Parallel adaptive GMRES implementations for homotopy methods

SIAM Journal on Optimization

Volumn 9, Issue 4, 1999, Pages 1149-1158

  Sosonkina, Maria ^a   Allison, Donald C S ^b   Watson, Layne T ^b  

^a UNIVERSITY OF MINNESOTA DULUTH   (United States)

^b VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

Globally convergent; GMRES method; Krylov subspace methods; Nonsymmetric linear systems; Probability one homotopy; Sparse matrix

Indexed keywords

EID: 0033420920     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623497329671     Document Type: Article

References (26)

1. E. L. ALLGOWER, C.-S. CHIEN, AND K. GEORG, Large sparse continuation problems, J. Comput. Appl. Math., 26 (1989), pp. 3-21.
2. D. C. S. ALLISON, A. CHAKRABORTY, AND L. T. WATSON, Granularity issues for solving polynomial systems via globally convergent algorithms on a hypercube, J. Supercomputing, 3 (1989), pp. 5-20.
3. D. C. S. ALLISON, K. M. IRANI, C. J. RIBBENS, AND L. T. WATSON, High-dimensional homotopy curve tracking on a shared-memory multiprocessor, J. Supercomputing, 5 (1992), pp. 347-366.
4. Z. BAI, D. Hu, AND L. REICHEL, A Newton basis GMRES implementation, IMA J. Numer. Anal., 4 (1994), pp. 563-581.
5. A. BJÖRCK, Solving linear least squares problems by Gram-Schmidt orthogonalization, BIT, 7 (1967), pp. 1-21.
6. C. H. BISCHOF AND P. T. P. TANG, Robust Incremental Condition Estimation, Tech. Report CS-91-133, LAPACK Working Note 33, Computer Science Department, University of Tennessee, Knoxville, TN, 1991.
7. P. N. BROWN AND H. F. WALKER, GMRES on (nearly) singular systems, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 37-51.
8. A. CHAKRABORTY, D. C. S. ALLISON, C. J. RIBBENS, AND L. T. WATSON, The parallel complexity of embedding algorithms for the solution of nonlinear equations, IEEE Trans. Parallel Distrib. Systems, 4 (1993), pp. 458-465.
9. C.-S. CHIEN, Z.-L. WENG, AND C.-L. SHEN, Lanczos-type methods jor continuation problems, Numer. Linear Algebra Appl., 4 (1997), pp. 23-41.
10. I. DUFF, R. G. GRIMES, AND J. G. LEWIS, Users' Guide for the Harwell-Boeing Sparse Matrix Collection (Release I), Tech. Report TR/PA/92/86, CERFACS, France, 1992.
11. J. ERTHEL, A parallel GMRES for general sparse matrices, Electron. Trans. Numer. Anal., 3 (1995), pp. 160-176.
12. 2/H ∞ controller synthesis, Optimal Control Appl. Methods, 17 (1996), pp. 187-208.
13. K. M. IRANI, M. P. KAMAT, C. J. RIBBENS, H. F. WALKER, AND L. T. WATSON, Experiments with conjugate gradient algorithms for homotopy curve tracking, SIAM J. Optim., 1 (1991), pp. 222-251.
14. G. KARYPIS AND V. KUMAR, MeTiS: Unstructured Graph Partitioning and Sparse Matrix Ordering System, User's Guide-Version 2.0, Department of Computer Science, University of Minnesota, Minneapolis, MN, 1995.
15. G.-C. LO AND Y. SAAD, Iterative Solution of General Sparse Linear Systems on Clusters of Workstations, Tech. Report, UMSI 96/117, Supercomputer Institute, University of Minnesota, Minneapolis, MN, August 1996.
16. W. D. MCQUAIN, R. C. MELVILLE, C. J. RIBBENS, AND L. T. WATSON, Preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation, Comput. Math. Appl., 27 (1994), pp. 25-45.
17. Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS, Boston, 1996.
18. Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal residual method for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869.
19. R. B. SlDJE, Alternatives for parallel Krylov subspace basis computation, Numer. Linear Algebra Appl., 4 (1997), pp. 305-331.
20. M. SOSONKINA, L. T. WATSON, R. K. KAPANIA, AND H. F. WALKER, A new adaptive GMRES algorithm for achieving high accuracy, Numer. Linear Algebra Appl., 5 (1998), pp. 275-297.
21. H. A. VAN DER VORST AND C. VUIK, The superlinear convergence behaviour of GMRES, J. Comput. Appl. Math., 48 (1993), pp. 327-341.
22. H. F. WALKER, Implementation of the GMRES method using Householder transformations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 152-163.
23. L. T. WATSON, Globally convergent homotopy algorithms for nonlinear systems of equations, Nonlinear Dynam., 1 (1990), pp. 143-191.
24. L. T. WATSON, Globally convergent homotopy methods: A tutorial, Appl. Math. Comput., 31 (1989), pp. 369-396.
25. L. T. WATSON, Numerical linear algebra aspects of globally convergent homotopy methods, SIAM Rev., 28 (1986), pp. 529-545.
26. L. T. WATSON, S. C. BILLUPS, AND A. P. MORGAN, Algorithm 652: HOMPACK: A suite of codes for globally convergent homotopy algorithms, ACM Trans. Math. Software, 13 (1987), pp. 281-310.