Hierarchical Kronecker tensor-product approximations

Hackbusch, Wolfgang; Khoromskij, Boris N.; Тыртышников, Е. Е.

doi:10.1515/1569395054012767

Cited by 106 publications

(41 citation statements)

References 33 publications

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“…For a summary and basic properties of the Kronecker product of matrices, we refer to [24]. For recent applications in fast computing in high dimensions [25,26]. Actually, the operator vec 2 maps a matrix of order N+1 into a vector of order (N−1) 2 using the lexical order on i and j for the interior entries of the matrix.…”

Section: Hermitian Box-scheme For the Poisson Problem In A Squarementioning

confidence: 99%

An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

Abbas

2014

Adv Appl Math Mech

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In this paper the problem −div(a(x,y)∇u)= f with Dirichlet boundary conditions on a square is solved iteratively with high accuracy for u and ∇u using a new scheme called "hermitian box-scheme". The design of the scheme is based on a "hermitian box", combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2h. The iterative technique is based on the repeated solution by a fast direct method of a discrete Poisson equation on a uniform rectangular mesh. The problem is suitably scaled before iteration. The numerical results obtained show the efficiency of the numerical scheme. This work is the extension to strongly elliptic problems of the hermitian box-scheme presented by Abbas and Croisille (J. Sci. Comput., 49 (2011), pp. 239-267).

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Section: Hermitian Box-scheme For the Poisson Problem In A Squarementioning

confidence: 99%

An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

Abbas

2014

Adv Appl Math Mech

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“…Let A and B be the two matrices, then the exponential kronecker product is described as: [19,20,21] has nice properties to imply the concept of vector logic theory. The properties are as:…”

Section: Exponential Kronecker Product (Ekp)mentioning

confidence: 99%

Fusion Based Multimodal Authentication in Biometrics Using Context-Sensitive Exponent Associative Memory Model : A Novel Approach

Prasad¹,

K²,

Ramakrishna

et al. 2013

Computer Science &Amp; Information Technology ( CS &Amp; IT )

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“…Formulation and proof of upper bounds on r were rst proposed in [23,24] and then, under different assumptions, in [9]. Kronecker approximations of low Kronecker rank can be computed ef ciently by a version of LU decomposition with a special dynamic choice of pivots (the approach was presented in [21] and an adapted detailed description can be found in [4]).…”

Section: (D) Apply Gmres To Solvementioning

confidence: 99%

Matrix approximations and solvers using tensor products and non-standard wavelet transforms related to irregular grids

Ford

Oseledets

Тыртышников

2004

Russian Journal of Numerical Analysis and Mathematical Modelling

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Dense large-scale matrices coming from integral equations and tensor-product grids can be approximated by a sum of Kronecker products with further sparsi cation of the factors via discrete wavelet transforms, which results in reduced storage and computational costs and also in good preconditioners in the case of uniform one-dimensional grids. However, irregular grids lead to a loss of approximation quality and, more signi cantly, to a severe deterioration in ef ciency of the preconditioners that have been considered previously (using a sparsi cation of the inverse to one Kronecker product or an incomplete factorizationapproach). In this paper we propose to use non-standard wavelet transforms related to the irregular grids involved and, using numerical examples, we show that the new transforms provide better compression than the Daubechies wavelets. A further innovation is a scaled two-level circulant preconditioner that performs well on irregular grids. The proposed approximation and preconditioning techniques have been applied to a hypersingularintegral equation modelling ow around a thin aerofoil and made it possible to solve linear systems with more than 1 million unknowns in 15-20 minutes even on a personal computer.

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Hierarchical Kronecker tensor-product approximations

Cited by 106 publications

References 33 publications

An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

Fusion Based Multimodal Authentication in Biometrics Using Context-Sensitive Exponent Associative Memory Model : A Novel Approach

Matrix approximations and solvers using tensor products and non-standard wavelet transforms related to irregular grids

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