Diagonal threshold techniques in robust multi‐level ILU preconditioners for general sparse linear systems

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. Copyright © 1999 John Wiley & Sons, Ltd.     

Multicolor low‐rank preconditioner for general sparse linear systems

This article presents a multilevel parallel preconditioning technique for solving general large sparse linear systems of equations. Subdomain coloring is invoked to reorder the coefficient matrix by multicoloring the adjacency graph of the subdomains, resulting in a two‐level block diagonal structure. A full binary tree structure $\mathrm{??}$ is then built to facilitate the construction of the preconditioner. A key property that is exploited is the observation that the difference between the inverse of the original matrix and that of its block diagonal approximation is often well approximated by a low‐rank matrix. This property and the block diagonal structure of the reordered matrix lead to a multicolor low‐rank (MCLR) preconditioner. The construction procedure of the MCLR preconditioner follows a bottom‐up traversal of the tree $\mathrm{??}$. All irregular matrix computations, such as ILU factorizations and related triangular solves, are restricted to leaf nodes where these operations can be performed independently. Computations in nonleaf nodes only involve easy‐to‐optimize dense matrix operations. In order to further reduce the number of iteration of the Preconditioned Krylov subspace procedure, we combine MCLR with a few classical block‐relaxation techniques. Numerical experiments on various test problems are proposed to illustrate the robustness and efficiency of the proposed approach for solving large sparse symmetric and nonsymmetric linear systems.     

Flexible and multi‐shift induced dimension reduction algorithms for solving large sparse linear systems

We give two generalizations of the induced dimension reduction (IDR) approach for the solution of linear systems. We derive a flexible and a multi‐shift quasi‐minimal residual IDR variant. These variants are based on a generalized Hessenberg decomposition. We present a new, more stable way to compute basis vectors in IDR. Numerical examples are presented to show the effectiveness of these new IDR variants and the new basis compared with existing ones and to other Krylov subspace methods. Copyright © 2014 John Wiley & Sons, Ltd.