Equivalence between GLT sequences and measurable functions

Barbarino, Giovanni

doi:10.1016/j.laa.2017.04.039

Cited by 22 publications

(46 citation statements)

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“…for {A n } n if, for all sufficiently large m, the sequence {B n,m } n approximates (asymptotically) the sequence {A n } n in the sense that A n is eventually equal to B n,m plus a small-rank matrix (with respect to the matrix size n) plus a small-norm matrix.Definition 1.2 (Asymptotic singular value and eigenvalue distribution of a matrix-sequence). Let {A n } n be a matrix-sequence and let f : D → C be a measurable function defined on a set D ⊂ R k with 0 < µ k (D) < ∞.It was proved in [2,7] that d a.c.s. is a distance on E which turns E into a complete topological (pseudometric) space (E , τ a.c.s. )…”

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confidence: 99%

“…It was proved in [2,7] that d a.c.s. is a distance on E which turns E into a complete topological (pseudometric) space (E , τ a.c.s. )…”

mentioning

confidence: 99%

“…The size n of this system diverges to infinity as the mesh discretization parameter tends to 0, and we are then in the presence of a matrix-sequence {A n } n . It is often observed in practice that {A n } n belongs to the class of the so-called generalized locally Toeplitz (GLT) sequences, and in particular it enjoys an asymptotic singular value and eigenvalue distribution as n → ∞; we refer the reader to [4] for a nice introduction to this subject and to [2,10,11,13,14,20,21,23] for more advanced studies. Another noteworthy example concerns the finite sections of an infinite Toeplitz matrix.…”

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confidence: 99%

“…After Definition 1.1, we provide the precise notions of asymptotic singular value and eigenvalue distribution for a matrix-sequence. It is worth stressing that the a.c.s., along with Theorems 1.3 and 1.4, form the basis of the theory of LT sequences [10,23] and GLT sequences [2,4,11,13,14,20,21]. For some of their concrete applications, we refer the reader to [4,, [10,Section 5.3], [11,Section 6.2.2], [13,Chapter 10], [14,Chapter 8], [15,Section 3], and also [8,9,12].…”

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confidence: 99%

“…and define τ measure to be the (pseudometric) topology of convergence in measure over M D ; then, Theorem 1.3 is equivalent to saying that the set of pairs ( Besides proving the existence and completeness of a pseudometric topology τ a.c.s. associated with the notion of a.c.s., papers [2,7] highlighted a strong connection between τ a.c.s. and τ measure , a connection which had somehow already been suggested in [19].…”

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confidence: 99%

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From convergence in measure to convergence of matrix-sequences through concave functions and singular values

Barbarino

Garoni

2017

ELA

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Abstract. Sequences of matrices with increasing size naturally arise in several areas of science, such as, for example, the numerical discretization of differential and integral equations. An approximation theory for sequences of this kind has recently been developed, with the aim of providing tools for computing their asymptotic singular value and eigenvalue distributions. The cornerstone of this theory is the notion of approximating classes of sequences (a.c.s.), which is also fundamental to the theory of generalized locally Toeplitz (GLT) sequences, and hence to the spectral analysis of PDE discretization matrices. Drawing inspiration from measure theory, here it is introduced a class of functions which are proved to be complete pseudometrics inducing the a.c.s. convergence. It is also shown that each of these pseudometrics gives rise to a natural isometry between the spaces of GLT sequences and measurable functions. Furthermore, it is highlighted that the a.c.s. convergence is an asymptotic matrix version of the convergence in measure, thus suggesting a way to obtain matrix theory results from measure theory results.

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“…It was proved in [2,7] that d a.c.s. is a distance on E which turns E into a complete topological (pseudometric) space (E , τ a.c.s. )…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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From convergence in measure to convergence of matrix-sequences through concave functions and singular values

Barbarino

Garoni

2017

ELA

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Non‐Hermitian perturbations of Hermitian matrix‐sequences and applications to the spectral analysis of the numerical approximation of partial differential equations

Barbarino

Serra‐Capizzano

2020

Numerical Linear Algebra App

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This article concerns the spectral analysis of matrix-sequences which can be written as a non-Hermitian perturbation of a given Hermitian matrix-sequence.The main result reads as follows. Suppose that for every n there is a Hermitian matrix X n of size n and that {X n } n ∼ f, that is, the matrix-sequence {X n } n enjoys an asymptotic spectral distribution, in the Weyl sense, described by a Lebesgue measurable function f ; if ||Y n || 2 = o( √ n) with || ⋅ || 2 being the Schatten 2 norm, then {X n + Y n } n ∼ f. In a previous article by Leonid Golinskii and the second author, a similar result was proved, but under the technical restrictive assumption that the involved matrix-sequences {X n } n and {Y n } n are uniformly bounded in spectral norm. Nevertheless, the result had a remarkable impact in the analysis of both spectral distribution and clustering of matrix-sequences arising from various applications, including the numerical approximation of partial differential equations (PDEs) and the preconditioning of PDE discretization matrices. The new result considerably extends the spectral analysis tools provided by the former one, and in fact we are now allowed to analyze linear PDEs with (unbounded) variable coefficients, preconditioned matrix-sequences, and so forth. A few selected applications are considered, extensive numerical experiments are discussed, and a further conjecture is illustrated at the end of the article. K E Y W O R D Sapproximation of PDEs, perturbation results, preconditioning, spectral distribution in the Weyl sense INTRODUCTIONA matrix-sequence {A n } n is an ordered collection of complex matrices such that A n ∈ C n×n and n belongs to N + or to an infinite subset of N + , with C being the complex field and N + being the set of positive integers. It is often observed in practice that matrix-sequences arising from the numerical discretization of linear differential equations possess a spectral symbol, that is, a measurable function f ∶ D ⊆ R q → C, q ≥ 1 describing the asymptotic distribution of the matrices eigenvalues Numer Linear Algebra Appl. 2020;27:e2286. wileyonlinelibrary.com/journal/nla

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Spectral Measures

Barbarino¹

2019

Structured Matrices in Numerical Linear Algebra

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Equivalence between GLT sequences and measurable functions

Cited by 22 publications

References 9 publications

From convergence in measure to convergence of matrix-sequences through concave functions and singular values

From convergence in measure to convergence of matrix-sequences through concave functions and singular values

Non‐Hermitian perturbations of Hermitian matrix‐sequences and applications to the spectral analysis of the numerical approximation of partial differential equations

Spectral Measures

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