INCOMPLETE CHOLESKY FACTORIZATION IN FIXED MEMORY WITH FLEXIBLE DROP-TOLERANCE STRATEGY
DOI:
https://doi.org/10.47839/ijc.2.2.200Keywords:
Large sparse system, drop-tolerance strategy, preconditioner, conjugate gradients, fixed memoryAbstract
We propose an incomplete Cholesky factorization for the solution of large positive definite systems of equations and for the solution of large-scale trust region sub-problems. The factorization is based on the two- parameter (m, p) drop-tolerance strategy for insignificant elements in the incomplete factor matrix. The factorization proposed essentially reduces the negative processes of irregular distribution and accumulation of errors in factor matrix and provides the optimal rate of memory filling with essential nonzero elements. On the contrary to the known p - retain and t - drop-tolerance strategies, the (m, p) strategy allows to form the factor matrix in fixed memory.References
A. Bouaricha, J. More, Z. Wu. Preconditioning Newton’s Method, Rice University, Center for Research on Parallel Computation, Houston, TX, May 1998, 21 p.
C. Lin, J. More. Incomplete Cholesky factorizations with limited memory, SIAM Journal on Sci. Comput., No.1 (1999). pp. 24–45.
E. Chow, Y. Saad. Experimental study of ILU Preconditioners for indefinite matrices, Department of Computer Science and Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN, June 1997. p. 32.
N. Li, Y. Saad, E. Chow. Crout versions of ILU for general sparse matrices. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN; Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA, April 2002. p. 17.
Y. Saad. Iterative Methods for Sparse Linear Systems. University of Minnesota, Department of Computer Science and Engineering, Minneapolis, MN, 2000, 447 p.
A. George, J. Liu. Computer Solution of Large Sparse Positive Definite Systems, Moscow: Mir, 1984. p. 333.
S. Pissanetzky. Sparse Matrix Technology. Moscow: Mir, 1988. p. 410.
B. Averick, J. More. Evaluation of large-scale optimization problems on vector and parallel architectures, SIAM Journal on Optimization, No.4, 1994, pp.708-721.
C. Schelthoff, A. Basermann, Polynomial Preconditioning for the Conjugate Gradient Method on Massively Parallel Systems, Informatik-Bericht, No.1, 1995, pp. 150-167.
J. Dennis, R. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Moscow: Mir, 1988. p. 440.
T. Stehiaug, The conjugate gradient method and trust regions in large-scale optimization, SIAM Journal on Numerical Analyses, No.20, 1983. pp.626–637.
M. Jones, P. Plassmann. An improved incomplete Cholesky factorizations. ACM Trans. Math. Software, No.21, 1995, pp.5-17.
N. Munksgaard. Solving sparse symmetric sets of linear equations by preconditional conjugate gradients, ACM, Trans. Math. Software, No.6, 1980, pp.206–219.
I. Gustafsson, A class of first order factorization methods. BIT. No.18, 1978. pp.142–156.
Downloads
Published
How to Cite
Issue
Section
License
International Journal of Computing is an open access journal. Authors who publish with this journal agree to the following terms:• Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
• Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
• Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.