Indefinite System Research Articles

Block preconditioners for symmetric indefinite linear systems

AbstractThis paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a Krylov subspace method for solving symmetric indefinite linear systems. The focus is on large‐scale real world problems where block approximations are a practical necessity. The main illustration is the performance of the block diagonal, constrained, and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large‐scale finite element discretization of Biot's consolidation equations. This system of equations is of fundamental importance to geomechanics. Numerical studies show that simple diagonal approximations to the (1,1) block K and inexpensive approximations to the Schur complement matrix S may not always produce the most spectacular time savings when K is explicitly available, but is able to deliver reasonably good results on a consistent basis. In addition, the block diagonal preconditioner with a negative (2,2) block appears to be reasonably competitive when compared to the more complicated ones. These observation are expected to remain valid for coefficient matrices whereby the (1,1) block is sparse, diagonally significant (a notion weaker than diagonal dominance), moderately well‐conditioned, and has a much larger block size than the (2,2) block. Copyright © 2004 John Wiley & Sons, Ltd.

Constraint preconditioning for linear systems in elasticity

AbstractThe mixed variational formulation of elastic deformation of a material body leads to a constraint minimisation (or saddle‐point) problem. Using the PEERS finite element space to discretise the problem one gets an indefinite system of linear equations. To solve this system, preconditioners with similar saddle‐point patterns will be applied. The results of numerical tests with this preconditioning strategy show its efficiency for linear systems with up to more than half a million unknowns.

Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems

AbstractThis paper is concerned with the numerical solution of a symmetric indefinite system which is a generalization of the Karush–Kuhn–Tucker system. Following the recent approach of Lukšan and Vlček, we propose to solve this system by a preconditioned conjugate gradient (PCG) algorithm and we devise two indefinite preconditioners with good theoretical properties. In particular, for one of these preconditioners, the finite termination property of the PCG method is stated. The PCG method combined with a parallel version of these preconditioners is used as inner solver within an inexact Interior‐Point (IP) method for the solution of large and sparse quadratic programs. The numerical results obtained by a parallel code implementing the IP method on distributed memory multiprocessor systems enable us to confirm the effectiveness of the proposed approach for problems with special structure in the constraint matrix and in the objective function. Copyright © 2002 John Wiley & Sons, Ltd.

Numerical Approximation of an SQP-Type Method for Parameter Identification

This paper deals with the numerical approximation of the Levenberg--Marquardt SQP (LMSQP) method for parameter identification problems, which has been presented and analyzed in [M. Burger and W. Mühlhuber, Inverse Problems, 18 (2002), pp. 943--969]. It is shown that a Galerkin-type discretization leads to a convergent approximation and that the indefinite system arising from the Karush--Kuhn--Tucker (KKT) system is well-posed. In addition, we present a multilevel version of the Levenberg--Marquardt method and discuss the simultaneous solution of the discretized KKT system by preconditioned iteration methods for indefinite problems. From a discussion of the numerical effort we conclude that these approaches may lead to a considerable speed-up with respect to standard iterative regularization methods that eliminate the underlying state equation. The numerical efficiency of the LMSQP method is confirmed by numerical examples.

Global‐basis two‐level method for indefinite systems. Part 2: computational issues

Algorithmic aspects and computational efficiency of the global-basis two-level method are investigated in the context of symmetric indefinite system of equations. The algorithm includes efficient construction of the global-basis prolongator using Lanczos vectors, predictor–corrector smoothing procedures, and a heuristic two-level feedback loop aimed at ensuring convergence. Numerical experiments consisting of 3D Helmholtz equations and shear banding problems with strain softening demonstrate the excellent performance of the method. Copyright © 2000 John Wiley & Sons, Ltd.

Alternative approach for solving indefinite symmetrical system of equation

In this work, an alternative approach is proposed for solving system of sparse, symmetrical, indefinite equations. The key idea here is first to transform the original indefinite system into a new (modified) system of symmetrical, “definite” equations which can be conveniently used to obtain the “intermediate solution”. This “intermediate” solution is then transformed back into the “original” space to obtain the “original” unknown vector. To validate the formulas developed in our paper, several NASA indefinite system of equations have been solved (ranging from 51 to 15,367 indefinite equations) on inexpensive workstations.

