On Minimal Subspaces in Tensor Representations

Falcó, Antonio; Hackbusch, Wolfgang

doi:10.1007/s10208-012-9136-6

Cited by 44 publications

(69 citation statements)

References 31 publications

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“…In [15] it was introduced the following definition. For a given v in the algebraic tensor space V, the minimal subspaces U j,min (v) ⊂ V j are given by the intersection of all subspaces U j ⊂ V j satisfying v ∈ a d j=1 U j .…”

Section: Remarkmentioning

confidence: 99%

Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces

Falcó

Nouy

2011

Numer. Math.

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Tensor-based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces. A family of methods called proper generalized decompositions (PGD) methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems. In this paper, we give a mathematical analysis of a family of progressive and updated PGDs for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space.

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Section: Remarkmentioning

confidence: 99%

Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces

Falcó

Nouy

2011

Numer. Math.

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“…Indeed, the application of such techniques has grown rapidly: micromagnetics [96][97][98], model order reduction [99,100], big data [101][102][103], signal processing [104][105][106][107], control design [108,109], and electronic design automation [93].…”

Section: Efficient Sampling Strategies For High-dimensional Problemsmentioning

confidence: 99%

Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits

2018

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Advances in manufacturing process technology are key ensembles for the production of integrated circuits in the sub-micrometer region. It is of paramount importance to assess the effects of tolerances in the manufacturing process on the performance of modern integrated circuits. The polynomial chaos expansion has emerged as a suitable alternative to standard Monte Carlo-based methods that are accurate, but computationally cumbersome. This paper provides an overview of the most recent developments and challenges in the application of polynomial chaos-based techniques for uncertainty quantification in integrated circuits, with particular focus on high-dimensional problems.

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“…In this paper, we call the TT formats (11) and (15) as the vector TT and matrix TT formats, respectively. Note that if the indices i n and j n are joined as k n = (i n , j n ) in (16), then the matrix TT format is reduced to the vector TT format.…”

Section: Tensor Train Formats For Vectors and Matricesmentioning

confidence: 99%

“…The TT and HT decompositions can avoid the curse-of-dimensionality by low-rank approximation, and possess the closedness property [10,11]. For numerical analysis, basic numerical operations such as addition and matrix-by-vector multiplication based on low-rank TT formats usually lead to TT-rank growth, so an efficient rank-truncation should be followed.…”

Section: Introductionmentioning

confidence: 99%

Estimating a Few Extreme Singular Values and Vectors for Large-Scale Matrices in Tensor Train Format

Lee¹,

Cichocki²

2015

SIAM J. Matrix Anal. & Appl.

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We propose new algorithms for singular value decomposition (SVD) of very large-scale matrices based on a low-rank tensor approximation technique called the tensor train (TT) format. The proposed algorithms can compute several dominant singular values and corresponding singular vectors for large-scale structured matrices given in a TT format. The computational complexity of the proposed methods scales logarithmically with the matrix size under the assumption that both the matrix and the singular vectors admit low-rank TT decompositions. The proposed methods, which are called the alternating least squares for SVD (ALS-SVD) and modified alternating least squares for SVD (MALS-SVD), compute the left and right singular vectors approximately through block TT decompositions. The very large-scale optimization problem is reduced to sequential small-scale optimization problems, and each core tensor of the block TT decompositions can be updated by applying any standard optimization methods. The optimal ranks of the block TT decompositions are determined adaptively during iteration process, so that we can achieve high approximation accuracy. Extensive numerical simulations are conducted for several types of TT-structured matrices such as Hilbert matrix, Toeplitz matrix, random matrix with prescribed singular values, and tridiagonal matrix. The simulation results demonstrate the effectiveness of the proposed methods compared with standard SVD algorithms and TT-based algorithms developed for symmetric eigenvalue decomposition.

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On Minimal Subspaces in Tensor Representations

Cited by 44 publications

References 31 publications

Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces

Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces

Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits

Estimating a Few Extreme Singular Values and Vectors for Large-Scale Matrices in Tensor Train Format

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