Papers by E. Tyrtyshnikov
Journal of Inverse and Ill-posed Problems, Oct 1, 2020
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subspace methods and minimal residuals
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Linear Algebra and its Applications, 2006
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Matrix Methods: Theory, Algorithms and Applications, 2010
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Journal of Numerical Mathematics, 2005
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Calcolo, 1996
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Journal of Inverse and Ill-Posed Problems, Nov 1, 2020
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Numerical Linear Algebra with Applications, 2011
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Recent experiments showed that the circulant preconditioners are eecient in many cases (especiall... more Recent experiments showed that the circulant preconditioners are eecient in many cases (especially if the system's matrix A is Toeplitz). There are two main types of circulant preconditioners: optimal, minimizing functional k A ? C k F , and super-optimal, minimizing functional k I ? C ?1 A k F. All circulant matrces C are diagonalized by the discrete Fourier transform: C = F F. Here F is the Fourier matrix and is a diagonal matrix. Thus, it is natural to generalize the idea of cir-culant preconditioners to a wider class of matrices such that every element C of this class can be presented in the form C = F F, where F is a given orthogonal matrix and is diagonal. In this paper we study spectral properties of the generalized optimal and super-optimal preconditioners and prove that if A is symmetric and positive deenite then both generalized preconditioners are symmetric and positive deenite as well. Some algebraic and geometric properties of operator c(A), which establishes the correspondence between matrices A and their generalized optimal preconditioners, are studied. c(A) turned out to be an ortho-projector from the space of Hermitian matrices to the subspace of pseudo-circulants. In the recent time a lot of attention is paid to the exploration of the iterative methods with preconditioning. Generally, the preconditioner should possess two main features. First, it should reeect the structure of the system's matrix A. It could be achieved, for example, by the minimization of functional k A ? C k E or functional k I ? C ?1 A k E by a class of matrices C. Second, it should be easily invertible in the case of explicit preconditioning and easily multiplicable by a vector in the case of implicit preconditioning. Such are the circulant matrices C, which can be expressed in the form C = F F; where is a diagonal matrix, F = f km ] n?1 k;m=0 is the Fourier matrix (f km = exp(i 2km n)), n is the order of the matrix and i is the imaginary unit. It is known that the inverted circulant matrix still is a circulant. So, while using circulants as preconditioners there is no diierence between explicit and implicit precon-ditioning. The multiplication of the circulant matrix by a vector can be performed in O(n log n) arithmetic operations. The review of the literature devoted to the circulant preconditioning is given …
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Linear Algebra and its Applications, 2013
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ABSTRACT
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SIAM Journal on Matrix Analysis and Applications, 2000
ABSTRACT
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Linear Algebra and its Applications, 2012
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Numerical Linear Algebra with Applications, 2011
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In the general case of multilevel Toeplitz matrices, we recently proved that any multilevel circu... more In the general case of multilevel Toeplitz matrices, we recently proved that any multilevel circulant preconditioner is not superlinear (a cluster it may provide cannot be proper). The proof was based on the concept of quasi-equimodular matrices, although this concept does not apply, for example, to the sine-transform matrices. In this paper, with a new concept of partially equimodular matrices, we cover all trigonometric matrix algebras widely used in the literature. We propose a technique for proving the non-superlinearity of certain frequently used preconditioners for some representative sample multilevel matrices. At the same time, we show that these preconditioners are, in a certain sense, the best among the sublinear preconditioners (with only a general cluster) for multilevel Toeplitz matrices.
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Most successful numerical algorithms for multi-dimensional problems usually involve multi-index a... more Most successful numerical algorithms for multi-dimensional problems usually involve multi-index arrays, also called tensors, and capitalize on those tensor decompositions that reduce, one way or another, to low-rank matrices associated with the given tensors. It can be argued that the most of recent progress is due to the TT and HT decompostions [1]. The differences between the two decompositions may look as rather subtle, because the both are based on the same dimensionality reduction tree and exploit seemingly the same idea. In this talk, we analyze the differences between the two decompositions and present them in a clear and simple way. Besides that, we demonstrate some new applciations of tensor approximations in numerical analysis [2].
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Computational Methods in Applied Mathematics
We show that the recent tensor-train (TT) decompositions of matrices come up from its recursive K... more We show that the recent tensor-train (TT) decompositions of matrices come up from its recursive Kronecker-product representations with a systematic use of common bases. The names TTM and QTT used in this case stress the relation with multilevel matrices or quantization that increases artificially the number of levels. Then we investigate how the tensor-train ranks of a matrix can be related to those of its inverse. In the case of a banded Toeplitz matrix, we prove that the tensor-train ranks of its inverse are bounded above by 1+(l+u)^2, where l and u are the bandwidths in the lower and upper parts of the matrix without the main diagonal.
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SAR and QSAR in Environmental Research
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Linear Algebra and its Applications
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Papers by E. Tyrtyshnikov