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Flexibly Enlarged Conjugate Gradient Methods

Sophie M. Moufawad

TL;DR

The Enlarged CG methods and their s-step versions converge in less iterations than the classical CG, but at the expense of requiring more memory storage than CG.

Abstract

Enlarged Krylov subspace methods and their s-step versions were introduced [7] in the aim of reducing communication when solving systems of linear equations Ax = b. These enlarged CG methods consist of enlarging the Krylov subspace by a maximum of t vectors per iteration based on the domain decomposition of the graph of A. As for the s-step versions, s iterations of the enlarged Conjugate Gradient methods are merged in one iteration. The Enlarged CG methods and their s-step versions converge in less iterations than the classical CG, but at the expense of requiring more memory storage than CG. Thus, in this paper we explore different options for reducing the memory requirements of these enlarged CG methods without affecting much their convergence.

Flexibly Enlarged Conjugate Gradient Methods

TL;DR

The Enlarged CG methods and their s-step versions converge in less iterations than the classical CG, but at the expense of requiring more memory storage than CG.

Abstract

Enlarged Krylov subspace methods and their s-step versions were introduced [7] in the aim of reducing communication when solving systems of linear equations Ax = b. These enlarged CG methods consist of enlarging the Krylov subspace by a maximum of t vectors per iteration based on the domain decomposition of the graph of A. As for the s-step versions, s iterations of the enlarged Conjugate Gradient methods are merged in one iteration. The Enlarged CG methods and their s-step versions converge in less iterations than the classical CG, but at the expense of requiring more memory storage than CG. Thus, in this paper we explore different options for reducing the memory requirements of these enlarged CG methods without affecting much their convergence.
Paper Structure (16 sections, 3 theorems, 27 equations, 6 figures, 13 tables, 5 algorithms)

This paper contains 16 sections, 3 theorems, 27 equations, 6 figures, 13 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Enlarged CG Methods
  3. SRE-CG2
  4. MSDO-CG
  5. Truncated and Restarted Enlarged CG methods
  6. Flexibly Enlarged CG methods
  7. Flexibly Enlarged Krylov Subspace
  8. Switch
  9. Flexibly SRE-CG2 Algorithm
  10. Flexibly MSDO-CG Algorithm
  11. Preconditioned versions
  12. Testing
  13. Truncated and Restarted Enlarged CG Methods
  14. Flexibly Enlarged CG Methods
  15. Preconditioned versions
  16. ...and 1 more sections

Key Result

Theorem 4.1

\newlabelthrm:kry1 Let $1 \leq k_F < k$. Then, the Krylov subspace where $r_{k_F}\in \mathcal{K}_{k_F+1}(A,r_0)$.

Figures (6)

  • Figure 4.1: Norm of residual vector for Ani3D matrix using SRE-CG2 for $32$ partitions over all the 156 iterations needed to convergence, zoomed view from iteration 45, and zoomed view from iteration 120.
  • Figure 5.1: Convergence of SRE-CG2(trunc) for different $trunc$ and $t$ values for matrices Sky3D (left), Sky2D (right), and Ani3D (bottom)
  • Figure 5.2: Convergence of preconditioned flexibly enlarged methods with switchTol = $10^{-5}$ for matrix Nh2D
  • Figure 5.3: Convergence of preconditioned flexibly enlarged methods with switchTol = $10^{-5}$ for matrix Sky3D
  • Figure 5.4: Convergence of preconditioned flexibly enlarged methods with switchTol = $10^{-5}$ for matrix Ani3D
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Corollary 4.3
  • proof