Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

. The problem of nding good preconditioners for the numerical solution of indenite linear systems is considered. Special emphasis is put on preconditioners that have a 2 2 block structure and which incorporate the (1; 2) and (2; 1) blocks of the original matrix. Results concerning the spectrum and form of the eigenvectors of the preconditioned matrix and its minimum polynomial are given. The consequences of these results are considered for a variety of Krylov subspace methods. Numerical experiments validate these conclusions. Key words. preconditioning, indenite matrices, Krylov subspace methods AMS subject classications. 65F10, 65F15, 65F50 1. Introduction. In this paper, we are concerned with investigating a new class of preconditioners for indenite systems of linear equations of a sort which arise in constrained optimization as well as in least-squares, saddle-point and Stokes problems. We attempt to solve the indenite linear system A B T B 0 | {z } A x 1 x...

Abstract. In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric/ skew-symmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The potential of this approach is illustrated by numerical

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.

ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linear-quadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linear-quadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that for a class of problems the performance of the two-level versions of our preconditioners is independent of the mesh size and of the subdomain size. 1.

We propose and analyze a class of penalty-function-free nonmonotone trust-region methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the Byrd--Omojokun class of algorithms, each step is composed of a quasinormal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trust-region subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filter-based methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach.

Abstract. We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions.

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

We propose and examine block-diagonal preconditioners and variants of indefinite preconditioners for block two-by-two generalized saddle-point problems. That is, we consider the nonsymmetric, nonsingular case where the (2,2) block is small in norm, and we are particularly concerned with the case where the (1,2) block is different from the transposed (2,1) block. We provide theoretical and experimental analyses of the convergence and eigenvalue distributions of the preconditioned matrices. We also extend the results of [de Sturler and Liesen 2005] to matrices with non-zero (2,2) block and to the use of approximate Schur complements. To demonstrate the effectiveness of these preconditioners we show convergence results, spectra and eigenvalue bounds for two model Navier-Stokes problems.

Abstract. For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications. 1.