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.

We present the main algorithmic features in the software package SuperLU DIST, a distributedmemory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software’s parallel performance and scalability on current machines. The solver is based on sparse Gaussian elimination, with an innovative static pivoting strategy proposed earlier by the authors. The main advantage of static pivoting over classical partial pivoting is that it permits a priori determination of data structures and communication patterns, which lets us exploit techniques used in parallel sparse Cholesky algorithms to better parallelize both LU decomposition and triangular solution on large-scale distributed machines.

This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.

This paper analyzes dropping strategies in a multilevel incomplete LU decomposition context and presents a few of strategies for obtaining related ILUs with enhanced robustness. The analysis shows that the Incomplete LU factorization resulting from dropping small entries in Gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large. As a consequence a few strategies are developed whose goal is to achieve this feature. A number of “templates” for enabling implementations of these factorizations are presented. Numerical experiments show that the resulting ILUs offer a good compromise between robustness and efficiency.

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. Maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems, especially in direct solvers. In this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for incomplete LDL T factorizations. The reorderings are constructed such that the matched entries form 1 × 1or2 × 2 diagonal blocks in order to increase the diagonal dominance of the system. During the incomplete factorization only tridiagonal pivoting is used. We report results for this approach and comparisons with other solution methods for a diverse set of symmetric indefinite matrices, ranging from nonlinear elasticity to interior point optimization.

This paper presents a preconditioning method based on combining two-sided permutations with a multilevel approach. The nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix. The method can be regarded as a complete pivoting version of the incomplete LU factorization. This leads to an effective incomplete factorization preconditioner for general nonsymmetric, irregularly structured, sparse linear systems.

Abstract. This paper addresses the parallelization of the preconditioned iterative methods that use explicit preconditioners such as approximate inverses. Parallelizing a full step of these methods requires the coefficient and preconditioner matrices to be well partitioned. We first show that different methods impose different partitioning requirements for the matrices. Then we develop hypergraph models to meet those requirements. In particular, we develop models that enable us to obtain partitionings on the coefficient and preconditioner matrices simultaneously. Experiments on a set of unsymmetric sparse matrices show that the proposed models yield effective partitioning results. A parallel implementation of the right preconditioned BiCGStab method on a PC cluster verifies that the theoretical gains obtained by the models hold in practice.

In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard LU/LDL T factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from the approach based on the Sherman-Morrison formula [16]. In contrast to the RIF algorithm [9], the direct and inverse factors here directly influence each other throughout the computation. Consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. Although we describe the theory behind the factorization for general nonsymmetric matrices, in implementation and experiments we restrict for clarity and conciseness only to the case when the system matrix is symmetric and positive definite. In this case, we call the new approximate LDL T factorization Balanced Incomplete Factorization (BIF). Our experimental results confirm that this factorization is very robust and may be useful in solving difficult ill-conditioned problems by preconditioned iterative methods. Moreover, the internal coupling of computation of direct and inverse factors results in much shorter setup times (times to compute approximate decomposition) than RIF, a method of a similar and very high level of robustness.

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