Volume 58, pp. 136-163, 2023.

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

Matthias Bolten, Sven-Erik Ekström, Isabella Furci, and Stefano Serra-Capizzano

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.

Full Text (PDF) [2.9 MB], BibTeX

Key words

Toeplitz matrices, asymptotic distribution of eigenvalues and singular values, numerical solution of discretized equations for boundary value problems involving PDEs, fractional differential equations

AMS subject classifications

15B05, 34L20, 65N22, 35R11

Links to the cited ETNA articles

[3]Vol. 53 (2020), pp. 28-112 Giovanni Barbarino, Carlo Garoni, and Stefano Serra-Capizzano: Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case
[4]Vol. 53 (2020), pp. 113-216 Giovanni Barbarino, Carlo Garoni, and Stefano Serra-Capizzano: Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case
[26]Vol. 54 (2021), pp. 499-513 Mariarosa Mazza, Stefano Serra-Capizzano, and Muhammad Usman: Symbol-based preconditioning for Riesz distributed-order space-fractional diffusion equations

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