A Note on Generalized Locally Toeplitz Operators

TL;DR

This paper reviews the operator-theoretic aspects of Generalized Locally Toeplitz (GLT) sequences, which are crucial for analyzing spectral properties of matrices from PDE discretizations, and proposes an automatic procedure for computing their spectral symbols.

Contribution

It provides a comprehensive overview of GLT sequences' operator theory and introduces an automatic method for spectral symbol computation under mild conditions.

Findings

GLT sequences effectively analyze spectral distributions of PDE-related matrices.

An automatic procedure for spectral symbol computation is proposed.

The method applies to matrices from various discretization techniques.

Abstract

Generalized Locally Toeplitz (GLT) matrix sequences arise from large linear systems that approximate Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), and Integro-Differential Equations (IDEs). GLT sequences of matrices have been developed to study the spectral/singular value behaviour of the numerical approximations to various PDEs, Fades and IDEs. These approximations can be achieved using any discretization method on appropriate grids through local techniques such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, and Discontinuous Galerkin methods. Spectral and singular value symbols are essential for analyzing the eigenvalue and singular value distributions of matrix sequences in the Weyl sense. In this article, we provide a comprehensive overview of the operator-theoretic aspect of GLT sequences. The theory of GLT sequences, along with findings on the asymptotic spectral distribution of perturbed matrix sequences, is a highly effective and successful method for calculating the spectral symbol f. Therefore, developing an automatic procedure to compute the spectral symbols of these matrix sequences would be advantageous, a task that Ahmed Ratnani, N S Sarathkumar, S. Serra-Capizzano have partially undertaken. As an application of the theory developed here, we propose an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming they form a GLT sequence that meets mild conditions.