TT-cross approximation for multidimensional arrays

Oseledets, Ivan; Тыртышников, Е. Е.

doi:10.1016/j.laa.2009.07.024

Cited by 454 publications

(495 citation statements)

References 26 publications

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“…Corollary 5.3 A statement analogous to Theorem 5.1 also holds for the set H r appearing for the hierarchical format from [12] and the format from [20]. The proof uses the fact that H r is weakly closed.…”

Section: Main Statementmentioning

confidence: 87%

“…[12], [20]), which are again related to subspaces. In these cases, further subspaces like, e.g., U 12 ⊂ U 1 ⊗ U 2 appear.…”

Section: (12)mentioning

confidence: 99%

“…The formats from [12] and, in particular, the approach in [20] using a linear tree correspond to the so-called matrix product systems used in quantum chemistry (cf. [25]), provided that the underlying graph structure is a tree.…”

Section: (12)mentioning

confidence: 99%

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On Minimal Subspaces in Tensor Representations

Falcó

Hackbusch

2012

Found Comput Math

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In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means a tensor given in a typical representation format (Tucker, hierarchical or tensor train). We show that this result holds in a tensor Banach space with a norm stronger that the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples by using topological tensor products of standard Sobolev spaces are given.

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Section: Main Statementmentioning

confidence: 87%

“…[12], [20]), which are again related to subspaces. In these cases, further subspaces like, e.g., U 12 ⊂ U 1 ⊗ U 2 appear.…”

Section: (12)mentioning

confidence: 99%

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On Minimal Subspaces in Tensor Representations

Falcó

Hackbusch

2012

Found Comput Math

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“…The DMRG approach to fast linear algebra in the TT-format was addressed in [75]. As the important step, in [80] the TT factorization is extended to the case of incomplete input data by using the TT-Cross approximation via interpolation on the entries of a TT-cross.…”

Section: Advances Of Tensor Methods In the Recent Decadementioning

confidence: 99%

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

Khoromskij¹

2012

Chemometrics and Intelligent Laboratory Systems

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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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“…Finally, such global metamodels may benefit from the set of the simulations done in past studies for its construction, allowing a more efficient management of the data. The link between such kind of method and reduced order models are the low rank approximations such as Tucker decomposition [28], High Order SVD [29], Train Tensor Cross decomposition [30], and CUR or Nystrom decomposition [31]. The interpolation of a new numerical experiment will indeed be performed using such a reduced order model together with a parametric function.…”

Section: Introductionmentioning

confidence: 99%