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

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. (English) Zbl 1516.15005

ETNA, Electron. Trans. Numer. Anal. 58, 136-163 (2023).
Summary: 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.

Cited in 3 Documents

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15B05 Toeplitz, Cauchy, and related matrices
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35R11 Fractional partial differential equations

Keywords:

Toeplitz matrices; asymptotic distribution of eigenvalues and singular values; numerical solution; discretized equations; boundary value problems; fractional differential equations
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References:

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