Spectral distribution of sequences of structured matrices: GLT theory and applications

Debora Sesana
University of Insubria - Como -
Tuesday 13th March 2018
Aula Seminari 'Saleri' VI Piano MOX-Dipartimento di Matematica, Politecnico di Milano - Edificio 14
Any discretization of a given differential problem for some sequence of stepsizes h tending to zero leads to a sequence of systems of linear equations {Am xm = bm}, where the dimension of {Am} depends on h and tends to infinity for h going to 0. To properly face the solution of such linear systems, it is important to deeply understand the spectral properties of the matrices {Am} in order to construct efficient preconditioners and to study the convergence of applied iterative methods. The spectral distribution of a sequence of matrices is a fundamental concept. Roughly speak- ing, saying that the sequence of matrices {Am} is distributed as the function f means that the eigenvalues of Am behave as a sampling of f over an equispaced grid of the domain of f, at least if f is smooth enough. The function f is called the symbol of the sequence. Many distribution results are known for particular sequences of structured matrices: diagonal matrices, Toeplitz matrices, etc. and an approximation theory for sequences of matrices has been developed to deduce spectral distributions of “complicated” matrix sequences from the spectral distribution of “simpler” matrix sequences. In this respect, recently, has played a fundamental role the theory of Generalized Locally Toeplitz (GLT) sequences (introduced by Tilli (1998) and Serra-Capizzano (2002, 2006)), which allows to deduce the spectral properties of matrix sequences obtained as a combination (linear combinations, products, inversion) of Toeplitz matrices and diagonal matrices; to this category belong many stiffness matrices arising from the discretization, using various methods, of PDEs. We present the main concepts of this theory with some applications. This is a joint work with Stefano Serra-Capizzano and Carlo Garoni. References [1] C. Garoni, C. Manni, S. Serra-Capizzano, D. Sesana, H. Speleers. Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods. Mathematics of Computation 85 (2016), pp. 1639–1680. [2] C. Garoni, C. Manni, S. Serra-Capizzano, D. Sesana, H. Speleers. Lusin theorem, GLT sequences and matrix computations: an application to the spectral analysis of PDE discretization matrices. Journal of Mathematical Analysis and Applications, 446 (2017), pp. 365–382. [3] C. Garoni, S. Serra-Capizzano. Generalized Locally Toeplitz Sequences: Theory and Applications. Springer 2017. [4] C. Garoni, S. Serra-Capizzano, D. Sesana. Block Locally Toeplitz Sequences: Construction and Properties. Springer INdAM Series: proceeding volume of the Cortona 2017 meeting, submitted. [5] C. Garoni, S. Serra-Capizzano, D. Sesana. Block Generalized Locally Toeplitz Sequences: Topological Construction, Spectral Distribution Results, and Star-Algebra Structure. Springer INdAM Series: proceeding volume of the Cortona 2017 meeting, submitted. Contact: franca.calio@polimi.it