A note on eigenvalues and singular values of variable Toeplitz matrices and matrix-sequences, with application to variable two-step BDF approximations to parabolic equations

TL;DR

The paper develops a GLT-based framework for a broad class of variable Toeplitz matrix-sequences arising from nonuniform-grid, two-step BDF discretizations of parabolic equations. By identifying explicit GLT symbols $\kappa(x,\theta)$ and their real parts, it proves $\{L_n\}_n\sim_\sigma\kappa$ and $\{S_n\}_n\sim_\lambda\Re(\kappa)$, and derives a first-order momentary expansion $f_n(x,\theta)=\kappa(\theta)+h l(x,\theta)$ to capture grid-induced corrections. A decomposition approach into low-rank blocks yields practical positivity conditions and links to matrix-valued LPOs, with both analytic results and extensive numerical evidence on distributional behavior and extremal eigenvalues. The findings enable robust spectral predictions and solver design for variable-grid parabolic problems, and point to open questions and potential refinements of positivity bounds under nonuniform grids.

Abstract

Here, we consider a more general class of matrix-sequences and we prove that they belong to the maximal $*$-algebra of generalized locally Toeplitz (GLT) matrix-sequences. Then, we identify the associated GLT symbols and GLT momentary symbols in the general setting and in the specific case, by providing in both cases a spectral and singular value analysis. More specifically, we use the GLT tools in order to study the asymptotic behaviour of the eigenvalues and singular values of the considered BDF matrix-sequences, in connection with the given non-uniform grids. Numerical examples, visualizations, and open problems end the present work.

Figures (9)

• Figure 1: The graph of the generating function (and GLT symbol) of $\mathbb{S}_n(r)=\Re(\mathbb{L}_n(r))=T_{n-1}(\Re(\kappa_r(\theta)))$ for $r=1$. The smoothing parameters are $\delta=0.9672$, $\eta=-0.1793$. The symbol is strictly positive with a minimum value $0.10605$ at zero.
• Figure 2: Generating functions (and GLT symbols) of $\mathbb{S}_n(r)=\Re(\mathbb{L}_n(r))=T_{n-1}(\Re(\kappa_r(\theta)))$ for various values of $r$: the largest is the limit value in BDF-Akrivis i.e. $r=1.9398$
• Figure 3: The graph of the generating function (and GLT symbol) of $\mathbb{S}_n(r)=\Re(\mathbb{L}_n(r))=T_{n-1}(\Re(\kappa_r(\theta)))$ for $r=1.9398$ along with the graph of $\theta^2$. The smoothing parameters are $\delta=0.9672$, $\eta=-0.1793$.
• Figure 4: A closer look at the generating function (and GLT symbol) with same parameters as in Figure \ref{['fig:symbol_vs_xsquare_r19398_distance']} reveals its two minima symmetrically located with respect to $\theta=0$.
• Figure 5: $\phi(x)=x^2$, $\delta=1$, $\eta=-0.5$ Top: The singular values of the matrix for $n=80$ (left) and $n=120$ (right) and the absolute value of the GLT symbol of the matrix-sequence Bottom: The eigenvalues of the symmetrized matrix for $n=80$ (left) and $n=120$ (right) and the GLT symbol of the matrix-sequence

Theorems & Definitions (16)

• Definition 1
• Theorem 1
• Definition 2
• Definition 3
• Remark 1
• Definition 4: Toeplitz momentary symbols
• Definition 5: GLT momentary symbols
• Theorem 2
• Proof
• Theorem 3