Exploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey

Böttcher, Albrecht; Garoni, Carlo; Serra‐Capizzano, Stefano

doi:10.1070/sm8823

Cited by 21 publications

(16 citation statements)

References 48 publications

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“…The matrix C n associated with the linear system (11) assuming the new orderings (18) and (19) is the 2 × 2 block tridiagonal matrix given by…”

Section: Fd Discretizationmentioning

confidence: 99%

“…The experience reveals that virtually any kind of numerical methods for the discretization of DEs gives rise to structured matrices A n whose asymptotic spectral distribution, as the mesh fineness parameter n tends to infinity, can be computed through the theory of GLT sequences. We refer the reader to ( [13] Section 10.5), ([14] Section 7.3), and [15,16,18] for applications of the theory of GLT sequences in the context of finite difference (FD) discretizations of DEs; to ( [13] Section 10.6), ( [14] Section 7.4), and [16,18,19] for the finite element (FE) case; to [20] for the finite volume (FV) case; to ( [13] Section 10.7), ( [14] Sections 7.5-7.7), and [21][22][23][24][25][26] for the case of isogeometric analysis (IgA) set", "measurable function", "a.e. ", etc.)…”

Section: Introductionmentioning

confidence: 99%

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Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

Garoni

Mazza

Serra‐Capizzano

2018

Axioms

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Abstract:The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices A n arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices A n give rise to a sequence {A n } n , which often turns out to be a GLT sequence or one of its "relatives", i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations.Keywords: spectral (eigenvalue) and singular value distributions; generalized locally Toeplitz sequences; discretization of systems of differential equations; higher-order finite element methods; discontinuous Galerkin methods; finite difference methods; isogeometric analysis; B-splines; curl-curl operator; time harmonic Maxwell's equations and magnetostatic problems MSC: 47B06; 15A18; 15B05; 65N30; 65N06; 65D07

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“…The matrix C n associated with the linear system (11) assuming the new orderings (18) and (19) is the 2 × 2 block tridiagonal matrix given by…”

Section: Fd Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

Garoni

Mazza

Serra‐Capizzano

2018

Axioms

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“…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%

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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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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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Exploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey

Cited by 21 publications

References 48 publications

Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

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

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