| 1. |
- Adriani, Andrea, et al.
(författare)
-
Asymptotic Spectra of Large (Grid) Graphs with a Uniform Local Structure (Part I) : Theory
- 2020
-
Ingår i: Milan Journal of Mathematics. - : Springer Science and Business Media LLC. - 1424-9286 .- 1424-9294. ; 88:2, s. 409-454
-
Tidskriftsartikel (refereegranskat)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
|
|
| 2. |
- Adriani, Andrea, et al.
(författare)
-
Asymptotic spectra of large (grid) graphs with a uniform local structure, Part II : Numerical applications
- 2024
-
Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 437
-
Tidskriftsartikel (refereegranskat)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.
|
|
| 3. |
|
|
| 4. |
- Adriani, Andrea, et al.
(författare)
-
Generalized Locally Toeplitz matrix-sequences and approximated PDEs on submanifolds : the flat case
- 2023
-
Ingår i: Linear and multilinear algebra. - : Informa UK Limited. - 0308-1087 .- 1563-5139. ; , s. 1-23
-
Tidskriftsartikel (refereegranskat)abstract
- In the present paper, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions where the operator is the Laplace-Beltrami operator Δ over Ω¯, Ω being an open and bounded submanifold of Rν, ν=2,3. We will take into consideration the classical Pk Finite Elements, in the case of Friedrichs-Keller triangulations, leading to sequences of matrices of increasing size. We are interested in carrying out a spectral analysis of the resulting matrix-sequences. The tools for our derivations are mainly taken from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz (GLT) matrix-sequences. The current contribution is only quite an initial step, where a general programme is provided, with partial answers leading to further open questions: indeed the analysis is performed on special flat submanifolds and hence there is room for wide generalizations, with a final picture which is still unclear with respect to, e.g. the role of the submanifold curvature.
|
|
