allows for convergence with expected rates. }