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On Optimal Algebraic Multigrid Methods

Luis García Ramos, Reinhard Nabben

TL;DR

The paper addresses constructing optimal interpolation operators for algebraic multigrid applied to Hermitian positive definite systems. It leverages a spectrum-based characterization of the two-grid error-propagation operator to derive interpolation operators that minimize the spectral radius (and thus the $A$-norm of the error) and to show these operators also minimize the condition number of the preconditioned system. For symmetric two-grid methods with pre- and post-smoothing, the $A$-norm- and spectral-radius-optimal interpolations coincide; for post-smoothing only, they can differ, but the spectrum-based approach still yields a principled operator based on the smallest generalized eigenvalues of $X^{-1}A$. The results extend to non-symmetric cases with scaling and provide explicit eigenvector-based interpolations, including bounds via $(I - X^{-1}A)^2$, thereby offering a cohesive, spectrum-driven route to optimal AMG components in the HPD setting. Overall, the work connects optimality across $A$-norm, spectral radius, and condition number within an abstract multigrid perspective.

Abstract

In this note we present an alternative way to obtain optimal interpolation operators for two-grid methods applied to Hermitian positive definite linear systems. Falgout and Vassilevski in [SIAM J. Numer. Anal, 42 (2004), pp. 1669-1693] and Zikatanov [Numer. Linear Algebra Appl., 15 (2008), pp. 439-454] have characterized the $A$-norm of the error propagation operator of algebraic multigrid methods. These results have been recently used by Xu and Zikatanov [Acta Numer., 26 (2017), pp. 591-721] and Brannick, Cao et al. [SIAM J. Sci. Comp, 40 (2018), pp. 591-721] to determine optimal interpolation operators. Here we use a characterization not of the $A$-norm but of the spectrum of the error propagation operator of two-grid methods, which was proved by García Ramos, Nabben and Kehl and holds for arbitrary matrices. For Hermitian positive definite systems this result leads to optimal interpolation operators with respect to the $A$-norm in a short way, moreover, it also leads to optimal interpolation operators with respect to the spectral radius. For the symmetric two-grid method (with pre- and post-smoothing) the optimal interpolation operators are the same. But for a two-grid method with only post-smoothing the optimal interpolations (and hence the optimal algebraic multigrid methods) can be different. Moreover, using the characterization of the spectrum, we can show that the found optimal interpolation operators are also optimal with respect to the condition number of the multigrid preconditioned system.

On Optimal Algebraic Multigrid Methods

TL;DR

The paper addresses constructing optimal interpolation operators for algebraic multigrid applied to Hermitian positive definite systems. It leverages a spectrum-based characterization of the two-grid error-propagation operator to derive interpolation operators that minimize the spectral radius (and thus the -norm of the error) and to show these operators also minimize the condition number of the preconditioned system. For symmetric two-grid methods with pre- and post-smoothing, the -norm- and spectral-radius-optimal interpolations coincide; for post-smoothing only, they can differ, but the spectrum-based approach still yields a principled operator based on the smallest generalized eigenvalues of . The results extend to non-symmetric cases with scaling and provide explicit eigenvector-based interpolations, including bounds via , thereby offering a cohesive, spectrum-driven route to optimal AMG components in the HPD setting. Overall, the work connects optimality across -norm, spectral radius, and condition number within an abstract multigrid perspective.

Abstract

In this note we present an alternative way to obtain optimal interpolation operators for two-grid methods applied to Hermitian positive definite linear systems. Falgout and Vassilevski in [SIAM J. Numer. Anal, 42 (2004), pp. 1669-1693] and Zikatanov [Numer. Linear Algebra Appl., 15 (2008), pp. 439-454] have characterized the -norm of the error propagation operator of algebraic multigrid methods. These results have been recently used by Xu and Zikatanov [Acta Numer., 26 (2017), pp. 591-721] and Brannick, Cao et al. [SIAM J. Sci. Comp, 40 (2018), pp. 591-721] to determine optimal interpolation operators. Here we use a characterization not of the -norm but of the spectrum of the error propagation operator of two-grid methods, which was proved by García Ramos, Nabben and Kehl and holds for arbitrary matrices. For Hermitian positive definite systems this result leads to optimal interpolation operators with respect to the -norm in a short way, moreover, it also leads to optimal interpolation operators with respect to the spectral radius. For the symmetric two-grid method (with pre- and post-smoothing) the optimal interpolation operators are the same. But for a two-grid method with only post-smoothing the optimal interpolations (and hence the optimal algebraic multigrid methods) can be different. Moreover, using the characterization of the spectrum, we can show that the found optimal interpolation operators are also optimal with respect to the condition number of the multigrid preconditioned system.

Paper Structure

This paper contains 4 sections, 8 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Optimal interpolation for Hermitian positive definite matrices
  3. The optimal interpolation with respect to the condition number
  4. Conclusion

Key Result

theorem 1

Let $A \in\mathbb{C}^{n\times n}$ be non-singular, and let $P \in\mathbb{C}^{n\times r}$ and $R \in\mathbb{C}^{r\times n}$ such that $RAP$ is non-singular. Moreover, let $M_1 \in\mathbb{C}^{n\times n}$ and $M_2 \in\mathbb{C}^{n\times n}$ be such that that the matrices $X$ in mgx and $RXP$ are non-si

Theorems & Definitions (13)

  • theorem 1
  • lemma 1
  • proof
  • theorem 2
  • proof
  • corollary 1
  • proof
  • corollary 2
  • proof
  • theorem 3
  • ...and 3 more