Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part I. Separable Approximation of Multi-variate Functions

By Wolfgang Hackbusch, Boris N. Khoromskij. 2005

131 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Computational 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.”
FFT-based Kronecker product approximation to micromagnetic long-range interactions (2012, methodology): “We state some basic facts from the theory of sinc function based approximation, see [29],[18].”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Multigrid Accelerated Tensor Approximation of Function Related Multi-dimensional Arrays Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays * (unknown): “…approximation methods have opened challenging perspectives for a feasible numerical solution of multidimensional problems arising in large scale electronic and molecular structure calculations [1] [2] [7] [8] [10] [12] [15] [4] [18] [28], in stochastic PDEs [24], and in financial mathematics.”
Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels (2012, background): “include Green’s kernels in Rd [ 5 , 7, 9]. Efficient computation of such integral transforms”
Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, unknown): “…(m) km ukm ⊗ vkm , the corresponding rank-pm approximation to Bm (by truncation of σ (m) pm+1, ..., σ (m) r ), we arrive at the rank-R canonical approximation to β, β(R) := r∑ m=1 Bpm ⊗ Zm, Zm ∈ Rr, (2.40) providing the error estimate ‖β − β(R)‖ ≤ r∑ m=1 ‖Bm − Bpm‖ = r∑ m=1 √√√√…”
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].”
Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝd (2009, methodology): “To our best knowledge, the low rank tensor-structured approximation for oscillating kernels have not been yet addressed in the literature (cf. the asymptotically optimal method in the volume size based on the fast multipole type representations [5, 6]).”
Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators (2005, methodology): “(we set cf = √ π) with ck, tk given in [16].”