Polynomial Approximation in Hierarchical Tucker Format by Vector – Tensorization

By Lars Grasedyck. 2010

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In Mathematical Optimization

Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “The exact and approximate ranks of X for various functions f have been discussed for the TT and HT decompositions in [104,108,154].”
Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs (2010, background): “Some theoretical results on the approximation in QTT-format were established in [12, 9].”
Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, unknown): “6.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140”
Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains (2014, background): “Note that the Hierarchical Tensor Representation [43,44] itself and coupled with the tensorization [45], an extensive overview of which is available in [40], are closely related counterparts of the TT and QTT formats respectively.”
Constructive Representation of Functions in Low-Rank Tensor Formats (2012, methodology): “However, there are only a few theoretical results in this direction [13, 21].”
QTT approximation of elliptic solution operators in higher dimensions (2011, background): “For piecewise-polynomial functions the ranks are bounded uniformly in n [14], and moreover it is possible to provide explicit decompositions for these function-related tensors [34].”
Low-rank tensor structure of linear diffusion operators in the TT and QTT formats☆ (2013, methodology): “Let us also note that the results on the TT and QTT structure can be applied straight-forwardly to the Hierarchical Tensor representation by Hackbusch and Kühn (see [19, 20]) with degenerate trees and its tensorized version [33], the counterpart of QTT.”
Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (2011, background): “Moreover, it can be shown [24] that Chebyshev polynomials sampled over Chebyshev grid can be represented by rank-2 quantics tensor, while the general polynomial vector sampled over uniform grid has the rank-(m + 1) QTT representation for any polynomial of degree m (see also [12] for related results)”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, background): “In the subsequent paper [27] the uniform rank bound of the quantized polynomial N -vector (renamed as “tensorization of vectors”) was proven in the case of hierarchical Tucker format.”
QTT-rank-one vectors with QTT-rank-one and full-rank Fourier images (2012, background): “can be accurately approximated by a low-rank vectors (x, e−αx2 , sin x x , 1 x et al) [9, 5, 11].”
O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling (2011, methodology): “For a class of analytic functions the basic approximability assumption (2.7) can be verified with ε = O(e−αM/ log M ), α > 0, by applying the sinc-approximation [17, 23, 25].”
Superfast Fourier Transform Using QTT Approximation (2012, background): “Some function-related examples are already mentioned, more examples of (piecewise) smooth functions and functions with singularities that have the low-rankQTT representation can be found in [16, 45, 37].”
Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity (2013, methodology): “After the TT format, the idea of “tensorization” of vectors was adopted for the Hierarchical Tucker format [39] in [40].”
Quantized-TT-Cayley Transform for Computing the Dynamics and the Spectrum of High-Dimensional Hamiltonians (2011, background): “Moreover, it can be shown [24] that Chebyshev polynomials sampled over the Chebyshev grid can be represented by rank-2 quantics tensor, while the general polynomial vector sampled over a uniform grid has the rank-(m+ 1) QTT representation for any polynomial of degree m (see also [12] for related res”