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- Barbarino, Giovanni, et al.
(author)
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Block generalized locally Toeplitz sequences : theory and applications in the multidimensional case
- 2020
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In: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften. - 1068-9613. ; 53, s. 113-216
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Journal article (peer-reviewed)abstract
- In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from the numerical discretization of a partial differential equation (PDE), knowledge of the spectral distribution of the associated matrix has proved to be useful information for designing/analyzing appropriate solvers-especially, preconditioned Krylov and multigrid solvers-for the considered problem. Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest, which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material). The theory of multilevel 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 PDEs. 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 multilevel GLT sequence or one of its "relatives", i.e., a multilevel block GLT sequence or a (multilevel) reduced GLT sequence. In particular, multilevel block GLT sequences are encountered in the discretization of systems of PDEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar/vectorial PDEs. In this work, we systematically develop the theory of multilevel block GLT sequences as an extension of the theories of (unilevel) GLT sequences [Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I., Springer, Cham, 2017], multilevel GLT sequences [Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. II., Springer, Cham, 2018], and block GLT sequences [Barbarino, Garoni, and Serra-Capizzano, Electron. Trans. Numer. A(n)al., 53 (2020), pp. 28-112]. We also present several emblematic applications of this theory in the context of PDE discretizations.
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- Barbarino, Giovanni, et al.
(author)
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Block generalized locally Toeplitz sequences : theory and applications in the unidimensional case
- 2020
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In: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften. - 1068-9613. ; 53, s. 28-112
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Journal article (peer-reviewed)abstract
- In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from the numerical discretization of a differential equation (DE), knowledge of the spectral distribution of the associated matrix has proved to be useful information for designing/analyzing appropriate solvers-especially, preconditioned Krylov and multigrid solvers-for the considered problem. Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest, which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material). 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 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/vectorial DEs. This work is a review, refinement, extension, and systematic exposition of the theory of block GLT sequences. It also includes several emblematic applications of this theory in the context of DE discretizations.
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- Barbarino, Giovanni, et al.
(author)
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Matrix-Less Eigensolver for Large Structured Matrices
- 2021
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Reports (other academic/artistic)abstract
- Sequences of structured matrices of increasing size arise in many scientific applications and especially in the numerical discretization of linear differential problems. We assume as a working hypothesis that the eigenvalues of a matrix X_n belonging to a sequence of this kind are given by a regular expansion. Based on this working hypothesis, which is illustrated to be plausible through numerical experiments, we propose an eigensolver for the computation of the eigenvalues of X_n for large n and we provide a theoretical analysis of its convergence. The eigensolver is called matrix-less because it does not operate on the matrix X_n but on a few similar matrices of smaller size combined with an interpolation-extrapolation strategy. Its performance is benchmarked on several numerical examples, with a special focus on matrices arising from the discretization of differential problems.
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- Barbarino, Giovanni, et al.
(author)
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Matrix-Less Eigensolver for Large Structured Matrices
- 2021
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Reports (other academic/artistic)abstract
- Sequences of structured matrices of increasing size arise in many scientific applications and especially in the numerical discretization of linear differential problems. We assume as a working hypothesis that the eigenvalues of a matrix X_n belonging to a sequence of this kind are given by a regular expansion. Based on the working hypothesis, which is proved to be plausible through numerical experiments, we propose an eigensolver for the computation of the eigenvalues of X_n for large n. The performance of the eigensolver—which is called matrix-less because it does not operate on the matrix X_n—is illustrated on several numerical examples, with a special focus on matrices arising from the discretization of differential problems, and turns out to be quite satisfactory in all cases. In a sense, this is an a posteriori proof of the reasonableness of the working hypothesis as well as a testimony of the fact that the spectra of large structured matrices are much more “regular” than one might expect.
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- Barbarino, Giovanni, et al.
(author)
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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
- 2020
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In: Numerical Linear Algebra with Applications. - : John Wiley & Sons. - 1070-5325 .- 1099-1506. ; 27:3
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Journal article (peer-reviewed)abstract
- 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 Xn of size n and that {Xn}n∼λf, that is, the matrix-sequence {Xn}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 {Xn+Yn}n∼λf. 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 {Xn}n and {Yn}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.
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- Barbarino, Giovanni, et al.
(author)
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Rectangular GLT sequences
- 2022
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In: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften, Verlag. - 1068-9613. ; 55, s. 585-617
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Journal article (peer-reviewed)abstract
- The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computingthe asymptotic spectral distribution of square matrices An arising from the discretization of differential problems.Indeed, as the mesh fineness parameter n increases to ∞, the sequence {An}n often turns out to be a GLT sequence.In this paper, motivated by recent applications, we further enhance the GLT apparatus by developing a full theory ofrectangular GLT sequences as an extension of the theory of classical square GLT sequences. We also provide twoexamples of application as an illustration of the potential of the theory presented herein
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