[Paper Review] A preconditioner based on the shift-splitting method for generalized saddle point problems

Davod Khojasteh Salkuyeh, Mohsen Masoudi+1 more|arXiv (Cornell University)|Jun 15, 2015

Matrix Theory and Algorithms4 references4 citations

TL;DR

This paper proposes a shift-splitting preconditioner for generalized saddle point problems with nonsymmetric positive definite (1,1)-blocks and symmetric positive semidefinite (2,2)-blocks. The method is based on an unconditionally convergent iterative scheme, and numerical experiments on Navier-Stokes problems show it significantly reduces GMRES iterations and CPU time compared to non-preconditioned GMRES.

ABSTRACT

In this paper, we propose a preconditioner based on the shift-splitting method for generalized saddle point problems with nonsymmetric positive definite (1,1)-block and symmetric positive semidefinite $(2,2)$-block. The proposed preconditioner is obtained from an basic iterative method which is unconditionally convergent. We also present a relaxed version of the proposed method. Some numerical experiments are presented to show the effectiveness of the method.

Motivation & Objective

• To develop an effective preconditioner for large, sparse generalized saddle point problems arising from finite element discretizations of the Navier-Stokes equations.
• To extend the shift-splitting method to cases where the (1,1)-block is nonsymmetric positive definite, not just symmetric.
• To ensure unconditional convergence of the proposed iterative method and derive a stable preconditioner for Krylov subspace solvers like GMRES.
• To evaluate the performance of the preconditioner on benchmark Navier-Stokes problems using GMRES(30).

Proposed method

• The method uses a matrix splitting of the saddle point system into M and N matrices based on shift parameters α and β.
• The iteration matrix Γα,β = M⁻¹N is proven to have spectral radius less than 1 for all α, β > 0, ensuring unconditional convergence.
• The preconditioner P_MGSS is defined as Mα,β = ½[(αI + A) Bᵀ; -B (βI + C)], which is symmetric positive definite and suitable for GMRES.
• A relaxed version, P_RMGSS, omits the (1,1)-block shift, simplifying implementation while retaining effectiveness.
• Inner systems are solved using GMRES(10) with a residual reduction tolerance of 10⁻² and a maximum of 40 iterations.
• The method is tested using the IFISS package for Q1-P0 finite element discretization of the steady-state Navier-Stokes equations.

Experimental results

Research questions

• RQ1 Can the shift-splitting method be extended to generalized saddle point problems with nonsymmetric positive definite (1,1)-blocks?
• RQ2 Does the proposed preconditioner ensure unconditional convergence for such problems?
• RQ3 How effective is the preconditioner in reducing GMRES iterations and CPU time for Navier-Stokes problems?
• RQ4 What is the impact of parameter choices α = 0.01 and β = 0.001 on convergence behavior?

Key findings

• The proposed MGSS method is unconditionally convergent, with spectral radius of the iteration matrix strictly less than 1 for all α, β > 0.
• For the 64×64 grid, the preconditioned GMRES(30) required only 28 iterations and 21.48 seconds, compared to 3554 iterations and 110.1 seconds without preconditioning.
• On the 32×32 grid, the preconditioner reduced iterations from 608 (non-preconditioned) to 12, cutting CPU time from 4.95s to 3.58s.
• The relaxed preconditioner RMGSS also showed strong performance, reducing iterations to 8 on the 16×16 grid and 12 on the 32×32 grid.
• The preconditioner significantly outperforms non-preconditioned GMRES across all grid sizes, especially on finer grids.
• Theoretical analysis confirms that the method remains convergent even when A is nonsymmetric positive definite, extending prior results limited to symmetric A.