Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators

By Wolfgang Hackbusch, Boris N. Khoromskij. 2005

95 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Numerical Analysis, Mathematical Optimization, Numerical Methods & Scientific Computing

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “In [93, 110, 111, 115], combinations of low-rank tensor formats with hierarchical matrices are investigated.”
Fast stray field computation on tensor grids (2011, unknown): “Here we use the exponentially convergent sinc-quadrature for numerical integration of the integral [9].”
Tensor-Structured Factorized Calculation of Two-Electron Integrals in a General Basis (2013, methodology): “In this aspect, the rank-| log ε log δ| approximation to the unfolding matrix E = [ek`,mn] follows by the well-known sinc-approximation to the Hilbert matrix { 1 i+j }, [12].”
Tensor-Product Approximation to Multidimensional Integral Operators and Green's Functions (2008, methodology): “Low separation rank decomposition to the multivariate functions 1/ρ, 1/ √ ρ and to the related Galerkin approximations were discussed in [2, 17, 18, 19, 24], while the kernel function ρ, μ ∈ R, was considered in [23].”
Møller-Plesset (MP2) energy correction using tensor factorization of the grid-based two-electron integrals (2014, background): “for xi ≥ 0,  xi > δ, which converges exponentially fast in M , see [32,33].”
Efficient time-stepping scheme for dynamics on TT-manifolds (2012, unknown): “It can be used to approximate high-dimensional operators and their inverses (see, for example, [2, 3, 11, 14, 15, 12]) using certain analytical expansions.”
Superfast Fourier Transform Using QTT Approximation (2012, unknown): “Similar problem was discussed for the representation ofNewton/Yukawa potential in canonical format [3], andwas solved using the sinc–quadrature from [19], which expresses r as the sum of Gaussian functions, i.”
Variational calculus with sums of elementary tensors of fixed rank (2012, unknown): “Mathematics Subject Classification 15A69 · 90C06 · 65F10”
Constructive Representation of Functions in Low-Rank Tensor Formats (2012, background): “The upper bound on the number of quadrature points is logarithmic in n and ε, and the details can be found, for example, in [14].”
Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity (2013, methodology): “We take advantage of the quadratures presented in [46], which represent the functions r2 7→ 1r and r2 7→ 1r erf ( 1 r ) , r > 0, as sums of exponentials, to decompose f and h in sums of Gaussians.”
Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels (2012, unknown): “3) leads to the exponential convergence in M , see [5], ‖G − G‖ ≤ C e−β √ ‖G‖ with C, β ∈ R+.”
QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations (2011, background): “Moreover, there are interesting results on the tensor-product approximation of the 3D Newton and Yukawa potentials in the framework of the waveletand polynomial-based multiresolution schemes [16, 4], as well as using the grid-based approximations [13, 8, 25, 29].”