Introduction to the theory of generalized locally Toeplitz sequences and its applications

TL;DR

This paper introduces the theory of generalized locally Toeplitz sequences, focusing on applications for spectral analysis of matrices from differential equation discretizations, and presents new efficient methods and a novel spectral analysis tool.

Contribution

It provides a self-contained overview of GLT sequences, introduces a new spectral analysis tool, and proposes more efficient approaches for spectral distribution computation.

Findings

New spectral analysis tool: modulus of integral continuity.

More efficient spectral analysis methods for DE discretization matrices.

Application-focused presentation suitable for master's level understanding.

Abstract

The theory of generalized locally Toeplitz (GLT) sequences was conceived as an apparatus for computing the spectral distribution of matrices arising from the numerical discretization of differential equations (DEs). The purpose of this review is to introduce the reader to the theory of GLT sequences and to present some of its applications to the computation of the spectral distribution of DE discretization matrices. We mainly focus on the applications, whereas the theory is presented in a self-contained tool-kit fashion, without entering into technical details. The exposition is supposed to be understandable to master's degree students in mathematics. It also discloses new more efficient approaches to the spectral analysis of DE discretization matrices as well as a novel spectral analysis tool that has not been considered in the GLT literature heretofore, i.e., the modulus of integral continuity.