Indefinite Systems Of Linear Equations Research Articles

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.

Fast iterative solution of stabilized Navier–Stokes systems

Discretization and linearization of the steady-state Navier–Stokes equations give rise to a nonsymmetric indefinite linear system of equations. In this paper we address the block triangular preconditioning iterative methods for solving this type of systems which result from stabilized mixed finite element discretization of the linearized Navier–Stokes equations. The convergence results are established and the results of the numerical experiments are presented.

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.

Coarse space orthogonalization for indefinite linear systems of equations arising in geometrically nonlinear elasticity

This paper is concerned with the solution of indefinite linear systems arising in the course of Newton iterations for problems of geometrically nonlinear elasticity by Krylov subspace methods. The convergence of these methods often slows down considerably due to the ill-conditioning or even indefiniteness of the matrix near simple limit or bifurcation points. We illustrate that orthogonalizing against a low-dimensional coarse finite element space is a key ingredient to get satisfactory convergence in all parameter ranges. This approach is combined with the usual hierarchical and multilevel preconditioners and implementation issues are discussed in some detail. Computational experiments performed for a realistic example from planar elasticity using isoparametric finite elements illustrate the effectiveness of the coarse space orthogonalization approach.

A preconditioning technique for indefinite linear systems

In this paper we discuss a new preconditioning technique developed for the solution of indefinite systems of linear equations. Such systems arise in finite element structural analysis using Lagrange multiplier based constraint processing. We use a specially constructed indefinite matrix as a preconditioner which is applied in every iteration step of the conjugate residual method. Numerical results for some test problems are presented.

Stability of Augmented System Factorizations in Interior-Point Methods

Some implementations of interior-point algorithms obtain their search directions by solving symmetric indefinite systems of linear equations. The conditioning of the coefficient matrices in these so-called augmented systems deteriorates on later iterations, as some of the diagonal elements grow without bound. Despite this apparent difficulty, the steps produced by standard factorization procedures are often accurate enough to allow the interior-point method to converge to high accuracy. When the underlying linear program is nondegenerate, we show that convergence to arbitrarily high accuracy occurs, at a rate that closely approximates the theory. We also explain and demonstrate what happens when the linear program is degenerate, where convergence to acceptable accuracy (but not arbitrarily high accuracy) is usually obtained.

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.

Fast Nonsymmetric Iterations and Preconditioning for Navier–Stokes Equations

Discretization and linearization of the steady–state Navier-Stokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization. We confirm and supplement these analytic results with a series of numerical experiments indicating that Krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter. In addition, we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner.

Conjugate Gradient-like solution algorithms for the mixed finite element approximation of the biharmonic equation, applied to plate bending problems

Discretization of the Biharmonic Equation with the Mixed Finite Element Method yields an indefinite linear system of equations with a special structure. In this paper two variants of the Conjugate Gradient method are formulated that are suited for solving such systems. They both require the solution of a system of linear equations in every iteration. Different strategies for doing this are examined. An Incomplete Choleski decomposition is used as a preconditioner. Both iterative methods and the preconditioner are chosen so that optimal use can be made of the special block structure of the global system of equations.

A Preconditioned GMRES Method for Nonsymmetric or Indefinite Problems

A preconditioning technique is proposed for nonsymmetric or indefinite linear systems of equations. The main idea in our theory, roughly speaking, is first to use some "coarser mesh" space to correct the nonpositive portion of the eigenvalues of the underlying operator and then switch to use a symmetric positive definite preconditioner. The generality of our theory allows us to apply any known preconditioners that were orginally designed for symmetric positive definite problems to nonsymmetric or indefinite problems, without losing the optimality that the original one has. Some numerical experiments based on GMRES are reported.

A preconditioned GMRES method for nonsymmetric or indefinite problems

A preconditioning technique is proposed for nonsymmetric or indefinite linear systems of equations. The main idea in our theory, roughly speaking, is first to use some "coarser mesh" space to correct the nonpositive portion of the eigenvalues of the underlying operator and then switch to use a symmetric positive definite preconditioner. The generality of our theory allows us to apply any known preconditioners that were orginally designed for symmetric positive definite problems to nonsymmetric or indefinite problems, without losing the optimality that the original one has. Some numerical experiments based on GMRES are reported.

A class of iterative methods for solving saddle point problems

We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations.

Two Conjugate-Gradient-Type Methods for Unsymmetric Linear Equations

We propose two new conjugate-gradient-type methods for the solution of sparse unsymmetric linear systems. We present a new tridiagonalization process for unsymmetric matrices that is closely related to the Lanczos process. We use orthogonal factorizations of the tridiagonal matrix to derive the new algorithms USYMLQ and USYMQR in the same fashion as SYMMLQ and MINRES [C. C. Paige and M. A. Saunders, “Solution of sparse indefinite systems of linear equations,” SIAM J. Numer. Anal., 12 (1975), pp. 617–629], for symmetric matrices. Some numerical results for the new methods and comparisons with other methods are presented.

Sparse matrices

One gives a survey of methods and programs for solving large sparse spectral problems based on the Lanczos algorithm. Practically all the important works on this topic are reflected in this survey. One also considers applications of the variants of the Lanczos method to the solution of symmetric indefinite systems of linear equations and to a series of other problems of linear algebra.

Practical Use of Some Krylov Subspace Methods for Solving Indefinite and Nonsymmetric Linear Systems

The main purpose of this paper is to develop stable versions of some Krylov subspace methods for solving linear systems of equations $Ax = b$. As in the case of Paige and Saunders’s SYMMLQ [SIAM J. Numer. Anal., 12 (1975), pp. 617–624], our algorithms are based on stable factorizations of the banded Hessenberg matrix representing the restriction of the linear application A to a Krylov subspace. We will show how an algorithm similar to the SYMMLQ can be derived for nonsymmetric problems and we will describe a more economical algorithm based upon the $LU$ factorization with partial pivoting. In the particular case where A is symmetric indefinite the new algorithm is theoretically equivalent to SYMMLQ but slightly more economical. As a consequence, an advantage of the new approach is that nonsymmetric or symmetric indefinite or both nonsymmetric and indefinite systems of linear equations can be handled by a single algorithm.

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.