[1] 
J. Hamaekers.
Tensor Product Multiscale ManyParticle Spaces with
FiniteOrder Weights for the Electronic Schödinger Equation.
Dissertation, Institut für Numerische Simulation, Universität
Bonn, July 2009. [ bib  .pdf (link)  http ] We study tensor product multiscale manyparticle spaces with finiteorder weights and their application for the electronic Schrödinger equation. Any numerical solution of the electronic Schrödinger equation using conventional discretization schemes is impossible due to its high dimensionality. Therefore, typically Monte Carlo methods (VMC/DMC) or nonlinear model approximations like HartreeFock (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 particlewise subspace splitting with respect to oneparticle functions which stem from a nonlinear rank1 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 manyparticle tensor product multiscale approximation spaces include the common configuration interaction (CI) spaces as a special case. To realize our new approach, we first introduce general manyparticle 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 twoelectron 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 oneparticle space we construct a subspace splitting of the manyparticle 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 manyparticle spaces in which the problem of an approximation of an Nparticle function reduces to the problem of the approximation of qparticle 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 finitedimensional subspaces of a weighted tensor product multiscale manyparticle approximation space. Furthermore, we construct a multiscale Gaussian frame and apply Gaussians and modulated Gaussians for the nonlinear rank1 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.
