| 1. |
- Arévalo, Carmen
(author)
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A note on numerically consistent initial values for high index differential-algebraic equations
- 2008
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In: Electronic Transactions on Numerical Analysis. - 1068-9613. ; 34, s. 14-19
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Journal article (peer-reviewed)abstract
- When differential-algebraic equations of index 3 or higher are solved with backward differentiation formulas, the solution in the first few steps can have gross errors, the solution can have gross errors in the first few steps, even if the initial values are equal to the exact solution and even if the step size is kept constant. This raises the question of what are consistent initial values for the difference equations. Here we study how to change the exact initial values into what we call numerically consistent initial values for the implicit Euler method.
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| 2. |
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| 3. |
- 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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| 4. |
- 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
-
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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| 5. |
- 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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| 6. |
- Bogoya, Manuel, et al.
(author)
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Upper Hessenberg and Toeplitz Bohemian matrix sequences: a note on their asymptotical eigenvalues and singular values
- 2022
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In: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften, Verlag. - 1068-9613. ; 55, s. 76-91
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Journal article (peer-reviewed)abstract
- In previous works, Bohemian matrices have attracted the attention of several researchers for theirrich combinatorial structure, and they have been studied intensively from several points of view, including height,determinants, characteristic polynomials, normality, and stability. Here we consider a selected number of examples ofupper Hessenberg and Toeplitz Bohemian matrix sequences whose entries belong to the population P = {0, ±1},and we propose a connection with the spectral theory of Toeplitz matrix sequences and Generalized Locally Toeplitz(GLT) matrix sequences in order to give results on the localization and asymptotical distribution of their spectra andsingular values. Numerical experiments that support the mathematical study are reported. A conclusion section endsthe note in order to illustrate the applicability of the proposed tools to more general cases.
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| 7. |
- Bolten, Matthias, et al.
(author)
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A note on the spectral analysis of matrix sequences via GLT momentary symbols : from all-at-once solution of parabolic problems to distributed fractional order matrices
- 2023
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In: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften, Verlag. - 1068-9613. ; 58, s. 136-163
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Journal article (peer-reviewed)abstract
- The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques.
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| 9. |
- Correnty, Siobhan, et al.
(author)
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Preconditioned Chebyshev BiCG method for parameterized linear systems
- 2023
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In: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften, Verlag. - 1068-9613. ; 58, s. 629-656
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Journal article (peer-reviewed)abstract
- We consider the problem of approximating the solution to A(μ)x(μ) = b for many different values of the parameter μ. Here, A(μ) is large, sparse, and nonsingular with a nonlinear dependence on μ. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of A(μ) on the interval [-a; a], a 2 R+, inspired by Effenberger and Kressner [BIT, 52 (2012), pp. 933-951]. The solution to the linearization is approximated in a preconditioned BiCG setting for shifted systems, as proposed in Ahmad et al. [SIAM J. Matrix Anal. Appl., 38 (2017), pp. 401-424], where the Krylov basis matrix is formed once. This process leads to a short-term recurrence method, where one execution of the algorithm produces the approximation of x(μ) for many different values of the parameter μ ∈ [-a; a] simultaneously. In particular, this work proposes one algorithm which applies a shift-and-invert preconditioner exactly as well as an algorithm which applies the preconditioner inexactly based on the work by Vogel [Appl. Math. Comput., 188 (2007), pp. 226-233]. The competitiveness of the algorithms is illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with a parameterized material coefficient. The software used in the simulations is publicly available online, and thus all our experiments are reproducible.
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| 10. |
- Elfving, Tommy, 1944-, et al.
(author)
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Semi-convergence and relaxation parameters for a class of SIRT algorithms
- 2010
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In: Electronic Transactions on Numerical Analysis. - : Kent State University Library. - 1068-9613. ; 37, s. 321-336
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Journal article (peer-reviewed)abstract
- This paper is concerned with the Simultaneous Iterative Reconstruction Technique (SIRT) class of iterative methods for solving inverse problems. Based on a careful analysis of the semi-convergence behavior of these methods, we propose two new techniques to specify the relaxation parameters adaptively during the iterations, so as to control the propagated noise component of the error. The advantage of using this strategy for the choice of relaxation parameters on noisy and ill-conditioned problems is demonstrated with an example from tomography
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