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42
Wavelet Approximation in Weighted Sobolev Spaces of Mixed Order with Applications to the Electronic Schrödinger Equation
, 2011
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DimensionAdaptive TensorProduct Quadrature
 Computing
, 2003
"... We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the highdimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approxi ..."
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Cited by 74 (13 self)
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We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the highdimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lowerdimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.
Sparse grids and related approximation schemes for higher dimensional problems
"... The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach ..."
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Cited by 45 (12 self)
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The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energynorm based sparse grids and demonstrate that, for functions with bounded mixed derivatives on the unit hypercube, the associated approximation rate in terms of the involved degrees of freedom shows no dependence on the dimension at all, neither in the approximation order nor in the order constant.
Numerical solutions of parabolic equations in high dimensions
 M2AN Math. Model. Numer. Anal
, 2004
"... Abstract. We consider the numerical solution of diusion problems in (0, T) × Ω for Ω ⊂ Rd and for T> 0 in dimension d ≥ 1. We use a wavelet based sparse grid space discretization with meshwidth h and order p ≥ 1, and hp discontinuous Galerkin timediscretization of order r = O(log h) on a geo ..."
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Cited by 43 (6 self)
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Abstract. We consider the numerical solution of diusion problems in (0, T) × Ω for Ω ⊂ Rd and for T> 0 in dimension d ≥ 1. We use a wavelet based sparse grid space discretization with meshwidth h and order p ≥ 1, and hp discontinuous Galerkin timediscretization of order r = O(log h) on a geometric sequence of O(log h) many time steps. The linear systems in each time step are solved iteratively by O(log h) GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L2(Ω)error of O(N−p) for u(x, T) where N is the total number of operations, provided that the initial data satises u0 ∈ Hε(Ω) with ε> 0 and that u(x, t) is smooth in x for t> 0. Numerical experiments in dimension d up to 25 conrm the theory.
Optimized general sparse grid approximation spaces for operator equations
 MATHEMATICS OF COMPUTATIONS
, 2008
"... This paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order. Based on the framework of tensorproduct biorthogonal wavelet bases and stable subspace splittings, we construct operatoradapted subspaces with a ..."
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Cited by 30 (5 self)
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This paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order. Based on the framework of tensorproduct biorthogonal wavelet bases and stable subspace splittings, we construct operatoradapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces, provided that certain additional regularity assumptions on the solution are fulfilled. Specifically for operators of positive order, their dimension is O(2^J) independent of the dimension n of the problem, compared to O(2^Jn) for the full grid space. Also, for operators of negative order the overall cost is significantly in favor of the new approximation spaces. We give cost estimates for the case of continuous linear information. We show these results in a constructive manner by proposing a Galerkin method together with optimal preconditioning. The theory covers elliptic boundary value problems as well as boundary integral equations.
On the regularity of the electronic Schrödinger equation in Hilbert spaces of . . .
, 2002
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Hierarchical tensorproduct approximation to the inverse and related operators for highdimensional elliptic problems
 Computing
"... The class of Hmatrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the Hmatrix technique combined with the Kronecker tensorproduct approximation (cf. [2, 20]) to represent the inverse of a discrete elliptic operator in a hypercube (0, 1)d ..."
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Cited by 19 (7 self)
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The class of Hmatrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the Hmatrix technique combined with the Kronecker tensorproduct approximation (cf. [2, 20]) to represent the inverse of a discrete elliptic operator in a hypercube (0, 1)d ∈ R d in the case of a high spatial dimension d. In this datasparse format, we also represent the operator exponential, the fractional power of an elliptic operator as well as the solution operator of the matrix LyapunovSylvester equation. The complexity of our approximations can be estimated by O(dn logq n), where N = nd is the discrete problem size.
Adaptive Wavelet Solvers for the Unsteady Incompressible NavierStokes Equations
 ADVANCED MATHEMATICAL THEORIES IN FLUID MECHANICS
, 2000
"... In this paper we describe adaptive waveletbased solvers for the NavierStokes equations. Our approach employs a PetrovGalerkin scheme with tensor products of Interpolet wavelets as ansatz functions. We present the fundamental algorithms for the adaptive evaluation of differential operators and non ..."
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Cited by 18 (3 self)
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In this paper we describe adaptive waveletbased solvers for the NavierStokes equations. Our approach employs a PetrovGalerkin scheme with tensor products of Interpolet wavelets as ansatz functions. We present the fundamental algorithms for the adaptive evaluation of differential operators and nonlinear terms. Furthermore, a simple but efficient preconditioning technique for the resulting linear systems is introduced. For the NavierStokes equations a Chorintype projection method with a stabilized pressure discretization is used. Numerical examples demonstrate the efficiency of our approach.
Lowrank Kronecker product approximation to multidimensional nonlocal operators. Part I. Separable approximation of multivariate functions
 Preprint 29, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig
, 2005
"... This article is the second part continuing Part I [16]. We apply theHmatrix techniques combined with the Kronecker tensorproduct approximation to represent integral operators as well as certain functions F (A) of a discrete elliptic operator A in a hypercube (0, 1)d ∈ Rd in the case of a high spat ..."
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Cited by 17 (13 self)
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This article is the second part continuing Part I [16]. We apply theHmatrix techniques combined with the Kronecker tensorproduct approximation to represent integral operators as well as certain functions F (A) of a discrete elliptic operator A in a hypercube (0, 1)d ∈ Rd in the case of a high spatial dimension d. We focus on the approximation of the operatorvalued functions A−σ, σ> 0, and sign(A) for a class of finite difference discretisations A ∈ RN×N. The asymptotic complexity of our datasparse representations can be estimated by O(np logq n), p = 1, 2, with q independent of d, where n = N1/d is the dimension of the discrete problem in one space direction.
SPARSE GRIDS FOR THE SCHRÖDINGER EQUATION
, 2005
"... We present a sparse grid/hyperbolic cross discretization for manyparticle problems. It involves the tensor product of a oneparticle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, differ ..."
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Cited by 16 (4 self)
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We present a sparse grid/hyperbolic cross discretization for manyparticle problems. It involves the tensor product of a oneparticle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for manyparticle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrödinger equation and compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system.