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Do you precondition on the left or on the right?

Nicole Spillane, Pierre Matalon, Daniel B Szyld

TL;DR

This paper reexamines whether left, right, or split preconditioning is preferable for GMRES by combining a theoretical analysis with a literature review and a social experiment. It clarifies that spectra are invariant across preconditioning forms and that observable residuals can differ mainly when the preconditioner is ill-conditioned, challenging some convergence-bound intuitions. Through illustrative bounds and a DD29 poll, the authors show that opposite predictions can arise from theory, yet practical performance is often similar. The work emphasizes choosing the preconditioning strategy based on the norm GMRES minimizes or on robust residual monitoring, rather than on general claims of superiority.

Abstract

This work is a follow-up to a poster that was presented at the DD29 conference. Participants were asked the question: ``Do you precondition on the left or on the right?''. Here we report on the results of this social experiment. We also provide context on left, right and split preconditioning, share our literature review on the topic, and analyze some of the finer points. Two examples illustrate that convergence bounds can sometimes lead to misleading conclusions.

Do you precondition on the left or on the right?

TL;DR

This paper reexamines whether left, right, or split preconditioning is preferable for GMRES by combining a theoretical analysis with a literature review and a social experiment. It clarifies that spectra are invariant across preconditioning forms and that observable residuals can differ mainly when the preconditioner is ill-conditioned, challenging some convergence-bound intuitions. Through illustrative bounds and a DD29 poll, the authors show that opposite predictions can arise from theory, yet practical performance is often similar. The work emphasizes choosing the preconditioning strategy based on the norm GMRES minimizes or on robust residual monitoring, rather than on general claims of superiority.

Abstract

This work is a follow-up to a poster that was presented at the DD29 conference. Participants were asked the question: ``Do you precondition on the left or on the right?''. Here we report on the results of this social experiment. We also provide context on left, right and split preconditioning, share our literature review on the topic, and analyze some of the finer points. Two examples illustrate that convergence bounds can sometimes lead to misleading conclusions.

Paper Structure

This paper contains 8 sections, 20 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Litterature Review
  3. Analysis
  4. Left and right preconditioned GMRES can only differ if $\mathbf{H}$ is ill-conditioned
  5. Different convergence bounds, same behaviour (1/2)
  6. Different convergence bounds, same behaviour (2/2)
  7. A social experiment at DD29
  8. Conclusion

Figures (2)

  • Figure 1: Attendees of DD29 were asked to vote for their favourite way of preconditioning during the poster session. See Section \ref{['sec:DD29']} and Table \ref{['tab:poll']} for more details.
  • Figure 2: Example of left-right preconditioning leading to similar convergence behaviour while presenting significantly different convergence bounds.