Abstract. A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. The first part with large diagonal dominance is reordered using a graph-based strategy, followed by an ILU factorization. A partial ILU factorization is applied to the second part to yield an approximate Schur complement matrix. The whole process is repeated on the Schur complement matrix and continues for a few times to yield a multilevel ILU factorization. Analyses are conducted to show how the Schur complement approach removes small diagonal elements of indefinite matrices and how the stability of the LU factor affects the quality of the preconditioner. Numerical results are used to compare the new preconditioning strategy with two popular ILU preconditioning techniques and a multilevel block ILU threshold preconditioner.

This paper introduces techniques based on diagonal threshold tolerance when developing multi-elimination and multi-level incomplete LU (ILUM) factorization preconditioners for solving general sparse linear systems. Existing heuristics solely based on the adjacency graph of the matrices have been used to find independent sets and are not robust for matrices arising from certain applications in which the matrices may have small or zero diagonals. New heuristic strategies based on the adjacency graph and the diagonal values of the matrices for finding independent sets are introduced. Analytical bounds for the factorization and preconditioned errors are obtained for the case of a two-level analysis. These bounds provide useful information in designing robust ILUM preconditioners. Extensive numerical experiments are conducted in order to compare robustness and efficiency of various heuristic strategies.

Incomplete factorization preconditioners such as ILU, ILUT and MILU are well-known robust general-purpose techniques for solving linear systems on serial computers. However, they are difficult to parallelize efficiently. Various techniques have been used to parallelize these preconditioners, such as multicolor orderings and subdomain preconditioning. These techniques may degrade the performance and robustness of ILU preconditionings. The purpose of this paper is to perform numerical experiments to compare these techniques in order to assess what are the most effective ways to use ILU preconditioning for practical problems on serial and parallel computers.

We investigate the use of sparse approximate inverse techniques in a multi-level block ILU preconditioner to design a robust and efficient parallelizable preconditioner for solving general sparse matrices. The resulting preconditioner retains robustness of the multi-level block ILU preconditioner (BILUM) and offers a new way to control the fill-in elements when large size blocks (subdomains) are used to form block independent set. Moreover, the new preconditioner affords maximum parallelism for operations within each level as well as for the coarsest level solution. Thus it has two advantages over the standard BILUM preconditioner: the ability to control sparsity and increased parallelism. Numerical experiments are used to show the effectiveness and efficiency of the new preconditioner. Key words: Sparse matrices, incomplete LU factorization, multi-level ILU preconditioner, sparse approximate inverse, Krylov subspace methods. AMS subject classifications: 65F10, 65N06. 1 Introduction ...

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.

A multi-level preconditioned iterative method based on a multi-level block ILU factorization preconditioning technique is introduced and is applied to the solution of unstructured sparse linear systems arising from the numerical simulation of coating problems. The coefficient matrices usually have several rows with zero diagonal values that may cause stability difficulty in standard ILU factorization techniques. The new preconditioning strategy employs a diagonal threshold tolerance and a local reordering of individual blocks to increase robustness of the multi-level block ILU factorization process. Keywords: sparse matrices, multi-level preconditioning, ILU factorization 1 Introduction In this paper, a multi-level block incomplete LU (ILU) preconditioning technique is designed for solving unstructured sparse linear systems from the numerical simulation of coating problems. Coating is a delicate process of putting a layer of one liquid material (film) over another solid material unifo...

A fast Newton-Krylov algorithm is presented for solving the compressible Navier-Stokes equations on structured multi-block grids with application to turbulent aerodynamic flows. The oneequation Spalart-Allmaras model is used to provide the turbulent viscosity. The optimization of the algorithm is discussed. ILU(4) is suggested for a preconditioner, operating on a modified Jacobian matrix. An efficient startup method to bring the system into the region of convergence of Newton's method is given. Three test cases are used to demonstrate convergence rates. Singleelement cases are solved in less than 100 seconds on an engineering workstation, while the solution of a multi-element case can be found in less than 25 minutes.

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This paper presents a class of preconditioning techniques which exploit rational function approximations to the original matrix. The matrix is first shifted and then an incomplete LU factorization of the resulting matrix is computed. The resulting factors are then used to compute a better preconditioner to the original matrix. Since the incomplete factorization is made on a shifted matrix, a good LU factorization is obtained without allowing much fill-in. The result needs to be extrapolated to the non-shifted matrix. Thus, the main motivation for this process is to save memory. The method is useful for matrices whose incomplete LU factorizations are poor, e.g., unstable. An error analysis for the conjugate gradient algorithm gives some guidance for choosing the shift of the matrix, in the special case where the shifted system is solved exactly. 1 Introduction Rational approximation preconditioners are targeted at extremely ill-conditioned linear systems. Examples of such systems are t...