Symmetric Indefinite Research Articles

Newton–Picard-Based Preconditioning for Linear-Quadratic Optimization Problems with Time-Periodic Parabolic PDE Constraints

We develop and investigate two preconditioners for a basic linear iterative splitting method for the numerical solution of linear-quadratic optimization problems with time-periodic parabolic PDE constraints. The resulting real-valued linear system to be solved is symmetric indefinite. We propose all-at-once symmetric indefinite preconditioners based on a Newton–Picard approach which divides the variable space into slow and fast modes. The division is performed either classically with eigenspace methods or with a novel two-grid approach. We prove mesh-independent convergence for the classical Newton–Picard preconditioner, present a complexity analysis, and show numerical results for the classical and the two-grid preconditioners. Moreover, the preconditioners compare favorably with existing symmetric positive definite Schur complement preconditioners in a Krylov method context.

An Optimal Iterative Solver for Symmetric Indefinite Systems Stemming from Mixed Approximation

We discuss the design and implementation of a suite of functions for solving symmetric indefinite linear systems associated with mixed approximation of systems of PDEs. The novel feature of our iterative solver is the incorporation of error control in the natural “energy” norm in combination with an a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. We describe a “proof of concept” MATLAB implementation of this algorithm, which we callEST_MINRES, and we illustrate its effectiveness when integrated into the Incompressible Flow Iterative Solution Software (IFISS) package (cf. ACM Transactions on Mathematical Software 33, Article 14, 2007).

Constraint Preconditioners for Symmetric Indefinite Matrices

We study the eigenvalue bounds of block two-by-two nonsingular and symmetric indefinite matrices whose $(1,1)$ block is symmetric positive definite and Schur complement with respect to its $(2,2)$ block is symmetric indefinite. A constraint preconditioner for this matrix is constructed by simply replacing the $(1,1)$ block by a symmetric and positive definite approximation, and the spectral properties of the preconditioned matrix are discussed. Numerical results show that, for a suitably chosen $(1,1)$ block-matrix, this constraint preconditioner outperforms the block-diagonal and the block-tridiagonal ones in iteration step and computing time when they are used to accelerate the GMRES method for solving these block two-by-two symmetric positive indefinite linear systems. The new results extend the existing ones about block two-by-two matrices of symmetric negative semidefinite $(2,2)$ blocks to those of general symmetric $(2,2)$ blocks.

Generalized least-squares polynomial preconditioners for symmetric indefinite linear equations

Polynomial preconditioners are frequently used in a parallel environment for the computation of the solution of large-scale sparse linear equations ( Ax= b) because of their easy implementation and trivial parallelization. With respect to symmetrical indefinite (SID) linear systems, the use of generalized least-squares (GLS) polynomial preconditioning is preferable to other polynomial preconditioning methods because of the ability to use a three-term recurrence relationship and the low implementation costs. The GLS preconditioning polynomial and its influence on the flexible generalized minimized residual (FGMRES) solver are discussed in this paper. The orthogonal polynomials required in the solution of the least-squares approximation problem are constructed using the Stieltjes procedure in multiple disjoint intervals which exclude the origin. The time-consuming numerical integration associated with this procedure is computed efficiently using Chebyshev polynomials of the first kind and the GLS polynomial reconditioned FGMRES algorithm is implemented using MPI in a highly parallel IBM SP2 environment. Experimental results using classical benchmark systems are presented and compared with those obtained using the recently developed SPAI preconditioned Bi-CGSTAB iterative method. The performance of the GLS preconditioned FGMRES solver is critically accessed.

Incomplete block factorization preconditioning for indefinite elliptic problems

The application of the finite difference method to approximate the solution of an indefinite elliptic problem produces a linear system whose coefficient matrix is block tridiagonal and symmetric indefinite. Such a linear system can be solved efficiently by a conjugate residual method, particularly when combined with a good preconditioner. We show that specific incomplete block factorization exists for the indefinite matrix if the mesh size is reasonably small, and that this factorization can serve as an efficient preconditioner. Some efforts are made to estimate the eigenvalues of the preconditioned matrix. Numerical results are also given.

Large Scale Optimal Power Flow Based on Interior Point Quadratic Programming of Solving Symmetric Indefinite System

This paper presents a new algorithm of the interior point quadratic programming which can solve large scale optimal power flow problems with profoundly less computational efforts and can start from either a feasible or an infeasible point. Numerical results on the IEEE test systems and a Japanese 344 bus system (2301 variables, 2222 equality constraints and 1534 inequality constraints) have verified that the proposed algorithm possesses enough robustness and reliability and is especially suitable for on-line use as well as off-line use for large scale systems

A Comparison of Algorithms for Solving Symmetric Indefinite Systems of Linear Equations

A Comparison of Algorithms for Solving Symmetric Indefinite Systems of Linear Equations

Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations

Methods for solving symmetric indefinite systems are surveyed including a new one which is stable and almost as fast as the Cholesky method.