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Blocking structures, approximation, and preconditioning

Nikos Barakitis, Marco Donatelli, Samuele Ferri, Valerio Loi, Stefano Serra-Capizzano, Rosita Luisa Sormani

TL;DR

The paper develops a GLT-based framework for block-structured matrices with block Toeplitz blocks, establishing asymptotic distributions for singular values (and eigenvalues when the symbol is Hermitian) and showing how simplified block-structured approximations preserve these distributions while allowing $O(n\log n)$ solves. It then designs preconditioners $S_n$ that render $S_n^{-1}A_n$ (or $A_nS_n^{\dagger}$) strongly cluster at $1$, enabling efficient Krylov solvers. The approach is validated through extensive numerical tests on three block-structure groups and fractional-diffusion-inspired problems, demonstrating zero- and $1$-clustering and substantial reductions in iteration counts. The work provides a practical, theoretically grounded strategy for preconditioning large block Toeplitz-like systems and outlines clear paths toward multilevel extensions and hybrid multigrid-Krylov methods.

Abstract

We consider block-structured matrices $A_n$, where the blocks are of (block) unilevel Toeplitz type with $s\times t$ matrix-valued generating functions. Under mild assumptions on the size of the (rectangular) blocks, the asymptotic distribution of the singular values of {the} associated matrix-sequences is identified and, when the related singular value symbol is Hermitian, it coincides with the spectral symbol. Building on the theoretical derivations, we approximate the matrices with simplified block structures that show two important features: a) the related simplified matrix-sequence has the same distributions as $\{A_{n}\}_{n}$; b) a generic linear system involving the simplified structures can be solved in $O(n\log n)$ arithmetic operations. The two key properties a) and b) suggest a natural way for preconditioning a linear system with coefficient matrix $A_n$. Under mild assumptions, the singular value analysis and the spectral analysis of the preconditioned matrix-sequences is provided, together with a wide set of numerical experiments.

Blocking structures, approximation, and preconditioning

TL;DR

The paper develops a GLT-based framework for block-structured matrices with block Toeplitz blocks, establishing asymptotic distributions for singular values (and eigenvalues when the symbol is Hermitian) and showing how simplified block-structured approximations preserve these distributions while allowing solves. It then designs preconditioners that render (or ) strongly cluster at , enabling efficient Krylov solvers. The approach is validated through extensive numerical tests on three block-structure groups and fractional-diffusion-inspired problems, demonstrating zero- and -clustering and substantial reductions in iteration counts. The work provides a practical, theoretically grounded strategy for preconditioning large block Toeplitz-like systems and outlines clear paths toward multilevel extensions and hybrid multigrid-Krylov methods.

Abstract

We consider block-structured matrices , where the blocks are of (block) unilevel Toeplitz type with matrix-valued generating functions. Under mild assumptions on the size of the (rectangular) blocks, the asymptotic distribution of the singular values of {the} associated matrix-sequences is identified and, when the related singular value symbol is Hermitian, it coincides with the spectral symbol. Building on the theoretical derivations, we approximate the matrices with simplified block structures that show two important features: a) the related simplified matrix-sequence has the same distributions as ; b) a generic linear system involving the simplified structures can be solved in arithmetic operations. The two key properties a) and b) suggest a natural way for preconditioning a linear system with coefficient matrix . Under mild assumptions, the singular value analysis and the spectral analysis of the preconditioned matrix-sequences is provided, together with a wide set of numerical experiments.
Paper Structure (24 sections, 14 theorems, 90 equations, 9 figures, 10 tables)

This paper contains 24 sections, 14 theorems, 90 equations, 9 figures, 10 tables.

Table of Contents

  1. Keywords.
  2. MSC codes.
  3. Introduction
  4. Problem setting and general theoretical tools
  5. Basic structured matrix-sequences and (rectangular) GLT sequences
  6. Structured matrices with block unilevel Toeplitz blocks
  7. Approximation and preconditioning
  8. Numerical Tests
  9. Matrix-sequences and example groups
  10. Group 1: $2 \times 2$ block case with scalar-valued $f_{i,j}$.
  11. Group 2: $2 \times 2$ block case with rectangular matrix-valued $f_{i,j}$.
  12. Group 3: $3 \times 3$ block case with square matrix-valued $f_{i,j}$
  13. Numerical results on distribution, clustering, and efficiency of preconditioning
  14. Zero clustering analysis
  15. Methodology:
  16. ...and 9 more sections

Key Result

Theorem 2.3

Let $\{X_n\}_n$ be a given (rectangular) matrix-sequence with $X_n$ of order $n^{(1)} \times n$, $n^{(1)} \geq n$. Let $P_n \in \mathbb{C}^{n \times n'}$, $P_{n^{(1)}} \in \mathbb{C}^{n^{(1)} \times n^{(1)'}}$ be two compression matrices with $n' < n$, ${n^{(1)'}} < n^{(1)}$ and $P_n^*P_n = I_{n'}$, i.e. $n = n' + o(n)$, ${n^{(1)}} = {n^{(1)'}} + o({n^{(1)}})$, we have

Figures (9)

  • Figure 1: The cluster at $0$ of the singular values of $\{A_n-S_n\}_n$ for Group 1. The first row is case (a), the second row is case (b), and the third row is case (c). The first column is for $\eta=100$ while the second column is for $\eta=500$.
  • Figure 2: The cluster at $0$ of the singular values of $\{A_n-S_n\}_n$ for Group 2. The first row is case (a), the second row is case (b), and the third row is case (c). The first column is for $\eta=100$ while the second column is for $\eta=500$.
  • Figure 3: The cluster at $0$ of the singular values of $\{A_n-S_n\}_n$ for Group 3. The plot on the left is for $\eta=100$ while the one on the right is for $\eta=500$.
  • Figure 4: The cluster at $1$ of the singular values of $\{S_n^{-1}A_n\}_n$ for Group 1. The first row is case (a), the second row is case (b), and the third row is case (c). The first column is for $\eta=100$ while the second column is for $\eta=500$.
  • Figure 5: The cluster at $1$ of the singular values of $\{A_nS_n^{\dag}\}_n$ for Group 2. The first row is case (a), the second row is case (b), and the third row is case (c). The first column is for $\eta=100$ while the second column is for $\eta=500$.
  • ...and 4 more figures

Theorems & Definitions (27)

  • Definition 2.1: GLT-blocks-d-dimGSIGSIIMR0890515TyZ
  • Definition 2.2
  • Theorem 2.3: pre-prequelprequel
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10: Se-Ti-LPO
  • ...and 17 more