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

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Abstract

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.

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Notes

  1. IgA is a modern paradigm for analyzing problems governed by PDEs; see [7].

  2. Note that two functions \(f,g\in L^1([-\pi ,\pi ]^d)\) which coincide a.e. give rise to the same multilevel Toeplitz matrices \(T_{\varvec{n}}(f)=T_{\varvec{n}}(g),\ {\varvec{n}}\in {\mathbb {N}}^d\), because the Fourier coefficients of f and g coincide.

  3. We refer the reader to the introduction of Tilli’s paper [26] for the origin and the meaning of this terminology.

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Acknowledgments

The authors express their sincere gratitude to the Editor Albrecht Böttcher and to the referee, who helped them to improve the paper. In particular, Sect. 5.4 originated from a specific remark by the referee.

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Authors and Affiliations

  1. Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Carlo Garoni

  2. Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100, Como, Italy

    Stefano Serra-Capizzano

Authors
  1. Carlo Garoni
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  2. Stefano Serra-Capizzano
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Correspondence to Carlo Garoni.

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figure a

Dedicated to Sergei M. Grudsky on his 60th birthday.

This work was supported by the Italian MIUR Program FIR 2013 through the Project DREAMS, by the INdAM GNCS (Gruppo Nazionale per il Calcolo Scientifico), and by the Donation KAW 2013.0341 from the Knut & Alice Wallenberg Foundation in collaboration with the Royal Swedish Academy of Sciences.

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Garoni, C., Serra-Capizzano, S. The theory of locally Toeplitz sequences: a review, an extension, and a few representative applications. Bol. Soc. Mat. Mex. 22, 529–565 (2016). https://doi.org/10.1007/s40590-016-0088-8

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  • DOI: https://doi.org/10.1007/s40590-016-0088-8

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