Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Robust iterative methods for linear systems with saddle point structure

Murat Manguoğlu, Volker Mehrmann

TL;DR

This paper tackles large sparse saddle point linear systems of the form $\mathcal{W}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}f\\g\end{bmatrix}$ with $A+A^T\ge 0$, introducing a purely algebraic, black-box three-layer iterative framework. The approach combines an approximate nullspace preconditioner (constructed via sparse bases $\tilde{Z}$ and $\tilde{U}$ using SAROC), iterative least-squares subproblems, and projection-based Krylov methods in an outer $fGMRES$ loop, with inner solvers such as LSQR and MRS; a key component is the construction and efficient handling of the projected nullspace matrices $\mathcal{N}^s$. Theoretical results establish positive definiteness of the symmetric part of the projected nullspace, stability under approximate nullspaces, and a structured block form enabling a robust preconditioner; numerically, the schemes demonstrate strong robustness and competitive sparsity across symmetric, generalized, and general saddle point problems, outperforming ILUTP baselines on several metrics. The work advances black-box, scalable solvers for saddle-point systems arising in applications like dissipative Hamiltonian discretizations, fluid dynamics, and constrained optimization, with potential for extensions to non-square blocks and parallel implementations.

Abstract

We propose a new class of multi-layer iterative schemes for solving sparse linear systems in saddle point structure. The new scheme consist of an iterative preconditioner that is based on the (approximate) nullspace method, combined with an iterative least squares approach and an iterative projection method. We present a theoretical analysis and demonstrate the effectiveness and robustness of the new scheme on sparse matrices from various applications.

Robust iterative methods for linear systems with saddle point structure

TL;DR

This paper tackles large sparse saddle point linear systems of the form with , introducing a purely algebraic, black-box three-layer iterative framework. The approach combines an approximate nullspace preconditioner (constructed via sparse bases and using SAROC), iterative least-squares subproblems, and projection-based Krylov methods in an outer loop, with inner solvers such as LSQR and MRS; a key component is the construction and efficient handling of the projected nullspace matrices . Theoretical results establish positive definiteness of the symmetric part of the projected nullspace, stability under approximate nullspaces, and a structured block form enabling a robust preconditioner; numerically, the schemes demonstrate strong robustness and competitive sparsity across symmetric, generalized, and general saddle point problems, outperforming ILUTP baselines on several metrics. The work advances black-box, scalable solvers for saddle-point systems arising in applications like dissipative Hamiltonian discretizations, fluid dynamics, and constrained optimization, with potential for extensions to non-square blocks and parallel implementations.

Abstract

We propose a new class of multi-layer iterative schemes for solving sparse linear systems in saddle point structure. The new scheme consist of an iterative preconditioner that is based on the (approximate) nullspace method, combined with an iterative least squares approach and an iterative projection method. We present a theoretical analysis and demonstrate the effectiveness and robustness of the new scheme on sparse matrices from various applications.

Paper Structure

This paper contains 13 sections, 5 theorems, 13 equations, 3 figures, 16 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical Analysis
  3. A new multi-layer iterative scheme
  4. The symmetric case
  5. Generalized case
  6. General case
  7. Test problems
  8. Experimental framework
  9. Numerical results
  10. Symmetric case
  11. Generalized case
  12. General case
  13. Conclusions

Key Result

Lemma 1

Consider $A=J+R\in\mathbb R^{n\times n}$, where $0\leq R=R^T$ and $0\neq J=-J^T$. Then there exist a real orthogonal matrix $Q\in\mathbb R^{n \times n}$, and integers $n_1 \geq n_2 \geq \cdots \geq n_{r-1} >0$ and $n_r,n_{r+1} \geq 0$, such that where $0<R_{11}=R_{11}^T \in\mathbb R^{n_1\times n_1}$ , $J_{ii}=-J_{ii}^T\in\mathbb R^{n_i\times n_i}$ for $i=1,\dots,r$, $J_{rr}$ invertible, and $J_{i

Figures (3)

  • Figure 1: Final true relative residual norms of the new scheme and the baseline method for the symmetric case
  • Figure 2: Final true relative residual norms of the new scheme and the baseline for the generalized case
  • Figure 3: Final true relative residual norms of the new scheme and the baseline for the generalized case

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Corollary 1
  • Lemma 3