A theory of pseudoskeleton approximations

Goreinov, Sergei A.; Тыртышников, Е. Е.; Замарашкин, Н. Л.

doi:10.1016/s0024-3795(96)00301-1

Cited by 401 publications

(240 citation statements)

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“…Such a method was proposed and called the TT-Cross method [55]. It heavily exploits the matrix cross interpolation [57], [58], [59], [60], [61] algorithm applied cleverly, although heuristically, to selected submatrices in the unfolding matrices of the given tensor. The matrix

A_{k} \in R^{n^{k} \times n^{d - k}}

, A k ( i 1 … i k , i k + 1 … i d ) = A ( i 1 , i 2 , … , i d ) is called the k -th unfolding matrix of the tensor A .…”

Section: Methodsmentioning

confidence: 99%

Evaluation of the novel algorithm of flexible ligand docking with moveable target-protein atoms

Сулимов

Zheltkov

Oferkin³

et al. 2017

Computational and Structural Biotechnology Journal

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We present the novel docking algorithm based on the Tensor Train decomposition and the TT-Cross global optimization. The algorithm is applied to the docking problem with flexible ligand and moveable protein atoms. The energy of the protein-ligand complex is calculated in the frame of the MMFF94 force field in vacuum. The grid of precalculated energy potentials of probe ligand atoms in the field of the target protein atoms is not used. The energy of the protein-ligand complex for any given configuration is computed directly with the MMFF94 force field without any fitting parameters. The conformation space of the system coordinates is formed by translations and rotations of the ligand as a whole, by the ligand torsions and also by Cartesian coordinates of the selected target protein atoms. Mobility of protein and ligand atoms is taken into account in the docking process simultaneously and equally. The algorithm is realized in the novel parallel docking SOL-P program and results of its performance for a set of 30 protein-ligand complexes are presented. Dependence of the docking positioning accuracy is investigated as a function of parameters of the docking algorithm and the number of protein moveable atoms. It is shown that mobility of the protein atoms improves docking positioning accuracy. The SOL-P program is able to perform docking of a flexible ligand into the active site of the target protein with several dozens of protein moveable atoms: the native crystallized ligand pose is correctly found as the global energy minimum in the search space with 157 dimensions using 4700 CPU ∗ h at the Lomonosov supercomputer.

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A_{k} \in R^{n^{k} \times n^{d - k}}

, A k ( i 1 … i k , i k + 1 … i d ) = A ( i 1 , i 2 , … , i d ) is called the k -th unfolding matrix of the tensor A .…”

Section: Methodsmentioning

confidence: 99%

Evaluation of the novel algorithm of flexible ligand docking with moveable target-protein atoms

Сулимов

Zheltkov

Oferkin³

et al. 2017

Computational and Structural Biotechnology Journal

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“…Thus, in practice we are interested to exploit the case of two levels as far as possible (cf Table 6.1 Optimal and quasi-optimal approximations Rank method Relative error [4,11,29]). Moreover, accurate low-rank approximation can be often obtained from picking up only a relatively small number of entries, which leads to very efficient practical algorithms (cf [9,10,30]). …”

Section: Examplementioning

confidence: 99%

Tensor properties of multilevel Toeplitz and related matrices

Olshevsky

Oseledets

Тыртышников

2006

Linear Algebra and its Applications

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A general proposal is presented for fast algorithms for multilevel structured matrices. It is based on investigation of their tensor properties and develops the idea recently introduced by Kamm and Nagy in the block Toeplitz case. We show that tensor properties of multilevel Toeplitz matrices are related to separation of variables in the corresponding symbol, present analytical tools to study the latter, expose truncation algorithms preserving the structure, and report on some numerical results confirming advantages of the proposal.

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“…Indeed, it may not even be possible to compute or store all of the entries of A. So we appeal to results proved in [11,12] which show that a low-rank approximation to a matrix of order n can be reliably obtained using only O(n) entries in some, appropriately chosen, positions. These results form the basis for the incomplete cross approximation algorithm (ICA) for approximating P(A) by r k=1 V(U k )(V(V k )) T , which can be found in [16].…”

Section: Approximating Dense Function-related Matricesmentioning

confidence: 99%

Solving linear systems using wavelet compression combined with Kronecker product approximation

Ford¹,

Тыртышников

2005

Numer Algor

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Discrete wavelet transform approximation is an established means of approximating dense linear systems arising from discretization of differential and integral equations defined on a one-dimensional domain. For higher dimensional problems, approximation with a sum of Kronecker products has been shown to be effective in reducing storage and computational costs. We have combined these two approaches to enable solution of very large dense linear systems by an iterative technique using a Kronecker product approximation represented in a wavelet basis. Further approximation of the system using only a single Kronecker product provides an effective preconditioner for the system. Here we present our methods and illustrate them with some numerical examples. This technique has the potential for application in a range of areas including computational fluid dynamics, elasticity, lubrication theory and electrostatics.

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A theory of pseudoskeleton approximations

Cited by 401 publications

References 1 publication

Evaluation of the novel algorithm of flexible ligand docking with moveable target-protein atoms

Evaluation of the novel algorithm of flexible ligand docking with moveable target-protein atoms

Tensor properties of multilevel Toeplitz and related matrices

Solving linear systems using wavelet compression combined with Kronecker product approximation

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