Conditioning analysis of block incomplete factorization and its application to elliptic equations

The paper deals with eigenvalue estimates for block incomplete fac- torization methods for symmetric matrices. First, some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue. Second, upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated, which presents a substantial im- provement of earlier results. Finally, the results are used to estimate bounds for every eigenvalue of the preconditioned matrices, in particular, for the maximum eigenvalue, when a modified block incomplete factorization is used to solve an elliptic equation with variable coefficients in two dimensions. The analysis yields a new upper bound of type γh−1 for the condition number of the preconditioned matrix and shows clearly how the coefficients of the differential equation influ- ence the positive constant γ

On implicit-factorization constraint preconditioners

Recently Dollar and Wathen [14] proposed a class of incomplete factorizations for saddle-point problems, based upon earlier work by Schilders [40]. In this paper, we generalize this class of preconditioners, and examine the spectral implications of our approach. Numerical tests indicate the efficacy of our preconditioners

MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems

CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.Comment: 26 pages, 6 figure

A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices

Eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods, such as preconditioned conjugate gradient methods. In recent years various estimates have been derived for (perturbed) modified (block) incomplete factorizations. We survey and extend some of these and derive new estimates. In particular we derive upper and lower estimates of individual eigenvalues and of condition number. This includes a discussion that the condition number of preconditioned second order elliptic difference matrices is O(h-1). Some of the methods are applied to compute certain parameters involved in the computation of the preconditioner

Some Preconditioning Techniques for Saddle Point Problems

Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud \ud The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336

Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections

This work focuses on the iterative solution of sequences of KKT linear systems arising in interior point methods applied to large convex quadratic programming problems. This task is the computational core of the interior point procedure and an efficient preconditioning strategy is crucial for the efficiency of the overall method. Constraint preconditioners are very effective in this context; nevertheless, their computation may be very expensive for large-scale problems, and resorting to approximations of them may be convenient. Here we propose a procedure for building inexact constraint preconditioners by updating a "seed" constraint preconditioner computed for a KKT matrix at a previous interior point iteration. These updates are obtained through low-rank corrections of the Schur complement of the (1,1) block of the seed preconditioner. The updated preconditioners are analyzed both theoretically and computationally. The results obtained show that our updating procedure, coupled with an adaptive strategy for determining whether to reinitialize or update the preconditioner, can enhance the performance of interior point methods on large problems.Comment: 22 page