Preconditioning Highly Indefinite and Nonsymmetric Matrices

Abstract

Abstract. Standard preconditioners, like incomplete factorizations, perform well when the coefficient matrix is diagonally dominant, but often fail on general sparse matrices. We experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices. The permutations and scalings are those developed by Olschowka and Neumaier [Linear Algebra Appl., 240 (1996), pp. 131-151] and by Duff and Koster [SIAM J. Matrix Anal. Appl., 20 (1999), pp. 889-901; Tech. report Ral-Tr-99-030, Rutherford Appleton Laboratory, Chilton, UK, 1999]. We target highly indefinite, nonsymmetric problems that cause difficulties for preconditioned iterative solvers. Our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing.

Fast Parallel Compositon of Dunhuang Murals based on Matrix Tearing

“…Gradient-domain techniques are not only used for compositing image regions, however. Various applications can be performed based on gradient domain constraints, ranging from shadow removal [19], intrinsic image recovery,1 high dynamic range (HDR) compression [20] …”

“…Pardiso [6,18], MUMPS [7,19], and SuperLU8 belong to direct methods. Though the direct methods are quite efficient for rmany applications, they might still be quite costly when matrixes are very large.…”

“…Let us assume that A has been reordered, for example using reverse CutHill-Mckee [17] or spectral [19][20][21] reordering schemes, into the overlapped diagonal blocks below , and (2) where, Mij,Xi and bi for i , j = 1 ,..., 3 are blocks of appropriate size. Let us define the partitions of M, delineated by lines in the illustration in (2), as (3) where, except for the overlap, = for ∂,ω= 0,1.…”

Max-plus singular values

“…In some cases H is close to being diagonally dominant, as the diagonal contains entries of modulus one and the off diagonal entries are all smaller. In these cases the performance of iterative methods is dramatically improved by applying the Hungarian scaling/reordering as a preprocessing step [14]. More generally the scaling/reordering can be shown to improve the speed of sparse direct linear system solvers through improved pivoting [15].…”

Heuristics for a Matrix Symmetrization Problem

“…Input: a bipartite graph G = (R ∪ C, E) corresponding to an n × n matrix A Input: a perfect matching M Output: another perfect matching M (1) where…”