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

TL;DR

This paper uses GLT theory and new momentary symbols to analyze the spectral properties of matrices from discretized parabolic and fractional order problems, providing insights into their eigenvalue and singular value distributions.

Contribution

It introduces the concept of GLT momentary symbols to describe spectral distributions of matrices from parabolic and fractional equations, extending classical GLT theory.

Findings

Characterization of spectrum and singular values of discretized matrices

Application of GLT and momentary symbols to small matrix sizes

Open problems in using this approach for iterative methods

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 with space-time grid for a parabolic diffusion problem and from the approximation of distributed order fractional equations. For this purpose we will use the classical GLT theory and the new concept of GLT momentary symbols. The first permits to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices, the latter permits to derive a function, which describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix-sizes but under given assumptions. The note 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.