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Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy

Vincent Darrigrand, Andrei Dumitrasc, Carola Kruse, Ulrich Ruede

TL;DR

This work analyzes inexact inner-outer Golub-Kahan bidiagonalization for saddle-point systems, showing that exact inner solves are unnecessary to reach a prescribed outer accuracy. It introduces dynamic relaxation strategies that adapt the inner tolerance per outer iteration based on the evolution of Krylov coefficients, backed by a perturbation/error analysis that identifies the outsized impact of early iterations. The authors demonstrate substantial reductions in inner iterations (roughly 33%–63%) on Stokes and Mixed Poisson problems, and show additional gains when combining relaxation with spectrum clustering preconditioning and augmented Lagrangian enhancements. The method is general, inexpensive, and compatible with black-box implementations, strengthening the practicality of GKB for large-scale PDE discretizations.

Abstract

We study an inexact inner-outer generalized Golub-Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.

Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy

TL;DR

This work analyzes inexact inner-outer Golub-Kahan bidiagonalization for saddle-point systems, showing that exact inner solves are unnecessary to reach a prescribed outer accuracy. It introduces dynamic relaxation strategies that adapt the inner tolerance per outer iteration based on the evolution of Krylov coefficients, backed by a perturbation/error analysis that identifies the outsized impact of early iterations. The authors demonstrate substantial reductions in inner iterations (roughly 33%–63%) on Stokes and Mixed Poisson problems, and show additional gains when combining relaxation with spectrum clustering preconditioning and augmented Lagrangian enhancements. The method is general, inexpensive, and compatible with black-box implementations, strengthening the practicality of GKB for large-scale PDE discretizations.

Abstract

We study an inexact inner-outer generalized Golub-Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.
Paper Structure (11 sections, 29 equations, 9 figures, 5 tables, 1 algorithm)

This paper contains 11 sections, 29 equations, 9 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Generalized Golub-Kahan algorithm
  3. Problem description
  4. Constant accuracy inner solver
  5. Perturbation and error study
  6. High initial accuracy followed by relaxation
  7. Relaxation strategy choices
  8. Increasing the savings by working on a simplified problem
  9. GKB with the augmented Lagrangian approach
  10. Conclusions
  11. Acknowledgments

Figures (9)

  • Figure 1: Exact solution to the Stokes problem in a channel of length 5. Plotted is the $1-y^2$ function, which represents the $x$ direction velocity, overlaid with the mesh resulting from the domain discretization (Q2-Q1 Finite Elements Method).
  • Figure 2: GKB convergence history for the IFISS channel problem. The length of each channel is given in the legend. Y-axis: Energy norm of the relative error for the velocity.
  • Figure 3: GKB convergence history for the IFISS Channel test case, depending on the CG tolerance (see legend). Y-axis: Energy norm of the relative error for the velocity. Target GKB tolerance 1.0e-7. Mixed precision: first two iterations 1.0e-3, afterwards 1.0e-14. The final value for each case with CG is: 3.0e-3 \ref{['fig:item:constCGtol_1e-3']}, 7.0e-7 \ref{['fig:item:constCGtol_1e-7']}, 8.0e-8 \ref{['fig:item:constCGtol_1e-8']} , 2.0e-3 \ref{['fig:item:constCGtol_first 2 steps 1e-3 then 1e-14']}. Only the cases \ref{['fig:item:constCGtol_1e-8']} and \ref{['fig:item:constCGtol_exact']} converge successfully, reducing the error norm below 1.0e-7.
  • Figure 4: Lower bound (\ref{['eq:lowBnd']}) for the error norm associated with the GKB iterates versus the cumulative number of inner CG iterations when solving the original problem from \ref{['sec:pbDesc']}. The parameter used in Optimal is $0.05$. See \ref{['eq:cst', 'eq:z', 'eq:zOptim', 'eq:adasquare', 'eq:hybrid']} for the strategies denoted by the labels.
  • Figure 5: GKB convergence curves for the IFISS channel test case before and after spectral clustering. Y-axis: Energy norm of the relative error for the velocity. Target GKB tolerance 1.0e-7. Using the Least Squares Commutator preconditioner or deflation of the smallest five spectral outliers.
  • ...and 4 more figures