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- 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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| 2. |
- 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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| 3. |
- Bogoya, Manuel, et al.
(author)
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On the extreme eigenvalues and asymptotic conditioning of a class of Toeplitz matrix-sequences arising from fractional problems
- 2022
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In: Linear and multilinear algebra. - : Informa UK Limited. - 0308-1087 .- 1563-5139. ; , s. 1-12
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Journal article (peer-reviewed)abstract
- The analysis of the spectral features of a Toeplitz matrix-sequence {Tn(f)}n∈N, generated by the function f∈L1([−π,π]), real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when f is nonnegative a.e. Here we consider a novel type of problem arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form Tn=c0Tn(f0)+c1hhTn(f1)+c2h2hTn(f2)+⋯+cn−1h(n−1)hTn(fn−1),c0,c1,…,cn−1 belong to the interval [c∗,c∗] with c∗⩾c∗>0 independent of n, h=1n, fj∼gj, and gj(θ)=|θ|2−jh for every j=0,…,n−1. For nonnegative functions or sequences, the notation s(x)∼t(x) means that there exist positive constants c, d, independent of the variable x in the definition domain such that cs(x)⩽t(x)⩽ds(x) for any x. Since the resulting generating function depends on n, the standard theory cannot be applied and the analysis has to be performed using new ideas. Few selected numerical experiments are presented, also in connection with matrices that come from distributed-order FDE problems, and the adherence with the theoretical analysis is discussed, together with open questions and future investigations.
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| 11. |
- Fuentes, Rafael Diaz, et al.
(author)
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A ømega-Circulant Regularization for Linear Systems Arising in Interpolation with Subdivision Schemes
- 2021
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Reports (other academic/artistic)abstract
- In the curve interpolation with primal and dual form of stationary subdivision schemes, the computation of the relevant parameters amounts in solving special banded circulant linear systems, whose coefficients are related to quantities arising from the used stationary subdivision schemes. In some important cases it happens that the associated generating function, which is a special Laurent polynomial called symbol, has zeros on the unit complex circle of the form exp(2\pi ı j/n), where n is the size of the matrix, ı^2=-1, and j is a non-negative integer bounded by n-1. When this situation occurs the discrete problem is ill-posed simply because the circulant coefficient matrix is singular and the problem has no solution (or infinitely many). Standard and nonstandard regularization techniques such as least square solutions or Tikhonov regularization have been tried, but the quality of the solution is not good enough. In this work we propose a structure preserving regularization in which the circulant matrix is replaced by the ømega-circulant counterpart, with ømega being a complex parameter. A careful choice of ømega close to 1 (recall that the set of 1-circulants coincides with standard circulant matrices) allows to solve satisfactorily the problem of the ill-posedness, even if the quality of the reconstruction is reasonable only in a restricted number of cases. Numerical experiments and further algorithmic proposals are presented and critically discussed.
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| 13. |
- Garoni, Carlo, et al.
(author)
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Block generalized locally Toeplitz sequences : From the theory to the applications
- 2018
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In: Axioms. - : MDPI AG. - 2075-1680. ; 7, s. 49:1-29
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Journal article (peer-reviewed)abstract
- The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}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 DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations.
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| 14. |
- Mazza, Mariarosa, et al.
(author)
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Algebra preconditionings for 2D Riesz distributed-order space-fractional diffusion equations on convex domains
- 2024
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In: Numerical Linear Algebra with Applications. - : John Wiley & Sons. - 1070-5325 .- 1099-1506. ; 31:3
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Journal article (peer-reviewed)abstract
- When dealing with the discretization of differential equations on non-rectangular domains, a careful treatment of the boundary is mandatory and may result in implementation difficulties and in coefficient matrices without a prescribed structure. Here we examine the numerical solution of a two-dimensional constant coefficient distributed-order space-fractional diffusion equation with a nonlinear term on a convex domain. To avoid the aforementioned inconvenience, we resort to the volume-penalization method, which consists of embedding the domain into a rectangle and in adding a reaction penalization term to the original equation that dominates in the region outside the original domain and annihilates the solution correspondingly. Thanks to the volume-penalization, methods designed for problems in rectangular domains are available for those in convex domains and by applying an implicit finite difference scheme we obtain coefficient matrices with a 2-level Toeplitz structure plus a diagonal matrix which arises from the penalty term. As a consequence of the latter, we can describe the asymptotic eigenvalue distribution as the matrix size diverges as well as estimate the intrinsic asymptotic ill-conditioning of the involved matrices. On these bases, we discuss the performances of the conjugate gradient with circulant and τ-preconditioners and of the generalized minimal residual with split circulant and τ-preconditioners and conduct related numerical experiments.
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