Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reduced rank extrapolation for multi-term Sylvester equations

Peter Benner, Pascal den Boef, Patrick Kürschner, Xiaobo Liu, Jens Saak

Abstract

We investigate the acceleration of stationary iterations for multi-term Sylvester equation by means of reduced rank extrapolation (RRE). Theoretical convergence results and implementations are provided for both small and large-scale problems. For the large-scale problems, an inexact non-stationary iteration is discussed, which makes use of low-rank matrix approximations. Numerical experiments illustrate the potential of the RRE acceleration which often leads to a substantial gain in convergence speed and therefore reducing the consumption of storage and computing time.

Reduced rank extrapolation for multi-term Sylvester equations

Abstract

We investigate the acceleration of stationary iterations for multi-term Sylvester equation by means of reduced rank extrapolation (RRE). Theoretical convergence results and implementations are provided for both small and large-scale problems. For the large-scale problems, an inexact non-stationary iteration is discussed, which makes use of low-rank matrix approximations. Numerical experiments illustrate the potential of the RRE acceleration which often leads to a substantial gain in convergence speed and therefore reducing the consumption of storage and computing time.
Paper Structure (20 sections, 61 equations, 4 figures, 1 table)

This paper contains 20 sections, 61 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Analysis and algorithm for small, dense problems
  3. Incorporation of RRE
  4. Convergence considerations
  5. Analysis and algorithm for large-scale problems
  6. Incorporation of RRE
  7. Iterative solution of the standard Sylvester equation
  8. Convergence considerations
  9. Further considerations w.r.t. the inner solves
  10. Right-hand side separation
  11. Rank truncation
  12. Estimation of the residual norm
  13. Numerical examples
  14. Code and data availability
  15. An example with dense coefficients
  16. ...and 5 more sections

Figures (4)

  • Figure 4.1: Residual norm history of dense algorithm with and without cycling RRE (RRE$_w$) for different values of the weighting parameter $\beta$, the number $\ell$ of terms in $\Pi$, and RRE window size $w$.
  • Figure 4.2: Results for the advection diffusion example. Residual norm history of low-rank non-stationary iteration algorithm without and with cycling RRE using window size $w$ (RRE$_w$) and tabular summary.
  • Figure 4.3: Results for the nonlinear circuit example for different $\beta$ values. Residual norm history of low-rank non-stationary iteration algorithm (for some selected cases) and tabular summary.
  • Figure 4.4: Results for the multiterm Sylvester example for different $w$ values. Residual norm history of low-rank non-stationary iteration algorithm and tabular summary.

Theorems & Definitions (1)

  • Remark 1