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On the Spectral Clustering of a Class of Multigrid Preconditioners

Jose Pablo Lucero Lorca

TL;DR

The paper proposes an algebraic two-level framework for block-partitioned, nonsymmetric systems and analyzes how smoothing and coarse-grid correction interact on a 2x2 invariant subspace basis. It derives a scalar relaxation response $r_m(\lambda)$ on each subspace and identifies smoothing parameters $\{\alpha_i\}$ that render $r_m(\lambda)$ independent of $\lambda$, achieving perfect spectral clustering to two values. Consequently, the preconditioned operator has spectrum $\{1,\,1-\frac{1}{(2m+1)^2}\}$, enabling a Krylov method to converge in two iterations, effectively acting as a direct solver. The method is demonstrated on finite-difference Poisson and discontinuous Galerkin discretizations, with discussions on extending to nonsymmetric cases where the field of values may widen, highlighting practical limits and guidance for robust implementation.

Abstract

This paper studies a common two-level multigrid construction for block-structured linear systems and identifies a simple way to describe how its smoothing and coarse-grid components interact. By examining the method through a collection of small coupling modes, we show that its behavior can be captured by a single scalar quantity for each mode. The main result is an explicit choice of smoothing parameters that makes all modes respond in the same way, causing the nontrivial eigenvalues of the preconditioned operator to collapse to a single value. This gives a clear and self-contained description of the ideal version of the method and provides a concrete target for designing related schemes. Although the exact spectral collapse requires ideal components, we also show that the same construction naturally produces operators that resemble those used in practical discretizations. Examples from finite-difference and discontinuous Galerkin settings illustrate how the ideal parameters can be used in practice.

On the Spectral Clustering of a Class of Multigrid Preconditioners

TL;DR

The paper proposes an algebraic two-level framework for block-partitioned, nonsymmetric systems and analyzes how smoothing and coarse-grid correction interact on a 2x2 invariant subspace basis. It derives a scalar relaxation response on each subspace and identifies smoothing parameters that render independent of , achieving perfect spectral clustering to two values. Consequently, the preconditioned operator has spectrum , enabling a Krylov method to converge in two iterations, effectively acting as a direct solver. The method is demonstrated on finite-difference Poisson and discontinuous Galerkin discretizations, with discussions on extending to nonsymmetric cases where the field of values may widen, highlighting practical limits and guidance for robust implementation.

Abstract

This paper studies a common two-level multigrid construction for block-structured linear systems and identifies a simple way to describe how its smoothing and coarse-grid components interact. By examining the method through a collection of small coupling modes, we show that its behavior can be captured by a single scalar quantity for each mode. The main result is an explicit choice of smoothing parameters that makes all modes respond in the same way, causing the nontrivial eigenvalues of the preconditioned operator to collapse to a single value. This gives a clear and self-contained description of the ideal version of the method and provides a concrete target for designing related schemes. Although the exact spectral collapse requires ideal components, we also show that the same construction naturally produces operators that resemble those used in practical discretizations. Examples from finite-difference and discontinuous Galerkin settings illustrate how the ideal parameters can be used in practice.

Paper Structure

This paper contains 18 sections, 8 theorems, 78 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Problem setting
  4. Two-level preconditioner
  5. Error operator
  6. Algebraic two-block inverse identity and its link to the two-level scheme
  7. Spectral optimization
  8. Preliminaries
  9. Invariant subspaces and scalar reduction
  10. Spectral clustering
  11. Diagonalization
  12. Diagonalization and explicit $r_m(\lambda)$
  13. Perfect clustering for arbitrary $m$
  14. Spectral consequence for Algorithm \ref{['alg:twolevel']}
  15. Clustering in practice
  16. ...and 3 more sections

Key Result

Proposition 3

For any $\alpha\in\mathbb{R}$ and any eigenpair $(w,\lambda)$ of $T$, possibly complex, the smoother $E_s(\alpha)$ defined in eq:Es satisfies

Figures (2)

  • Figure 1: Spectral radius $\rho(E_m)$ versus the number of smoothing steps $m$ for the two–dimensional Laplacian preconditioned system on a $16\times16$ interior finite–difference grid. Gray curves: constant $\alpha\in\{0.1,\dots,1.0\}$; dashed: constant $\alpha=2/3$; thick curve with filled markers: relaxation parameters given by Theorem \ref{['thm:alg1-clustering-final']}.
  • Figure 2: Field of values and spectra for the original matrix $B$ and the three preconditioned operators used in the non-normality experiment.

Theorems & Definitions (20)

  • Definition 1: Coupling operator
  • Definition 2: Invariant subspaces
  • Proposition 3: Invariance under the smoother
  • Proof 1
  • Proposition 4: Invariance under the coarse correction
  • Proof 2
  • Lemma 5: Matrix form in the basis $\{v_1,v_2\}$
  • Proof 3
  • Definition 6: Symmetric scaling
  • Remark 1
  • ...and 10 more