Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Spectral Preconditioner for the Conjugate Gradient Method with Iteration Budget

Youssef Diouane, Selime Gürol, Oussama Mouhtal, Dominique Orban

Abstract

We study the solution of large symmetric positive-definite linear systems in a matrix-free setting with a limited iteration budget. We focus on the preconditioned conjugate gradient (PCG) method with spectral preconditioning. Spectral preconditioners map a subset of eigenvalues to a positive cluster via a scaling parameter, and leave the remainder of the spectrum unchanged, in hopes to reduce the number of iterations to convergence. We formulate the design of the spectral preconditioners as a constrained optimization problem. The optimal cluster placement is defined to minimize the error in energy norm at a fixed iteration. This optimality criterion provides new insight into the design of efficient spectral preconditioners when PCG is stopped short of convergence. We propose practical strategies for selecting the scaling parameter, hence the cluster position, that incur negligible computational cost. Numerical experiments highlight the importance of cluster placement and demonstrate significant improvements in terms of error in energy norm, particularly during the initial iterations.

A Spectral Preconditioner for the Conjugate Gradient Method with Iteration Budget

Abstract

We study the solution of large symmetric positive-definite linear systems in a matrix-free setting with a limited iteration budget. We focus on the preconditioned conjugate gradient (PCG) method with spectral preconditioning. Spectral preconditioners map a subset of eigenvalues to a positive cluster via a scaling parameter, and leave the remainder of the spectrum unchanged, in hopes to reduce the number of iterations to convergence. We formulate the design of the spectral preconditioners as a constrained optimization problem. The optimal cluster placement is defined to minimize the error in energy norm at a fixed iteration. This optimality criterion provides new insight into the design of efficient spectral preconditioners when PCG is stopped short of convergence. We propose practical strategies for selecting the scaling parameter, hence the cluster position, that incur negligible computational cost. Numerical experiments highlight the importance of cluster placement and demonstrate significant improvements in terms of error in energy norm, particularly during the initial iterations.

Paper Structure

This paper contains 23 sections, 4 theorems, 57 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Conjugate Gradient
  4. Convergence properties of CG
  5. Properties of a good preconditioner
  6. A scaled spectral preconditioner
  7. An optimal choice for $D$
  8. A practical choice for $\pi_k^\ast$
  9. On the choice of the scaling parameter $\theta$
  10. theta as the mid-range between lambdak and lambdan
  11. theta with respect to the initial iteration
  12. theta providing lower error in energy norm
  13. $\theta$ as the smallest eigenvalue
  14. Numerical experiments
  15. Approximating largest and smallest eigenvalues
  16. ...and 8 more sections

Key Result

corollary 1

Solution of eq:Prec_Opt can be expressed as where $U^\ast= {U^{\ast}}^\top := I_n + \sum\limits_{i\in \pi^{*}_k }\left(\sqrt{\frac{\theta^\ast}{\lambda_i}}-1\right)s_{i}s_{i}^\top$, and $\pi^{*}_k$ and $\theta^\ast$ are defined in th:PrecOptimale.

Figures (2)

  • Figure 1: Top: Eigenvalue distributions of $F_{\theta}A$ for $k = 30$, $40$, and $50$ (from left to right). Bottom: Relative errors in energy norm versus iteration $\ell$, including DefCG.
  • Figure 2: Eigenvalues of the preconditioned matrix, the values of $\zeta_i$, and the energy norm of the relative error along iterations for fast decay and fast growth of $\zeta_i$.

Theorems & Definitions (6)

  • corollary 1
  • corollary 2
  • proposition 1
  • proof
  • proposition 2
  • proof