Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances

By Boris N. Khoromskij. 2012

221 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Computational Theory and Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Surveys with a similar scope are the lecture notes by Grasedyck [103], Khoromskij [148,154,156], and Schneider [234], as well as the monograph by Hackbusch [109].”
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, methodology): “We refer to survey and lecture notes [24] [22] addressing the recent advances in tensor numerical methods.”
Tensor-Structured Factorized Calculation of Two-Electron Integrals in a General Basis (2013, methodology): “Further promising generalizations can be based on a combination of the Tucker, TT and QTT formats [26, 11].”
Low-rank tensor structure of linear diffusion operators in the TT and QTT formats☆ (2013, background): “Recent surveys [1, 2] and the monograph [3] present a variety of tensor decompositions, i.”
Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation (2011, methodology): “The most commonly used are the canonical, Tucker formats as well as the class of so-called matrix product states (MPS) representations [35, 30, 31] commonly used in high-dimensional quantum computations (see survey paper [16] for more details).”
Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity (2013, background): “tensor structure, provide a dramatic leverage in various computational problems [1, 2, 3], including evaluation of convolution [4, 5].”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, background): “…Garreis and Ulbrich, 2017, Liao et al., 2015, Lubich et al., 2015], • TT/HT approximation of nonlinear functions and systems [Dolgov and Khoromskij, 2013, Khoromskij, 2011b, Khoromskij and Miao, 2014], • CP decomposition and its applications [Choi and Vishwanathan, 2014, Kamal et al., 2016,…”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, unknown): “For example: • If Ar are rank-1 tensors of size I1 ˆ I2 ˆ ̈ ̈ ̈ ˆ IN , and Br are scalars, @r, then (37) expresses the rank-R CP decomposition; • If Ar are rank-Lr tensors of size I1 ˆ I2 ˆ ̈ ̈ ̈ ˆ IR ˆ 1 ˆ ̈ ̈ ̈ ˆ 1, in the Kruskal (CP) format, and Br are rank-1 tensors of size 1ˆ ̈ ̈ ̈”
Superfast Fourier Transform Using QTT Approximation (2012, background): “Among many low-parametric formats, tensor decomposition methods appear to be the most promising for high-dimensional data, see reviews [3, 32, 38] and mono-graph [16].”
Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis (2014, background): “…data into easy to interpret components (i.e., the rank-1 terms), while the goal of unconstrained Tucker decompositions is most often to compress data into a tensor of smaller size (i.e., the core tensor) or to ﬁnd the subspaces spanned by the ﬁbers (i.e., the column spaces of the factor matrices).”
Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs (2015, background): “In [28, 29, 30, 31], other low rank tensor approaches for the solution of evolution equations are proposed.”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Generalized Cross Approximation for 3d-tensors Generalized Cross Approximation for 3d-tensors (background): “For the latest developements in this area, one can look at [3] [6] [7] [8] [11].”
Tensor-product approach to global time-space-parametric discretization of chemical master equation (2012, methodology): “Traditional and commonly used tensor representations in multilinear algebra and numerical analysis include canonical and Tucker formats, see the surveys and lecture notes [26, 39, 40].”
Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (2011, methodology): “Numerical examples on O(d log n) complexity scaling for QTT representation of the highdimensional Laplacian can be found [25].”