| 1. |
- Barakitis, Nikos, et al.
(författare)
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Preconditioners for fractional diffusion equations based on the spectral symbol
- 2022
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Ingår i: Numerical Linear Algebra with Applications. - : John Wiley & Sons. - 1070-5325 .- 1099-1506. ; 29:5
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Tidskriftsartikel (refereegranskat)abstract
- It is well known that the discretization of fractional diffusion equations with fractional derivatives , using the so-called weighted and shifted Grünwald formula, leads to linear systems whose coefficient matrices show a Toeplitz-like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so-called generalized locally Toeplitz class. Conversely, when the given FDE has constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function, called the spectral symbol, having a unique zero at zero of real positive order between one and two. For the fast solution of such systems by preconditioned Krylov methods, several preconditioning techniques have been proposed in both the one- and two-dimensional cases. In this article we propose a new preconditioner denoted bywhich belongs to the -algebra and it is based on the spectral symbol. Comparing with some of the previously proposed preconditioners, we show that although the low band structure preserving preconditioners are more effective in the one-dimensional case, the new preconditioner performs better in the more challenging multi-dimensional setting.
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| 2. |
- Bogoya, Manuel, et al.
(författare)
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Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory : The preconditioned setting
- 2024
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Ingår i: Applied Mathematics and Computation. - : Elsevier. - 0096-3003 .- 1873-5649. ; 466
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Tidskriftsartikel (refereegranskat)abstract
- Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form with real-valued, g nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that is even and monotonic over , matrix-less algorithms have been developed for the fast eigenvalue computation of large preconditioned matrices of the type above, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions as in the case , combined with the extrapolation idea, and hence we conjecture that the simple-loop theory has to be extended in such a new setting, as the numerics strongly suggest.Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we consider new matrix-less algorithms ad hoc for the current case.Numerical experiments show a much higher accuracy till machine precision and the same linear computational cost, when compared with the matrix-less procedures already proposed in the literature.
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| 3. |
- Bogoya, Manuel, et al.
(författare)
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Matrix-less methods for the spectral approximation of large non-Hermitian Toeplitz matrices : A concise theoretical analysis and a numerical study
- 2024
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Ingår i: Numerical Linear Algebra with Applications. - : Wiley. - 1070-5325 .- 1099-1506.
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Tidskriftsartikel (refereegranskat)abstract
- It is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non-Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non-Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix-size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non-Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high-precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented.
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| 4. |
- Ekström, Sven-Erik, et al.
(författare)
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A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues
- 2022
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Ingår i: Numerical Algorithms. - : Springer Nature. - 1017-1398 .- 1572-9265. ; 89:2, s. 701-720
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Tidskriftsartikel (refereegranskat)abstract
- It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol f, appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function f. The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of f. Future research directions are outlined at the end of the paper.
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