Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse

Kazeev, Vladimir; Khoromskij, Boris N.

doi:10.1137/100820479

Cited by 90 publications

(132 citation statements)

References 33 publications

(35 reference statements)

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“…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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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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“…The more general QTT-rank estimate in terms of the separation rank of a generating function f (x + y) was obtained in [74]. Explicit low-rank representation of the discrete quantized Laplacian and its inverse was derived in [45].…”

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

confidence: 99%

“…The general concept on the explicit QTT representation of vectors and matrices can be found in [74], [45], [20], [46], and [56].…”

Section: And [74]) (Withmentioning

confidence: 99%

“…where the rank product operation "⋊ ⋉" is defined as a regular matrix product of the two corresponding core matrices, their blocks being multiplied by means of tensor product (see [45]). The similar bound is true for the Tucker rank rank T uck (∆ d ) = 2.…”

Section: Elliptic Operator and Its Inversementioning

confidence: 99%

“…The similar bound is true for the Tucker rank rank T uck (∆ d ) = 2. The explicit rank-4 QTT representation of ∆ d for d ≥ 2 is obtained in [45], while for d = 1, we have…”

Section: Elliptic Operator and Its Inversementioning

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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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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Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse

Cited by 90 publications

References 33 publications

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

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

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

A literature survey of low‐rank tensor approximation techniques

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