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- Bolten, Matthias, et al.
(author)
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A note on the spectral analysis of matrix sequences via GLT momentary symbols : from all-at-once solution of parabolic problems to distributed fractional order matrices
- 2023
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In: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften, Verlag. - 1068-9613. ; 58, s. 136-163
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Journal article (peer-reviewed)abstract
- The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques.
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- Bolten, Matthias, et al.
(author)
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Toeplitz momentary symbols : definition, results, and limitations in the spectral analysis of structured matrices
- 2022
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In: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 651, s. 51-82
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Journal article (peer-reviewed)abstract
- A powerful tool for analyzing and approximating the singular values and eigenvalues of structured matrices is the theory of Generalized Locally Toeplitz (GLT) sequences. By the GLT theory one can derive a function, called the symbol, which describes the singular value or the eigenvalue distribution of the sequence, the latter under precise assumptions. However, for small values of the matrix-size of the considered sequence, the approximations may not be as good as it is desirable, since in the construction of the GLT symbol one disregards small norm and low-rank perturbations. On the other hand, Local Fourier Analysis (LFA) can be used to construct polynomial symbols in a similar manner for discretizations, where the geometric information is present, but the small norm perturbations are retained. The main focus of this paper is the introduction of the concept of sequence of "Toeplitz momentary symbols", associated with a given sequence of truncated Toeplitz-like matrices. We construct the symbol in the same way as in the GLT theory, but we keep the information of the small norm contributions. The low-rank contributions are still disregarded, and we give an idea on the reason why this is negligible in certain cases and why it is not in other cases, being aware that in presence of high nonnormality the same low-rank perturbation can produce a dramatic change in the eigenvalue distribution. Moreover, a difference with respect to the LFA symbols is that GLT symbols and Toeplitz momentary symbols are more general -just Lebesgue measurable -and are applicable to a larger class of matrices, while in the LFA setting only trigonometric polynomials are considered and more specifically those related to the approximation stencils. We show the applicability of the approach which leads to higher accuracy in some cases, when approximating the singular values and eigenvalues of Toeplitz-like matrices using Toeplitz momentary symbols, compared with the GLT symbol. Finally, since for many applications and their analysis it is often necessary to consider non-square Toeplitz matrices, we formalize and provide some useful definitions, applicable for non-square Toeplitz momentary symbols.
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- Ferrari, Paola, et al.
(author)
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Multilevel symmetrized Toeplitz structures and spectral distribution results for the related matrix sequences
- 2021
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In: The Electronic Journal of Linear Algebra. - : University of Wyoming Libraries. - 1537-9582 .- 1081-3810. ; 37, s. 370-386
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Journal article (peer-reviewed)abstract
- In recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that f belongs to L-1 ([-pi, pi]) and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence {YnTn[f]}(n), has been identified, where n is the matrix size, Y-n is the anti-identity matrix, and T-n [f] is the Toeplitz matrix generated by f. In this note, the authors consider the multilevel Toeplitz matrix T-n [f] generated by f is an element of L-1 ([-pi, pi](k)), n being a multi-index identifying the matrix-size, and they prove spectral and singular value distribution results for the matrixsequence {YnTn [f]}(n) with Y-n being the corresponding tensorization of the anti-identity matrix.
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