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Please use this identifier to cite or link to this item: http://hdl.handle.net/10277/414

Authors: Ngondiep, Eric
Title: Approximation and spectral analysis for large structured linear systems.
Abstract: In this work we are interested in standard and less standard structured linear systems coming from applications in various _elds of computational mathematics and often modeled by integral and/or di_erential equations. Starting from classical Toeplitz and Circulant structures, we consider some extensions as g-Toeplitz and g-Circulants matrices appearing in several contexts in numerical analysis and applications. Then we consider special matrices arising from collocation methods for di_erential equations: also in this case, under suitable assumptions we observe a Toeplitz structure. More in detail we _rst propose a detailed study of singular values and eigenvalues of g-circulant matrices and then we provide an analysis of distribution of g-Toeplitz sequences. Furthermore, when possible, we consider Krylov space methods with special attention to the minimization of the computational work. When the involved dimensions are large, the Preconditioned Conjugate Gradient (PCG) method is recommended because of the much stronger robustness with respect to the propagation of errors. In that case, crucial issues are the convergence speed of this iterative solver, the use of special techniques (preconditioning, multilevel techniques) for accelerating the convergence, and a careful study of the spectral properties of such matrices. Finally, the use of radial basis functions allow of determining and studying the asymptotic behavior of the spectral radii of collocation matrices approximating elliptic boundary value problems.
Keywords: circulant matrices, Toeplitz sequences, spectral properties, approximations, preconditioning, g-circulant matrices, g-Toeplitz sequences, singular values, eigenvalues, distribution, linear systems, Krylov space methods, multigrid methods, regularizing techniques, collocation matrices, elliptic boundary value problems, RBFs, spectral radii, block Toeplitz matrices with unbounded generating functions.
Issue Date: 2011
Language: eng
Doctoral course: Matematica del Calcolo: Modelli, Strutture, Algoritmi e Applicazioni 
Academic cycle: 23
Publisher: Università degli Studi dell'Insubria
Citation: Ngondiep, E.Approximation and spectral analysis for large structured linear systems. (Doctoral Thesis, Università degli Studi dell'Insubria, 2011).

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