The GLT theory: perspectives and beyond

The idea of Generalized Locally Toeplitz (GLT) sequences has been introduced [94, 95] 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 [27, 102] in terms of a function (the symbol) that can be easily identified, via specific topological notions [101, 92] (see also [12, 14, 52, 4] and references therein). It goes back to the seminal paper by Tilli [101], where the most important tools were also established, while related concepts were anticipated in [88]. 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 (see [29]).
For every $r, d \ge 1$ 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, Linear Positive Operators and Korovkin theory, spectral detection of branches, fast ’matrix-less’ computation of eigenvalues, stability issues, and challenges such as the GLT use in tensors, stochastic, machine learning algorithms. We will discuss also the impact and the further potential of the theory with special attention to new tools and to new directions as those based on symmetrization tricks [81, 44, 45, 72, 73, 61, 62], on the extra-dimensional approach [103, 74, 4, 11], and on blocking structures/operations [48, 3].
The seminal papers on the considered spectral theory are [29, 102, 101, 94, 95].
The GLT theory is organized in books and revue papers [52, 53, 54, 15, 16, 17].
The other references contain applications and theoretical results, where either the GLT machinery has been used, or new related techniques are introduced for the asymptotic eigenvalue analysis in a non-Hermitian/non-normal setting [6, 18, 25, 39, 44, 61, 62, 80, 83, 84].
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