The theory of locally Toeplitz sequences: a review, an extension, and a few representative applications

• Published in: Boletín de la Sociedad Matemática Mexicana Vol. 22; no. 2; pp. 529 - 565
• Main Authors: Garoni, Carlo, Serra-Capizzano, Stefano
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-10-2016; Springer Nature B.V
• Subjects: Mathematics; Mathematics and Statistics; Original Article; Partial differential equations; 47B06; Discretization of partial differential equations; 65N06; Locally Toeplitz sequence; Singular value and eigenvalue distribution; Approximating class of sequences; 15B05; 65N30
• ISSN: 1405-213X; 2296-4495
• DOI: 10.1007/s40590-016-0088-8

The theory of locally Toeplitz (LT) sequences is a powerful apparatus for computing the asymptotic singular value and eigenvalue distribution of the discretization matrices
A
n
arising from the numerical approximation of partial differential equations (PDEs). Indeed, when the discretization parameter
n
tends to infinity, the matrices
A
n
give rise to a sequence
{
A
n
}
n
, which often can be expressed as a finite sum of LT sequences. In this work, we review and extend the theory of LT sequences, which dates back to the pioneering work by Tilli in 1998 and was partially developed by the second author during the last decade. We also present some applications of the theory to the finite difference and finite element approximation of PDEs.