Tensors-structured numerical methods in scientific computing: Survey on recent advances

Khoromskij, Boris N.

doi:10.1016/j.chemolab.2011.09.001

Cited by 161 publications

(155 citation statements)

References 87 publications

(164 reference statements)

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“…We conclude with a more general concept from scientific computing and quantum information theory called tensor networks. For more theory and applications we direct the reader to the references, especially [4]- [6], [9]- [11].…”

Section: Tensor Decompositionsmentioning

confidence: 99%

Breaking the Curse of Dimensionality Using Decompositions of Incomplete Tensors: Tensor-based scientific computing in big data analysis

Vervliet

Debals

Sorber

et al. 2014

IEEE Signal Process. Mag.

148

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Higher-order tensors and their decompositions are abundantly present in domains such as signal processing (e.g. higher-order statistics [1], sensor array processing [2]), scientific computing (e.g. discretized multivariate functions [3]- [6]) and quantum information theory (e.g. representation of quantum many-body states [7]). In many applications the, possibly huge, tensors can be approximated well by compact multilinear models or decompositions. Tensor decompositions are more versatile tools than the linear models resulting from traditional matrix approaches.Compared to matrices, tensors have at least one extra dimension. The number of elements in a tensor increases exponentially with the number of dimensions, and so do the computational and memory requirements. The exponential dependency (and the problems that are caused by it) is called the curse of dimensionality. The curse limits the order of the tensors that can be handled.Even for modest order, tensor problems are often large-scale. Large tensors can be handled, and the curse can be alleviated or even removed, by using a decomposition that represents the tensor, instead of the tensor itself. However, most decomposition algorithms require full tensors, which renders these algorithms infeasible for large datasets. If a tensor can be represented by a decomposition, this hypothesized structure can be exploited by using compressed sensing type methods working on incomplete tensors, i.e. tensors with only a few known elements.In domains such as scientific computing and quantum information theory, tensor decompositions such as the Tucker decomposition and tensor trains have successfully been applied to represent large tensors. In the latter case, the tensor can contain more elements than the number of atoms in the universe [8]

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Section: Tensor Decompositionsmentioning

confidence: 99%

Breaking the Curse of Dimensionality Using Decompositions of Incomplete Tensors: Tensor-based scientific computing in big data analysis

Vervliet

Debals

Sorber

et al. 2014

IEEE Signal Process. Mag.

148

View full text Add to dashboard Cite

show abstract

“…Notice that all coefficients are presented by one-dimensional integrals, which can be efficiently computed with the help of special (tensor-type) methods (see, e.g., [15][16][17][18][19]). It is not difficult to see that…”

Section: Examplesmentioning

confidence: 99%

“…Literature survey on the modern tensor numerical methods for multi-dimensional PDEs can be found in [13,14,16]. In the context of problems considered in the paper, we are mainly concerned with another specific feature: very complicated material structure.…”

Section: Introductionmentioning

confidence: 99%

“…The use of tensor-structured preconditioned iteration with the adaptive QTT rank truncation may lead to the logarithmic complexity in the grid size, O(log p n), see [14,16,23] for the rank-truncated iterative methods, [4,[10][11][12] for various examples of the QTT tensor approximation to lattice structured systems, and [2] for tensor approximation of complicated functions with multiple cusps in ℝ d . In Section 6, we conclude with the discussion on further perspectives of the presented approach for 2D and 3D elliptic PDEs with quasi-periodic coefficients.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems

Khoromskij

Repin

2017

Computational Methods in Applied Mathematics

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Abstract:We consider an iteration method for solving an elliptic type boundary value problem Au = f , where a positive definite operator A is generated by a quasi-periodic structure with rapidly changing coefficients (a typical period is characterized by a small parameter ϵ). The method is based on using a simpler operator A (inversion of A is much simpler than inversion of A), which can be viewed as a preconditioner for A. We prove contraction of the iteration method and establish explicit estimates of the contraction factor q. Certainly the value of q depends on the difference between A and A . For typical quasi-periodic structures, we establish simple relations that suggest an optimal A (in a selected set of "simple" structures) and compute the corresponding contraction factor. Further, this allows us to deduce fully computable two-sided a posteriori estimates able to control numerical solutions on any iteration. The method is especially efficient if the coefficients of A admit low-rank representations and if algebraic operations are performed in tensor structured formats. Under moderate assumptions the storage and solution complexity of our approach depends only weakly (merely linear-logarithmically) on the frequency parameter ϵ .

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“…Modern grid-based tensor methods [12,13] achieve linear memory costs O(dn) with respect to dimension d and grid size n. The novel method of quantized tensor approximation is proven to provide a logarithmic data-compression for a wide class of discrete functions and operators [11]. It allows to discretize and to solve multi-dimensional steady-state and dynamical problems with a logarithmic complexity in the volume size of the computational grid.…”

Section: Introductionmentioning

confidence: 99%

Matrix Generation in Isogeometric Analysis by Low Rank Tensor Approximation

Mantzaflaris

Jüttler²,

Khoromskij³

et al. 2015

Curves and Surfaces

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Abstract. It has been observed that the task of matrix assembly in Isogeometric Analysis (IGA) is more challenging than in the case of traditional finite element methods. The additional difficulties associated with IGA are caused by the increased degree and the larger supports of the functions that occur in the integrals defining the matrix elements. Recently we introduced an interpolation-based approach that approximately transforms the integrands into piecewise polynomials and uses look-up tables to evaluate their integrals [15,16]. The present paper relies on this earlier work and proposes to use tensor methods to accelerate the assembly process further. More precisely, we show how to represent the matrices that occur in IGA as sums of a small number of Kronecker products of auxiliary matrices that are defined by univariate integrals. This representation, which is based on a low-rank tensor approximation of certain parts of the integrands, makes it possible to achieve a significant speedup of the assembly process without compromising the overall accuracy of the simulation.

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Tensors-structured numerical methods in scientific computing: Survey on recent advances

Cited by 161 publications

References 87 publications

Breaking the Curse of Dimensionality Using Decompositions of Incomplete Tensors: Tensor-based scientific computing in big data analysis

Breaking the Curse of Dimensionality Using Decompositions of Incomplete Tensors: Tensor-based scientific computing in big data analysis

Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems

Matrix Generation in Isogeometric Analysis by Low Rank Tensor Approximation

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