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Domain decomposition methods and preconditioning strategies using generalized locally Toepltiz tools: proposals, analysis, and numerical validation

Abdessadek Rifqui, Ahmed Ratnani, Stefano Serra-Capizzano

TL;DR

The paper analyzes the spectral behavior of additive and multiplicative Schwarz preconditioners for domain-decomposed discretizations using generalized locally Toeplitz (GLT) sequences. It derives GLT symbols for convergence factors and shows that, for fixed numbers of blocks, Schwarz-type preconditioners preserve the underlying GLT symbol $oldsymbol{ kappa}$ and drive preconditioned systems $oxed{P_n^{-1}A_n}$ to cluster at $1$, enabling rapid Krylov convergence. It also treats restricted variants (RAS/RMS) and provides a rigorous GLT-based justification for their performance, including symmetric and overlapping cases. Numerical experiments in 1D and 2D IgA/FE settings corroborate the theory, demonstrating eigenvalue clustering and accelerated convergence across a range of discretizations and overlaps.

Abstract

In the current work we present a spectral analysis of the additive and multiplicative Schwarz methods within the framework of domain decomposition techniques, by investigating the spectral properties of these classical Schwarz preconditioning matrix-sequences, with emphasis on their convergence behavior and on the effect of transmission operators. In particular, after a general presentation of various options, we focus on restricted variants of the Schwarz methods aimed at improving parallel efficiency, while preserving their convergence features. In order to rigorously describe and analyze the convergence behavior, we employ the theory of generalized locally Toeplitz (GLT) sequences, which provides a robust framework for studying the asymptotic spectral distribution of the discretized operators arising from Schwarz iterations. By associating each operator sequence with the appropriate GLT symbol, we derive explicit expressions for the GLT symbols of the convergence factors, for both additive and multiplicative Schwarz methods. The GLT-based spectral approach offers a unified and systematic understanding of how the spectrum evolves with mesh refinement and overlap size (in the algebraic case). Our analysis not only deepens the theoretical understanding of classical Schwarz methods, but also establishes a foundation for examining future restricted or hybrid Schwarz variants using symbolic spectral tools. These results enable the prediction of the remarkable efficiency of block Jacobi/Gauss--Seidel and block additive/multiplicative Schwarz preconditioners for GLT sequences, as further illustrated through a wide choice of numerical experiments.

Domain decomposition methods and preconditioning strategies using generalized locally Toepltiz tools: proposals, analysis, and numerical validation

TL;DR

The paper analyzes the spectral behavior of additive and multiplicative Schwarz preconditioners for domain-decomposed discretizations using generalized locally Toeplitz (GLT) sequences. It derives GLT symbols for convergence factors and shows that, for fixed numbers of blocks, Schwarz-type preconditioners preserve the underlying GLT symbol and drive preconditioned systems to cluster at , enabling rapid Krylov convergence. It also treats restricted variants (RAS/RMS) and provides a rigorous GLT-based justification for their performance, including symmetric and overlapping cases. Numerical experiments in 1D and 2D IgA/FE settings corroborate the theory, demonstrating eigenvalue clustering and accelerated convergence across a range of discretizations and overlaps.

Abstract

In the current work we present a spectral analysis of the additive and multiplicative Schwarz methods within the framework of domain decomposition techniques, by investigating the spectral properties of these classical Schwarz preconditioning matrix-sequences, with emphasis on their convergence behavior and on the effect of transmission operators. In particular, after a general presentation of various options, we focus on restricted variants of the Schwarz methods aimed at improving parallel efficiency, while preserving their convergence features. In order to rigorously describe and analyze the convergence behavior, we employ the theory of generalized locally Toeplitz (GLT) sequences, which provides a robust framework for studying the asymptotic spectral distribution of the discretized operators arising from Schwarz iterations. By associating each operator sequence with the appropriate GLT symbol, we derive explicit expressions for the GLT symbols of the convergence factors, for both additive and multiplicative Schwarz methods. The GLT-based spectral approach offers a unified and systematic understanding of how the spectrum evolves with mesh refinement and overlap size (in the algebraic case). Our analysis not only deepens the theoretical understanding of classical Schwarz methods, but also establishes a foundation for examining future restricted or hybrid Schwarz variants using symbolic spectral tools. These results enable the prediction of the remarkable efficiency of block Jacobi/Gauss--Seidel and block additive/multiplicative Schwarz preconditioners for GLT sequences, as further illustrated through a wide choice of numerical experiments.
Paper Structure (10 sections, 7 theorems, 87 equations, 29 figures, 42 tables)

This paper contains 10 sections, 7 theorems, 87 equations, 29 figures, 42 tables.

Table of Contents

  1. Introduction
  2. Iterative methods with classical block preconditioners
  3. Particular case: $\boldsymbol{\nu = 2}$.
  4. Overview of the Theory of GLT Sequences
  5. Schwarz Methods with GLT Preconditioning
  6. One-Level GLT Sequences
  7. Numerical experiments
  8. The 1D setting
  9. The 2D Setting
  10. Conclusions and perspectives

Key Result

Lemma 3.2

Let $f : D \subseteq \mathbb{R}^k \to \mathbb{C}^{s \times s}$ be a measurable function, and let $g : \mathbb{C}^s \to \mathbb{C}$ be continuous and symmetric in its $s$ arguments, that is, Then, the mapping is well-defined (independently of the ordering of the eigenvalues of $f(x)$) and measurable. As a consequence:

Figures (29)

  • Figure 1: A $320 \times 320$ band matrix partitioned into $4 \times 4$ blocks, with block sizes ${n_1} = {n_4} = 130$ and ${n_2} = {n_3} = 30$.
  • Figure 2: Comparison of iterative and preconditioner methods using GMRES. The left plot shows the residuals for standard iterative methods, while the right plot shows residuals for preconditioned methods. The simulation is performed with matrix size $n=1280$, number of subdomains $\nu=2$, and overlap $o=30$.
  • Figure 3: Comparison of eigenvalues for different block methods with matrix size $n=40$, number of subdomains $\nu=2$, and overlap $o=10$. The markers represent the eigenvalues of $A_n$, $P_n^{-1}$, and $P_n^{-1}A_n$ for each method.
  • Figure 4: Comparison between the GLT symbol $(2 - 2 \cos \theta)$ and the eigenvalues of the block additive Schwarz preconditioner $P_{\mathrm{BAS},n}^{(\nu)}$ without overlap. Left: $n=400$, $\nu=2$. Right: $n=800$, $\nu=2$.
  • Figure 5: Comparison between the GLT symbol $(2 - 2 \cos \theta)$ and the eigenvalues of the block additive Schwarz preconditioner $P_{\mathrm{BAS},n}^{(\nu)}$ with overlap $o=10$. Left: $n=400$, $\nu=2$. Right: $n=800$, $\nu=2$.
  • ...and 24 more figures

Theorems & Definitions (26)

  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1: Singular value and spectral distribution of a sequence of matrices
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 5.1
  • ...and 16 more