Schur complement reduction in the mixed-hybrid approximation of Darcy's law: rounding error analysis

Mixed-hybrid finite element approximation of the potential fluid flow problem leads to the solution of a large symmetric indefinite system for the velocity and potential head vector components. Such discretization gives rise to a very accurate approximation of the continuity equation in every element, and for low-order discretizations, the structural properties of the discrete matrix blocks allow cheap block elimination of the positive-definite diagonal block and subsequent reduction to the Schur complement system for the pressure and Lagrangian vector components. This system is then frequently solved by the iterative conjugate gradient-type method. Whereas this approach is well known, considerably less attention has been paid to the numerical stability aspects of such transformation. It was shown in [5] that block LU factorization can be unstable even when the system matrix is symmetric positive definite. In this paper we examine this type of conditional stability for a particular application in the underground water flow modelling. We show that the actual error of the computed approximate solution depends not only on the user-defined tolerance in the conjugate gradient process but also on the spectral properties of the corresponding matrix blocks eliminated during the Schur complement reduction. It is often observed that although the backward error of the approximate solution in the iterative part is reduced to the level of machine accuracy, the total residual norm after the back-substitution process remains at certain accuracy level. We give a bound for this maximal attainable accuracy and illustrate our theoretical results on a model example.

Sparse algorithms for indefinite system of linear equations

Sparse LDLT algorithms, based upon mixed forward-backward factorization strategies, are developed for direct solution of indefinite system of linear equations. A simple rotation matrix is also introduced and incorporated into 2×2 pivoting strategies. Several test problems have been conducted in order to evaluate the numerical performance of the proposed algorithms, and its associated FORTRAN codes.

Global-basis two-level method for indefinite systems. Part 1: convergence studies

A robust two-level solver for high indefinite system of equations arising from the finite element discretization is developed. It is shown that the optimal coarse model is spanned by the spectrum of the highest eigenmodes of the smoothing iteration matrix. Convergence studies conducted on a model prolongation operator reveal pathological sensitivity to any deviation from the optimal coarse model. Copyright © 2000 John Wiley & Sons, Ltd.

Domain decomposition preconditioners for p and hp finite element approximation of Stokes equations

Domain decomposition preconditioning techniques are developed in the context of hp finite element approximation of the Stokes problem. Two basic types of preconditioner are considered: a block diagonal scheme based on decoupling the velocity and pressure components, and a scheme based on an indefinite system similar to the original Stokes system. For each type of scheme, theoretical estimates are obtained for the location of the eigenvalues of the preconditioned operators in terms of the polynomial degree, the mesh sizes on the coarse and fine grids, and the inf—sup constant for the method. Theoretical estimates show that the growth of the bounds is modest as the mesh is refined and the polynomial order is increased. The preconditioners are shown to be applicable to various iterative schemes for the Stokes systems. The theoretical bounds are compared with actual quantities obtained in practical computations for several representative problems.

Linear Algebra Methods in a Mixed Approximation of Magnetostatic Problems

In the solution of magnetostatic problems the use of a mixed formulation based on both the magnetic and magnetic displacement fields is particularly appropriate as it allows us to impose the physical conditions exactly and to maintain the continuity properties of the two fields, together with an efficient treatment of boundary conditions. The discretization by means of a proper finite element method yields a strongly structured algebraic linear system. This paper is concerned with the solution of this large, very sparse indefinite linear system. In particular, we present the implementation of some known direct and preconditioned iterative methods and discuss their performance on two-dimensional (2-D) and three-dimensional (3-D) specific models. We show that the 2-D system can be efficiently handled by appropriate variants of these schemes, while preliminary tests on the 3-D system give some insight in the understanding of the analysis that needs to be done.

Interior point method and indefinite sparse solver for linear programming problems

In 1984, N. Karmarkar at AT&T Bell Labs, proposed a new method of solving the linear programming problem. It was claimed that this method, an interior point method (IPM), would be able to solve certain large-scale linear programming problems many times faster on average than the existing Simplex method. Recent studies have indicated that the interior point methods do seem to offer the computational advantages claimed. Furthermore, tremendous progress has also been made in recent years in developing highly efficient sparse solvers, an essential component of IPM. It is the purpose of this paper to explain a version of IPM and a version of direct sparse solver which has the capability of solving indefinite system of equations that arise from IPM.

