Spectral Analysis and Spectral Symbol of $d$-variate $\mathbb Q_{\boldsymbol p}$ Lagrangian FEM Stiffness Matrices

Garoni, Carlo; Serra‐Capizzano, Stefano; Sesana, Debora

doi:10.1137/140976480

Cited by 42 publications

(35 citation statements)

References 13 publications

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“…Through the theory of block GLT sequences, we show that the corresponding sequence of (normalized) FE discretization matrices enjoys a spectral distribution described by a (p − k) × (p − k) matrix-valued function, where p and k represent, respectively, the degree and the smoothness of the piecewise polynomial functions involved in the FE approximation. Note that this result represents a remarkable argument in support of ( [35] Conjecture 2).…”

Section: Higher-order Fe Discretization Of the Diffusion Equationsupporting

confidence: 64%

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

Garoni

Mazza

Serra‐Capizzano

2018

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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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Section: Higher-order Fe Discretization Of the Diffusion Equationsupporting

confidence: 64%

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

Garoni

Mazza

Serra‐Capizzano

2018

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“…In Example 1 of [18] the case p = 2 is considered, and explicit formulas for the two eigenvalue functions are given, with their notation,…”

Section: Now We Define the Fourier Coefficients Of F(θ ) That Iŝmentioning

confidence: 99%

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

2018

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Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix T n ( f ), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f . A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where f is an s × s matrix-valued trigonometric polynomial with s ≥ 1, and T n (f) is the block Toeplitz matrix generated by f, whose size is N (n, s) = sn. The case s = 1 corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrixless. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the Q p Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.

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“…It is worth mentioning that multilevel block Toeplitz matrices arise in important applications such as Markov chains (with k = 1 and s > 1), in the reconstruction of signals with missing data (with k = 1 and s = 2), in the inpainting problem (with k = 2 and s = 2), and, of course, in the (finite difference) numerical approximation of constant‐coefficient r × r systems of PDEs over d ‐dimensional domains (with k = d and s = r ). Recently, it was discovered that multilevel block Toeplitz matrices also arise in the approximation by classical

{double-struckQ}_{p}

finite element methods of classical convection diffusion PDEs over d ‐dimensional domains (with k = d and s = p d ).…”

Section: Multilevel Block Toeplitz Matricesmentioning

confidence: 99%

“…In Theorem , we prove the linearity of the Toeplitz operator T n (·) with respect to the direct sum, modulo permutation transformations that depend only on the dimensions of the involved matrices. The proof of Theorem is based on a distributive property of the tensor product with respect to the direct sum (, Lemma 2]). Theorem Let

t 0 = (t_{1}, \dots, t_{p}) \in {double-struckN}^{p}

, let

n \in {double-struckN}^{k}

, and let f v ∈ L 1 ( k , t v ), v = 1,…, p .…”

Section: Multilevel Block Toeplitz Matricesmentioning

confidence: 99%

“…The paper is organized as follows. In Section 2, we introduce multilevel block Toeplitz matrices, and we provide some of their properties needed later on; we recall that there is a wide literature on multilevel block Toeplitz structures (see the works in [14][15][16][17] for the applicative viewpoint, [18][19][20][21] for analytic results on their spectral features, and [9,18,[22][23][24] for a numerical treatment of the associated linear systems, in the context of multigrid and preconditioned Krylov solvers). In Sections 3 and 4, we report and extend previous theoretical tools, developed in [8][9][10]20], for analyzing the asymptotic spectrum of sequences of matrices (matrix sequences).…”

Section: The Best Parameter˛that Minimizes the Upper Boundmentioning

confidence: 99%

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Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix‐valued symbol

Donatelli

Garoni

Mazza

et al. 2015

Numerical Linear Algebra App

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We perform a spectral analysis of the preconditioned Hermitian/skew-Hermitian splitting (PHSS) method applied to multilevel block Toeplitz linear systems in which the coefficient matrix Tn.f / is associated with a Lebesgue integrable matrix-valued function f . When the preconditioner is chosen as a Hermitian positive definite multilevel block Toeplitz matrix Tn.g/, the resulting sequence of PHSS iteration matrices Mn belongs to the generalized locally Toeplitz class. In this case, we are able to compute the symbolø.f; g/ describing the asymptotic eigenvalue distribution ofMn when n→∞and the matrix size diverges. By minimizing the infinity norm of the spectral radius of the symbolø.f; g/, we are also able to identify effective PHSS preconditioners Tn.g/ for the matrix Tn.f /. A number of numerical experiments are presented and commented, showing that the theoretical results are confirmed and that the spectral analysis leads to efficient PHSS methods

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Spectral Analysis and Spectral Symbol of $d$-variate $\mathbb Q_{\boldsymbol p}$ Lagrangian FEM Stiffness Matrices

Cited by 42 publications

References 13 publications

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

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

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix‐valued symbol

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