Abstract not found

This research presents new preconditioners for linear systems. We proceed from the most general case to the very specific problem area of sparse optimal control. In the first most general approach, we assume only that the coefficient matrix is nonsingular. We target highly indefinite, nonsymmetric problems that cause difficulties for preconditioned iterative solvers, and where standard preconditioners, like incomplete factorizations, often fail. We experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of precon-ditioning for general sparse matrices. Our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing. Secondly, we present two new preconditioners for KKT systems. KKT systems arise in areas such as quadratic programming, sparse optimal control, and mixed finite element formulations. Our preconditioners approximate a constraint precon-ditioner with incomplete factorizations for the normal equations. Numerical experiments compare these two preconditioners with exact constraint preconditioning and the approach described above of permuting large entries to the diagonal. Finally, we turn to a specific problem area: sparse optimal control. Many optimal control problems are broken into several phases, and within a phase, most variables and constraints depend only on nearby variables and constraints. However, free initial and final times and time-independent parameters impact variables and constraints throughout a phase, resulting in dense factored blocks in the KKT matrix. We drop fill due to these variables to reduce density within each phase. The resulting preconditioner is tightly banded and nearly block tridiagonal. Numerical experiments demonstrate that the preconditioners are effective, with very little fill in the factorization.

Each step of an interior point method for nonlinear optimization requires the solution of a symmetric indefinite linear system known as a KKT system, or more generally, a saddle point problem. As the problem size increases, direct methods become prohibitively expensive to use for solving these problems; this leads to iterative solvers being the only viable alternative. In this thesis we consider iterative methods for solving saddle point systems and show that a projected preconditioned conjugate gradient method can be applied to these indefinite systems. Such a method requires the use of a specific class of preconditioners, (extended) constraint preconditioners, which exactly replicate some parts of the saddle point system that we wish to solve. The standard method for using constraint preconditioners, at least in the optimization community, has been to choose the constraint

Constraint-style preconditioners for regularized saddle point problems

We discuss the research activities of the Numerical Analysis Group in the Computational Science and Engineering Department at the Rutherford Appleton Laboratory of CLRC for the period January 1998 to December 1999. Keywords: sparse matrices, frontal and multifrontal methods, numerical linear algebra, large-scale eigenvalue computations, large-scale optimization, Fortran, Harwell Subroutine Library, HSL AMS(MOS) subject classifications: 65F05, 65F50. Current reports available by anonymous ftp to ftp.numerical.rl.ac.uk in directory pub/reports. This report is in file duffRAL2000001.ps.gz. Report also available through URL http://www.numerical.rl.ac.uk/reports/reports.html. Computational Science and Engineering Department Atlas Centre Rutherford Appleton Laboratory Oxon OX11 0QX January 21, 2000 Contents 1 Introduction (I. S. Duff) 1 2 Frontal and multifrontal methods 6 2.1 MUMPS - a distributed memory multifrontal solver (P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent and J. Kos...