Short Communication: A note on optimal hybrid V-cycle multilevel algorithms for mixed finite element systems with penalty term

In this paper, we investigate some cost-effective hybrid V-cycle multilevel algorithms for the discrete systems that arise when a mixed finite element approach is used to solve certain second-order elliptic boundary value problems. By introducing a small penalty parameter, the perturbed indefinite system can be reduced to a symmetric positive definite system involving the unknown flux alone. We study the numerical behaviour of some hybrid V-cycle multilevel algorithms with optimal computational complexity based on the hierarchical spatial decomposition approach proposed by Cai, Goldstein and Pasciaks for the reduced system. © 1997 by John Wiley & Sons, Ltd.

Short Communication: A note on optimal hybridV-cycle multilevel algorithms for mixed finite element systems with penalty term

In this paper, we investigate some cost-effective hybrid V-cycle multilevel algorithms for the discrete systems that arise when a mixed finite element approach is used to solve certain second-order elliptic boundary value problems. By introducing a small penalty parameter, the perturbed indefinite system can be reduced to a symmetric positive definite system involving the unknown flux alone. We study the numerical behaviour of some hybrid V-cycle multilevel algorithms with optimal computational complexity based on the hierarchical spatial decomposition approach proposed by Cai, Goldstein and Pasciaks for the reduced system. © 1997 by John Wiley & Sons, Ltd.

The Potential Fluid Flow Problem and the Convergence Rate of the Minimal Residual Method

In the paper the potential fluic flow problem in porous media using Darcy's law and the continuity equation is solved. Mixed-hybrid finite element formulation based on general trilateral prismatic elements is considered. Spectral properties of resulting symmetric indefinite system of linear equations are examined. Minimal residual method for the solution of systems with a symmetric indefinite matrix is applied. The rate of convergence and the asymptotic convergence factor which depend on the eigenvalue distribution of the system matrix are estimated.

MULTIGRID AND KRYLOV SUBSPACE METHODS FOR THE DISCRETE STOKES EQUATIONS

Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper we compare the performance of four such methods, namely variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual and multigrid methods, for solving several two-dimensional model problems. The results indicate that multigrid with smoothing based on incomplete factorization is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantage of being independent of iteration parameters.

An application of interior point quadratic programming algorithm to power system optimization problems

This paper presents a new interior point quadratic programming algorithm which can solve power system optimization problems with significantly less computational efforts. The proposed algorithm has the following two special features. First, it is based on the path-following interior point algorithm whose search direction is the Newton direction, and therefore the algorithm has quadratic convergence. In the second place, it solves directly a symmetric indefinite system and thus the algorithm avoids the formation of [AD/sup -1/A/sup T/] and as a result generates fewer fill-ins than the case of factorizing the positive definite system matrix for large scale systems. This has brought about a profound speed-up. Since the formula of the interior point method have been deduced more generally, the proposed algorithm can start from either a feasible (interior point) or an infeasible point (noninterior point). Numerical results on the IEEE test systems and a Japanese 344 bus system have verified that the proposed algorithm possesses enough robustness and needs significantly less solution time compared with already reported applications of the interior point method.

Mixed-hybrid finite element approximation of the potential fluid flow problem

In the paper a mixed-hybrid approximation of the potential fluid flow problem based on prismatic discretization of the domain is presented. Trilateral prismatic elements with vertical faces and nonparallel bases suitable for the modelling of real geological circumstances are considered. The set of linearly independent vector basis functions is defined and existence and uniqueness of the approximate solution from the resulting symmetric indefinite system are examined. Possible approaches to the solution of the discretized system are discussed.

Operator weighing in a multigrid method for locally refined grids

A multigrid method for an elliptic linear boundary value problem is presented. A consistent lowest-order mixed-finite-element discretisation is used on Cartesian locally refined grids in up to three space dimensions. The solution of the indefinite system of equations is reformulated as a problem of constrained minimisation. The constraint is satisfied exactly after one multigrid cycle; the functional is reduced iteratively by smoothing and coarse-grid corrections. By a suitable choice of prolongation and restriction operators, all corrections on coarser levels also reduce the functional within the constraint. This approach leads to a non-standard convergence proof which also holds for the variable-coefficient case. The proof does not predict the actual convergence rate, but shows that the functional will never increase after a multigrid cycle, while the constraint is satisfied exactly after the first multigrid cycle. The conditions required for convergence allow some freedom in choosing the restriction and prolongation operators, which can be used to improve the convergence rate. This leads to operator weighed restriction and prolongation operators in a novel manner. Some numerical examples are presented to demonstrate the robustness of the method.

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