O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling

Khoromskij, Boris N.

doi:10.1007/s00365-011-9131-1

Cited by 222 publications

(341 citation statements)

References 35 publications

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“…The arising discretizations via "formatted" function related tensors typically inherit the separability properties of the initial solutions on the continuous level, usually providing fast exponential convergence in the separation rank. This favorable feature combined with the modern multilinear algebra methods of nonlinear tensor approximation [21,1,42,79,57,55] lead to the new concept of numerical schemes in higher dimensions which scale linearly in the dimension parameter d.…”

Section: Methods Of Separation Of Variablesmentioning

confidence: 99%

“…until the irreducible mode-size of a tensor, 2 × 2 × ... × 2, is achieved [73,55]. It was found in [73] by numerical tests that in some cases the dyadic reshaping of 2 L × 2 L matrix leads to a small TT-rank of the resultant quantized matrix of size (2 × 2) ⊗L .…”

Section: Quantized N-d Tensors Lead To D Log N Complexity: "Blessing mentioning

confidence: 99%

“…In [55] the diadically quantized representation of vectors (called there quantics 1 ) was generalized to q-adic folding applied to the general N -d tensors. There the principal question why the folding of a vector to a higher dimensional tensor might lead to an essential data compression was first addressed and rigorously analyses from the approximation aspects.…”

Section: Quantized N-d Tensors Lead To D Log N Complexity: "Blessing mentioning

confidence: 99%

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

Khoromskij¹

2012

Chemometrics and Intelligent Laboratory Systems

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160

153

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In the present paper, we give a survey of the recent results and outline future prospects of the tensor-structured numerical methods in applications to multidimensional problems in scientific computing. The guiding principle of the tensor methods is an approximation of multivariate functions and operators relying on certain separation of variables. Along with the traditional canonical and Tucker models, we focus on the recent quantics-TT tensor approximation method that allows to represent N -d tensors with log-volume complexity, O(d log N ). We outline how these methods can be applied in the framework of tensor truncated iteration for the solution of the high-dimensional elliptic/parabolic equations and parametric PDEs. Numerical examples demonstrate that the tensor-structured methods have proved their value in application to various computational problems arising in quantum chemistry and in the multi-dimensional/parametric FEM/BEM modeling-the tool apparently works and gives the promise for future use in challenging high-dimensional applications.

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Section: Methods Of Separation Of Variablesmentioning

confidence: 99%

Section: Quantized N-d Tensors Lead To D Log N Complexity: "Blessing mentioning

confidence: 99%

Section: Quantized N-d Tensors Lead To D Log N Complexity: "Blessing mentioning

confidence: 99%

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

Khoromskij¹

2012

Chemometrics and Intelligent Laboratory Systems

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160

153

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“…, i l ) = y(i), and the TT approximation can be applied to y. The resulting TT format was called the Quantized TT (QTT) [42,59,60,86]. If the TT ranks of y are moderate, the total storage reduces to a logarithmic amount O(lr 2 ) = O(log n).…”

Section: The Tensor-train Decompositionmentioning

confidence: 99%

“…If the TT ranks of y are moderate, the total storage reduces to a logarithmic amount O(lr 2 ) = O(log n). For many elementary functions and operators, their TT/QTT formats can be written analytically, for example, the discretized Laplace operator [41], the sine, exponential and polynomial functions, sampled on uniform grids in one [42,62] and many dimensions [20,43].…”

Section: The Tensor-train Decompositionmentioning

confidence: 99%

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Dolgov

Pearson

Savostyanov

et al. 2016

Applied Mathematics and Computation

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Fractional differential equations have recently received much attention within computational mathematics and applied science, and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms. An optimization problem with constraints given by fractional differential equations is considered, which in its discretized form leads to a high-dimensional tensor equation. To reduce the computation time and storage, the solution is sought in the tensor-train format. We compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned Krylov solvers or tailored alternating schemes. The competitiveness of these approaches is presented using several examples with constant and variable coefficients.

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A literature survey of low‐rank tensor approximation techniques

2013

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Key words tensor, low rank, multivariate functions, linear systems, eigenvalue problems MSC (2010) 15A69, 65F10, 65F15During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors.

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O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling

Cited by 222 publications

References 35 publications

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

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

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

A literature survey of low‐rank tensor approximation techniques

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