Stefano Serra-Capizzano: Maximal GLT *-algebras: theoretical framework and applications in the days of AI

Abstract: The idea of Generalized Locally Toeplitz (GLT) sequences has been introduced as a generalization both of classical Toeplitz sequences and of variable coefficient differential operators and, for every sequence of the class, it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function (the symbol) that can be easily identified.

This generalizes the notion of a symbol for differential operators (discrete and continuous) or for Toeplitz sequences, where for the latter it is identified through the Fourier coefficients and is related to the classical Fourier Analysis [1,2,3].

For every positive integers \(r\), \(d\) the \(r\)-block \(d\)-level GLT class has nice *-algebra features and indeed it has been proven that it is stable under linear combinations, products, and inversion when the sequence which is inverted shows a sparsely vanishing symbol (sparsely vanishing symbol = a symbol whose minimal singular value vanishes at most in a set of zero Lebesgue measure). Furthermore, the GLT *-algebras virtually include any approximation of partial differential equations (PDEs), fractional differential equations (FDEs), integro-differential equations (IDEs) by local methods (Finite Difference, Finite Element, Isogeometric Analysis etc) and, based on this, we demonstrate that our results on GLT sequences can be used in a PDE/FDE/IDE setting in various directions, including preconditioning, multigrid, spectral detection of branches, fast 'matrix-less' computation of eigenvalues, stability issues, asymptotic low-rank structures, and challenges such as the GLT use in geometric means, tensors, stochastic, machine learning algorithms; see [4,5,6,7,8] and references therein for a systematic treatment and [9] for a programme for the future.

We discuss these issues also in connection with the tools provided by the new models of Artificial Intelligence (AI) and after that a GLT related conjecture by Barbarino and myself [10] was solved in a very interesting manner via AI. As a general initial comment on this matter, one can safely say that the advancement of AI will upheave the role of mathematicians and academics in general and in a profound way. Regarding Research probably, among mathematicians, those using imagination can have advantages, while those who base their scientific success solely on technical skills risk being overwhelmed.

References:

[1] P. Tilli. Locally Toeplitz sequences: spectral properties and applications. Linear Algebra Appl. 278 (1998), no. 1-3, 91-120.

[2] S. Serra Capizzano. Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl. 366 (2003), 371-402.

[3] S. Serra-Capizzano. The GLT class as a generalized Fourier analysis and applications. Linear Algebra Appl. 419 (2006), no. 1, 180-233.

[4] C. Garoni, S. Serra-Capizzano. Generalized locally Toeplitz sequences: theory and applications. Vol. I. Springer, Cham, 2017.

[5] C. Garoni, S. Serra-Capizzano. Generalized locally Toeplitz sequences: theory and applications. Vol. II. Springer, Cham, 2018.

[6] G. Barbarino, C. Garoni, S. Serra-Capizzano. Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case. Electron. Trans. Numer. Anal. 53 (2020), 28-112.

[7] G. Barbarino, C. Garoni, S. Serra-Capizzano. Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case. Electron. Trans. Numer. Anal. 53 (2020), 113-216.

[8] G. Barbarino. A systematic approach to reduced GLT. BIT 62 (2022), no. 3, 681-743.

[9] S. Serra-Capizzano. Spectral theory of matrix-sequences: perspectives of the GLT analysis and beyond. Proc. Numerical Analysis and Applications to Data Science - N2ADS 2025, Springer Proceedings on Mathematics and Statistics (January 2027), in press.

[10] G. Barbarino, S. Serra-Capizzano. Non-Hermitian perturbations of Hermitian matrix-sequences and applications to the spectral analysis of the numerical approximation of partial differential equations. Numer. Linear Algebra Appl. 27 (2020), no. 3, e2286, 31 pp.