# Seminars

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

 Speaker: Debora Sesana Affiliation: University of Insubria - Como - When: Tuesday 13th March 2018 Time: 15:00:00 Abstract: 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  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.  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.  C. Garoni, S. Serra-Capizzano. Generalized Locally Toeplitz Sequences: Theory and Applications. Springer 2017.  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.  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 