Alec Jacopo Almo Schiavoni-Piazza

arXiv2026

The $*$-algebra of unbounded GLT: construction and theoretical foundations

In the present paper, we are concerned with the study of matrix-sequences arising from the discretization of PDEs and FDEs on domains $Ω\subset \mathbb{R}^d$ with finite measure. When $Ω$ is either a hypercube or a bounded domain, the theory of Generalized Locally Toeplitz (GLT) sequences and of reduced GLT sequences cover the spectral analysis of the matrix-sequences derived from the approximation of the continuous problem. This work aims to extend the machinery and tools of the GLT apparatus to the case of unbounded domains with finite measure. For any unbounded domain $Ω\subset \mathbb{R}^d$ with finite measure, we define a new class of sequences, which we call unbounded GLT, and study their spectral properties.

math.NA↗

arXiv2024

Can the a.c.s. notion and the GLT theory handle approximated PDEs/FDEs with either moving or unbounded domains?

In the current note we consider matrix-sequences $\{B_{n,t}\}_n$ of increasing sizes depending on $n$ and equipped with a parameter $t>0$. For every fixed $t>0$, we assume that each $\{B_{n,t}\}_n$ possesses a canonical spectral/singular values symbol $f_t$ defined on $D_t\subset \R^{d}$ of finite measure, $d\ge 1$. Furthermore, we assume that $ \{ \{ B_{n,t}\} : \, t > 0 \} $ is an approximating class of sequences (a.c.s.) for $ \{ A_n \} $ and that $ \bigcup_{t > 0} D_t = D $ with $ D_{t + 1} \supset D_t $. Under such assumptions and via the notion of a.c.s, we prove results on the canonical distributions of $ \{ A_n \} $, whose symbol, when it exists, can be defined on the, possibly unbounded, domain $D$ of finite or even infinite measure. We then extend the concept of a.c.s. to the case where the approximating sequence $ \{ B_{n,t}\}_n $ has possibly a different dimension than the one of $ \{ A_n\} $. This concept seems to be particularly natural when dealing, e.g., with the approximation both of a partial differential equation (PDE) and of its (possibly unbounded, or moving) domain $D$, using an exhausting sequence of domains $\{ D_t \}$. Examples coming from approximated PDEs/FDEs with either moving or unbounded domains are presented in connection with the classical and the new notion of a.c.s., while numerical tests and a list of open questions conclude the present work.

math.NA↗