Numerically-aware orderings for sparse symmetric indefinite linear systems

Sparse symmetric indefinite problems arise in a large number of important application areas; they are often solved through the use of an LDLT factorization via a sparse direct solver. Whilst for many problems, prescaling the system matrix A is sufficient to maintain stability of the factorization, for a small but important fraction of problems numerical pivoting is required. Pivoting often incurs a significant overhead and consequently a number of techniques have been proposed to try and limit the need for pivoting. In particular, numerically-aware ordering algorithms may be used, that is, orderings that depend not only on the sparsity pattern of A but also on the values of its (scaled) entries. Current approaches identify large entries of A and symmetrically permute them onto the subdiagonal where they can be used as part of a 2x2 pivot. This is numerically effective, but the fill in the factor L and hence the runtime of the factorization and subsequent triangular solves may be significantly increased over a standard ordering if no pivoting is required. We present a new algorithm that combines a matching-based approach with a numerically-aware nested dissection ordering. Numerical comparisons with current approaches for some tough symmetric indefinite problems are given

The design and use of a sparse direct solver for skew symmetric matrices

AbstractWe consider the LDLT factorization of sparse skew symmetric matrices. We see that the pivoting strategies are similar, but simpler, to those used in the factorization of sparse symmetric indefinite matrices, and we briefly describe the algorithms used in a forthcoming direct code based on multifrontal techniques for the factorization of real skew symmetric matrices. We show how this factorization can be very efficient for preconditioning matrices that have a large skew component

Improving the stability and robustness of incomplete symmetric indefinite factorization preconditioners

Sparse symmetric indefinite linear systems of equations arise in numerous practical applications. In many situations, an iterative method is the method of choice but a preconditioner is normally required for it to be effective. In this paper, the focus is on a class of incomplete factorization algorithms that can be used to compute preconditioners for symmetric indefinite systems. A limited memory approach is employed that incorporates a number of new ideas with the goal of improving the stability, robustness and efficiency of the preconditioner. These include the monitoring of stability as the factorization proceeds and the incorporation of pivot modifications when potential instability is observed. Numerical experiments involving test problems arising from a range of real-world applications demonstrate the effectiveness of our approach

Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization

Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly ill-conditioned symmetric saddle-point problems have to be solved. We present combinatorial methods to preprocess these matrices in order to establish more favorable numerical properties for the subsequent factorization. Our approach is based on symmetric weighted matchings and is used in a sparse direct LDL T factorization method where the pivoting is restricted to static supernode data structures. In addition, we will dynamically expand the supernode data structure in cases where additional fill-in helps to select better numerical pivot elements. This technique can be seen as an alternative to the more traditional threshold pivoting techniques. We demonstrate the competitiveness of this approach within an interior-point method on a large set of test problems from the CUTE and COPS sets, as well as large optimal control problems based on partial differential equations. The largest nonlinear optimization problem solved has more than 12 million variables and 6 million constraint

Comparing pivoting strategies for almost strictly sign regular matrices

In this paper some properties of two-determinant pivoting for Neville elimination are presented. In particular, we consider a zero-increasing property and we show an optimal normwise growth factor. Comparisons with other pivoting strategies for Neville elimination and with Gaussian elimination with partial pivoting of almost strictly sign regular matrices are performed. Numerical examples are included

Data Structures and Algorithms for Efficient Solution of Simultaneous Linear Equations from 3-D Ice Sheet Models

Two current software packages for solving large systems of sparse simultaneous l~neare equations are evaluated in terms of their applicability to solving systems of equations generated by the University of Maine Ice Sheet Model. SuperLU, the first package, has been developed by researchers at the University of California at Berkeley and the Lawrence Berkeley National Laboratory. UMFPACK, the second package, has been developed by T. A. Davis of the University of Florida who has ties with the U. C. Berkeley researchers as well as European researchers. Both packages are direct solvers that use LU factorization with forward and backward substitution. The University of Maine Ice Sheet Model uses the finite element method to solve partial differential equations that describe ice thickness, velocity,and temperature throughout glaciers as functions of position and t~me. The finite element method generates systems of linear equations having tens of thousands of variables and one hundred or so non-zero coefficients per equation. Matrices representing these systems of equations may be strictly banded or banded with right and lower borders. In order to efficiently Interface the software packages with the ice sheet model, a modified compressed column data structure and supporting routines were designed and written. The data structure interfaces directly with both software packages and allows the ice sheet model to access matrix coefficients by row and column number in roughly 100 nanoseconds while only storing non-zero entries of the matrix. No a priori knowledge of the matrix\u27s sparsity pattern is required. Both software packages were tested with matrices produced by the model and performance characteristics were measured arid compared with banded Gaussian elimination. When combined with high performance basic linear algebra subprograms (BLAS), the packages are as much as 5 to 7 times faster than banded Gaussian elimination. The BLAS produced by K. Goto of the University of Texas was used. Memory usage by the packages varted from slightly more than banded Gaussian elimination with UMFPACK, to as much as a 40% savings with SuperLU. In addition, the packages provide componentwise backward error measures and estimates of the matrix\u27s condition number. SuperLU is available for parallel computers as well as single processor computers. UMPACK is only for single processor computers. Both packages are also capable of efficiently solving the bordered matrix problem

Distribution of the number of pivots needed using Gaussian elimination with partial pivoting on random matrices

Gaussian elimination with partial pivoting (GEPP) is a widely used method to solve dense linear systems. Each GEPP step uses a row transposition pivot movement if needed to ensure the leading pivot entry is maximal in magnitude for the leading column of the remaining untriangularized subsystem. We will use theoretical and numerical approaches to study how often this pivot movement is needed. We provide full distributional descriptions for the number of pivot movements needed using GEPP using particular Haar random ensembles, as well as compare these models to other common transformations from randomized numerical linear algebra. Additionally, we introduce new random ensembles with fixed pivot movement counts and fixed sparsity, $\alpha$. Experiments estimating the empirical spectral density (ESD) of these random ensembles leads to a new conjecture on a universality class of random matrices with fixed sparsity whose scaled ESD converges to a measure on the complex unit disk that depends on $\alpha$ and is an interpolation of the uniform measure on the unit disk and the Dirac measure at the origin

On using Cholesky-based factorizations and regularization for solving rank-deficient sparse linear least-squares problems

By examining the performance of modern parallel sparse direct solvers and exploiting our knowledge of the algorithms behind them, we perform numerical experiments to study how they can be used to efficiently solve rank-deficient sparse linear least-squares problems arising from practical applications. The Cholesky factorization of the normal equations breaks down when the least-squares problem is rank-deficient, while applying a symmetric indefinite solver to the augmented system can give an unacceptable level of fill in the factors. To try to resolve these difficulties, we consider a regularization procedure that modifies the diagonal of the unregularized matrix. This leads to matrices that are easier to factorize. We consider both the regularized normal equations and the regularized augmented system. We employ the computed factors of the regularized systems as preconditioners with an iterative solver to obtain the solution of the original (unregularized) problem. Furthermore, we look at using limited-memory incomplete Cholesky-based factorizations and how these can offer the potential to solve very large problems