Block structured matrix-sequences and their spectral and singular value canonical distributions: a general theory

TL;DR

The paper develops a general Weyl-distribution theory for block-structured matrix-sequences with block unilevel Toeplitz blocks, enabling eigenvalue and singular value distributions to be characterized by a matrix-valued symbol $F=(f_{i,j})$. It shows that, under suitable block-size ratios and Hermitianity of $F$, ${A_n}_n$ distributes as $F$ in the eigenvalue and/or singular-value sense, with compression-extradimensional arguments used to relate complex block structures to their symbols. Numerical experiments across 2×2 and 3×3 block cases, including scalar, rectangular, and matrix-valued blocks tied to PDE discretizations, confirm the theoretical distributions and reveal only $o(d_n)$ outliers. The work also discusses practical implications for preconditioning and Krylov solvers, and outlines future directions toward multilevel generalizations and variable-coefficient problems, which would broaden applicability to multidimensional PDEs and fractional problems.

Abstract

In recent years more and more involved block structures appeared in the literature in the context of numerical approximations of complex infinite dimensional operators modeling real-world applications. In various settings, thanks the theory of generalized locally Toeplitz matrix-sequences, the asymptotic distributional analysis is well understood, but a general theory is missing when general block structures are involved. The central part of the current work deals with such a delicate generalization when blocks are of (block) unilevel Toeplitz type, starting from a problem of recovery with missing data. Visualizations, numerical tests, and few open problems are presented and critically discussed.

Figures (10)

• Figure 4.1: Comparison between the eigenvalue functions $\lambda_l(F)$, $l=1,\ldots,3$ and the eigenvalues of $A_n$, for $\nu=2$, $t,s=1$, $n_1=\eta$, $n_2=2\eta$, $\eta= 20, 40, 80$.
• Figure 4.2: Comparison between the eigenvalue functions $\lambda_l(F)$, $l=1,\ldots,3$ and the eigenvalues of $A_n$, for $\nu=2$, $t,s=1$, $n_1=\eta$, $n_2=2\eta+4$, $\eta= 20, 40, 80$.
• Figure 4.3: Comparison between the eigenvalue functions $\lambda_l(F)$, $l=1,\ldots,3$ and the eigenvalues of $A_n$, for $\nu=2$, $t,s=1$, $n_1=\eta$, $n_2=2\eta+\lceil{\sqrt{\eta}\,\rceil}$, $\eta= 25, 49, 81$.
• Figure 4.4: Comparison between the singular value functions $\sigma_l(F)$, $l=1,\ldots,3$ and the singular values of $A_n$, for $\nu=2$, $t,s=1$, $n_1=\eta$, $n_2=2\eta+\lceil{\sqrt{\eta}\,\rceil}$, $\eta= 256$.
• Figure 4.5: Comparison between the singular value functions $\sigma_l(F)$, $l=1,\ldots,3$ and the singular values of $A_n$, for $\nu=2$, $t,s=1$, $n_1=\eta$, $n_2=2\eta+\lceil{\sqrt{\eta}\,\rceil}$, $\eta= 25, 49, 81$.

Theorems & Definitions (27)

• Definition 2.1
• remark 1
• Theorem 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• Theorem 2.5
• remark 2
• Definition 3.1
• remark 3