Eugene E. Tyrtyshnikov

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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)
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.
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)
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)
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)
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)