Mathematik in den Naturwissenschaften Leipzig Tensor-Product Approximation to Operators and Functions in High Dimensions

By Wolfgang Hackbusch, Boris N. Khoromskij. 2007

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…example, they have been used for addressing: • the approximation of parameter-dependent integrals [15, 164, 190], multi-dimensional integrals [48, 112,153], and multi-dimensional convolution [152]; • computational tasks in electronic structure calculations, e.g., based on Hartree-Fock or DFT…”
A New Scheme for the Tensor Representation (2009, methodology): “Concerning the construction of tensor representations for certain application we refer to [ 8 ] and [9].”
On Fast Computation of Gradients for CANDECOMP/PARAFAC Algorithms (2012, methodology): “ization which has found applications such as in chemometrics, telecommunication, analysis of fMRI data, time-varying EEG spectrum, data mining [7,13], classiﬁcation, clustering [29], stochastic PD Es [15]. For example, CP was applied to analyze the auditory tones by Carroll and Chang [8], or to vowe”
Approximation of the electron density of Aluminium clusters in tensor-product format (2012, unknown): “Recent tensor-structured methods [17, 17, 10, 16, 18, 19, 14] reduce dramatically the complexity of the numerical evaluation of the multivariate functions and operators in R, d ≥ 3.”
Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, unknown): “6.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140”
Generic and typical ranks of multi-way arrays (2009, unknown): “Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the Curse of Dimensionality [21, p. 125] [22] [ 23 ].”
A New Truncation Strategy for the Higher-Order Singular Value Decomposition (2012, background): “Such function-related tensors can be compressed greatly [20, 27].”
CANDECOMP/PARAFAC Decomposition of High-Order Tensors Through Tensor Reshaping (2012, background): “Hackbusch, Khoromskij, and Tyrtyshnikov [19] and Hackbusch and Khorom skij [20] investigated CP approximation to operators and functions in high dimensions.”
Tensor Decompositions and Applications (2009, methodology): “Khoromskij and Khoromskaia [115] have applied CANDELINC (which they call “Tucker-to-canonical”) to approximations of classical potentials.”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, methodology): “2 ([32]) Suppose that a multi-variate function g : Ω ⊂ R → R can be approximated by a separable expansion”
Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness (2008, background): “Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the curse of dimensionality [2, 3, 24, 25, 33].”