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An inexact augmented Lagrangian algorithm for unsymmetric saddle-point systems

N. Huang, Y. -H. Dai, D. Orban, M. A. Saunders

TL;DR

This work addresses solving unsymmetric saddle-point systems that arise in discretized constrained problems by using augmented Lagrangian methods (SPAL). It develops an exact SPAL and an inexact SPAL (SPAL) with inner solves performed by a Barzilai–Borwein (BB) gradient step, yielding SPALBB; the authors provide convergence and semi-convergence analyses under full-column-rank and rank-deficient B, and they establish a convergence condition for SPALBB based on eigenvalue spectra. Through numerical tests on steady Navier–Stokes and coupled Stokes–Darcy problems, SPALBB demonstrates robust performance and often lower CPU time than GMRES and BICGSTAB, especially for large-scale, highly unsymmetric systems, while requiring fixed storage. The results highlight the practical viability of SPAL and SPALBB for large-scale saddle-point computations in PDE-constrained optimization and fluid-flow simulations, and point to future work on parameter selection for inner/outer iteration balance and improved inner solvers.

Abstract

Augmented Lagrangian (AL) methods are a well known class of algorithms for solving constrained optimization problems. They have been extended to the solution of saddle-point systems of linear equations. We study an AL (SPAL) algorithm for unsymmetric saddle-point systems and derive convergence and semi-convergence properties, even when the system is singular. At each step, our SPAL requires the exact solution of a linear system of the same size but with an SPD (2,2) block. To improve efficiency, we introduce an inexact SPAL algorithm. We establish its convergence properties under reasonable assumptions. Specifically, we use a gradient method, known as the Barzilai-Borwein (BB) method, to solve the linear system at each iteration. We call the result the augmented Lagrangian BB (SPALBB) algorithm and study its convergence. Numerical experiments on test problems from Navier-Stokes equations and coupled Stokes-Darcy flow show that SPALBB is more robust and efficient than BICGSTAB and GMRES. SPALBB often requires the least CPU time, especially on large systems.

An inexact augmented Lagrangian algorithm for unsymmetric saddle-point systems

TL;DR

This work addresses solving unsymmetric saddle-point systems that arise in discretized constrained problems by using augmented Lagrangian methods (SPAL). It develops an exact SPAL and an inexact SPAL (SPAL) with inner solves performed by a Barzilai–Borwein (BB) gradient step, yielding SPALBB; the authors provide convergence and semi-convergence analyses under full-column-rank and rank-deficient B, and they establish a convergence condition for SPALBB based on eigenvalue spectra. Through numerical tests on steady Navier–Stokes and coupled Stokes–Darcy problems, SPALBB demonstrates robust performance and often lower CPU time than GMRES and BICGSTAB, especially for large-scale, highly unsymmetric systems, while requiring fixed storage. The results highlight the practical viability of SPAL and SPALBB for large-scale saddle-point computations in PDE-constrained optimization and fluid-flow simulations, and point to future work on parameter selection for inner/outer iteration balance and improved inner solvers.

Abstract

Augmented Lagrangian (AL) methods are a well known class of algorithms for solving constrained optimization problems. They have been extended to the solution of saddle-point systems of linear equations. We study an AL (SPAL) algorithm for unsymmetric saddle-point systems and derive convergence and semi-convergence properties, even when the system is singular. At each step, our SPAL requires the exact solution of a linear system of the same size but with an SPD (2,2) block. To improve efficiency, we introduce an inexact SPAL algorithm. We establish its convergence properties under reasonable assumptions. Specifically, we use a gradient method, known as the Barzilai-Borwein (BB) method, to solve the linear system at each iteration. We call the result the augmented Lagrangian BB (SPALBB) algorithm and study its convergence. Numerical experiments on test problems from Navier-Stokes equations and coupled Stokes-Darcy flow show that SPALBB is more robust and efficient than BICGSTAB and GMRES. SPALBB often requires the least CPU time, especially on large systems.
Paper Structure (10 sections, 16 theorems, 75 equations, 2 figures, 12 tables, 3 algorithms)

This paper contains 10 sections, 16 theorems, 75 equations, 2 figures, 12 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Augmented Lagrangian algorithm
  3. Convergence analysis when B has full column rank
  4. Convergence analysis when B is rank-deficient
  5. Inexact augmented Lagrangian algorithm
  6. Convergence analysis when B has full column rank
  7. Convergence analysis when B is rank-deficient
  8. Augmented Lagrangian BB algorithm
  9. Numerical experiments
  10. Conclusions

Key Result

Lemma 2.1

\newlabell100 Let $G\in \mathds{R}^{n\times n}$ be unsymmetric but positive definite on $\mathop{\mathrm{Null}}(B^T\!\,)$, and For any SPD $Q\in \mathds{R}^{m\times m}$, if $0<\omega<1/(-\eta)_+$, then $S_Q$ is positive definite.

Figures (2)

  • Figure 1: Evolution of the relative residual of SPALBB tested on \ref{['ex1']} (left) with $n=8450$, $m=1089$, and on \ref{['ex2']} (right) with $n=8416$, $m=1096$ and $\omega$ as in \ref{['a11']}.
  • Figure 2: Evolution of the relative residual of SPALBB tested on \ref{['ex3']} (left) with $n=5890$, $m=769$, and on \ref{['ex:sd']} with $n=12675$, $m=1089$ and $\omega$ as in \ref{['a11']}.

Theorems & Definitions (41)

  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Lemma 2.3
  • Proof 2
  • Theorem 2.1
  • Proof 3
  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • ...and 31 more