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- Donatelli, M., et al.
(author)
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Multigrid methods for cubic spline solution of two point (and 2D) boundary value problems
- 2016
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In: Applied Numerical Mathematics. - : Elsevier BV. - 0168-9274 .- 1873-5460. ; 104, s. 15-29
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Journal article (peer-reviewed)abstract
- In this paper we propose a scheme based on cubic splines for the solution of the second order two point boundary value problems. The solution of the algebraic system is computed by using optimized multigrid methods. In particular the transformation of the stiffness matrix essentially in a block Toeplitz matrix and its spectral analysis allow to choose smoothers able to reduce error components related to the various frequencies and to obtain an optimal method. The main advantages of our strategy can be listed as follows: (i) a fourth order of accuracy combined with a quadratic conditioning matrix, (ii) a resulting matrix structure whose eigenvalues can be compactly described by a symbol (this information is the key for designing an optimal multigrid method). Finally, some numerics that confirm the predicted behavior of the method are presented and a discussion on the two dimensional case is given, together with few 2D numerical experiments. (C) 2014 IMACS. Published by Elsevier B.V. All rights reserved.
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| 2. |
- Aceto, Lidia, et al.
(author)
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Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis
- 2020
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In: Calcolo. - : Springer Science and Business Media LLC. - 0008-0624 .- 1126-5434. ; 57
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Journal article (peer-reviewed)abstract
- In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.
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| 3. |
- Adriani, Andrea, et al.
(author)
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Asymptotic Spectra of Large (Grid) Graphs with a Uniform Local Structure (Part I) : Theory
- 2020
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In: Milan Journal of Mathematics. - : Springer Science and Business Media LLC. - 1424-9286 .- 1424-9294. ; 88:2, s. 409-454
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Journal article (peer-reviewed)abstract
- We are mainly concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain omega subset of Rd, d >= 1. When omega=[0,1] , such graphs include the standard Toeplitz graphs and, for omega=[0,1](d), the considered class includesd-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and we show that we can associate to it a symbol f. The knowledge of the symbol and of its basic analytical features provides many information on the eigenvalue structure, of localization, spectral gap, clustering, and distribution type.Few generalizations are also considered in connection with the notion of generalized locally Toeplitz sequences and applications are discussed, stemming e.g. from the approximation of differential operators via numerical schemes. Nevertheless, more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks
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| 4. |
- Adriani, Andrea, et al.
(author)
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Asymptotic spectra of large (grid) graphs with a uniform local structure, Part II : Numerical applications
- 2024
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In: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 437
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Journal article (peer-reviewed)abstract
- In the current work we are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain Ω ⊂ Rd , d ≥ 1. When Ω = [0, 1], such graphs include the standard Toeplitz graphs and, for Ω = [0,1]d, the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown in the theoretical part of this work that we can associate to it a symbol f. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. In the present paper, many different applications are discussed and various numerical examples are presented in order to underline the practical use of the developed theory. Tests and applications are mainly obtained from the approximation of differential operators via numerical schemes such as Finite Differences, Finite Elements, and Isogeometric Analysis. Moreover, we show that more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks, whenever the involved matrices enjoy a uniform local structure.
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| 10. |
- Akbari, Hesam, et al.
(author)
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Recognizing seizure using Poincaré plot of EEG signals and graphical features in DWT domain
- 2023
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In: Bratislava Medical Journal. - : AEPress. - 0006-9248 .- 1336-0345. ; 124:1, s. 12-24
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Journal article (peer-reviewed)abstract
- Electroencephalography (EEG) signals are considered one of the oldest techniques for detecting disorders in medical signal processing. However, brain complexity and the non-stationary nature of EEG signals represent a challenge when applying this technique. The current paper proposes new geometrical features for classification of seizure (S) and seizure-free (SF) EEG signals with respect to the Poincaré pattern of discrete wavelet transform (DWT) coefficients. DWT decomposes EEG signal to four levels, and thus Poincaré plot is shown for coefficients. Due to patterns of the Poincaré plot, novel geometrical features are computed from EEG signals. The computed features are involved in standard descriptors of 2-D projection (STD), summation of triangle area using consecutive points (STA), as well as summation of shortest distance from each point relative to the 45-degree line (SSHD), and summation of distance from each point relative to the coordinate center (SDTC). The proposed procedure leads to discriminate features between S and SF EEG signals. Thereafter, a binary particle swarm optimization (BPSO) is developed as an appropriate technique for feature selection. Finally, k-nearest neighbor (KNN) and support vector machine (SVM) classifiers are used for classifying features in S and SF groups. By developing the proposed method, we have archived classification accuracy of 99.3 % with respect to the proposed geometrical features. Accordingly, S and SF EEG signals have been classified. Also, Poincaré plot of SF EEG signals has more regular geometrical shapes as compared to S group. As a final remark, we notice that the Poincaré plot of coefficients in S EEG signals has occupied more space as compared to SF EEG signals (Tab. 3, Fig. 11, Ref. 57).
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