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Article history: Received 19 July 2009 Accepted 27 July 2009 Available online 21 August 2009 Submitted by V. Sergeichuk AMS classification: 15A12 65F10 65F15

- Ivan V. Oseledets, Eugene E. Tyrtyshnikov
- SIAM J. Scientific Computing
- 2009

- Ivan V. Oseledets, Dmitry V. Savostyanov, Eugene E. Tyrtyshnikov
- SIAM J. Matrix Analysis Applications
- 2008

Abstract. We consider Tucker-like approximations with an r × r × r core tensor for threedimensional n×n×n arrays in the case of r ¿ n and possibly very large n (up to 104−106). As the approximation contains only O(rn + r3) parameters, it is natural to ask if it can be computed using only a small amount of entries of the given array. A similar question for… (More)

- Eugene E. Tyrtyshnikov
- Computing
- 2000

The mosaic-skeleton method was bred in a simple observation that rather large blocks in very large matrices coming from integral formulations can be approximated accurately by a sum of just few rank-one matrices (skeletons). These blocks might correspond to a region where the kernel is smooth enough, and anyway it can be a region where the kernel is… (More)

- Stefano Serra Capizzano, Eugene E. Tyrtyshnikov
- SIAM J. Matrix Analysis Applications
- 2000

- Bernhard Beckermann, S. A. Goreinov, Eugene E. Tyrtyshnikov
- SIAM J. Matrix Analysis Applications
- 2005

Starting from an GMRES error estimate proposed by Elman in terms of the ratio of the smallest eigenvalue of the hermitian part and the norm of some non-symmetric matrix, we propose some asymptotically tighter bound in terms of the same ratio. Here we make use of a recent deep result of Crouzeix et al. on the norm of functions of matrices.

- Wolfgang Hackbusch, Boris N. Khoromskij, Eugene E. Tyrtyshnikov
- J. Num. Math.
- 2005

The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integrodifferential) operators, especially for high-dimensional problems. These new formats elaborate on a sum of few terms of Kronecker products of smaller-sized… (More)

- Vladimir A. Kazeev, Boris N. Khoromskij, Eugene E. Tyrtyshnikov
- SIAM J. Scientific Computing
- 2013

We consider two operations in the QTT format: composition of a multilevel Toeplitz matrix generated by a given multidimensional vector and convolution of two given multidimensional vectors. We show that low-rank QTT structure of the input is preserved in the output and propose efficient algorithms for these operations in the QTT format. For a d-dimensional… (More)

- S. A. Goreinov, Eugene E. Tyrtyshnikov, Alex Yu. Yeremin
- Numerical Lin. Alg. with Applic.
- 1997

A purely algebraic approach to solving very large general unstructured dense linear systems, in particular, those that arise in 3D boundary integral applications is suggested. We call this technique the matrix-free approach because it allows one to avoid the necessity of storing the whole coefficient matrix in any form, which provides significant memory and… (More)

- Vasily Strela, Eugene E. Tyrtyshnikov
- Math. Comput.
- 1996

The eigenvalue clustering of matrices S−1 n An and C −1 n An is experimentally studied, where An, Sn and Cn respectively are Toeplitz matrices, Strang, and optimal circulant preconditioners generated by the Fourier expansion of a function f(x). Some illustrations are given to show how the clustering depends on the smoothness of f(x) and which preconditioner… (More)