Abstract
In this work we describe a portable sequential and parallel algorithm based on Newton’s method, for solving nonlinear systems. We used the GMRES iterative method to solve the inner iteration. To control the inner iteration as much as possible and avoid the oversolving problem, we also parallelized several forcing term criterions. We implemented the parallel algorithms using the parallel numerical linear algebra library SCALAPACK based on the MPI environment. Experimental results have been obtained using a cluster of Pentium II PC’s connected through a Myrinet network. To test our algorithms we used three different test problems, the H-Chandrasekhar problem, computing the intersection point of several hyper-surfaces, and the Extended Rosenbrock Problem. The latter requires some improvements for the method to work with structured sparse matrices and chaotic techniques. The algorithm obtained shows a good scalability in most cases. This work is included in a framework tool we are developing where, given a problem that implies solving a nonlinear system, the best nonlinear method must be chosen to solve the problem. The method we present here is one of the methods we implemented.
Work supported by Spanish CICYT. Project TIC 2000-1683-C03-03.
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References
Anderson E., Bai Z., Bischof C., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., Mackenney A., Ostrouchov S., Sorensen D. (1995). LAPACK Users’ Guide. Second edition. SIAM Publications, Philadelphia.
Balay S., Gropp W.D, Curfman L., and Smith B.F. (2001). PETSc Users Manual. T. Report ANL-95/11. Revision 2.11. Argonne National Laboratory.
Blackford L.S., Choi J., Cleary A., D'Azevedo E., Demmel J. Dhillon I., Dongarra J., Hammarling S., Henry G., Petitet A., Stanley K., Walker D., Whaley R.C. (1997). SCALAPACK Users’ Guide. SIAM Publications, Philadelphia.
Bru R, Elsner L, Neumann M. (1988). Models of Parallel Chaotic Iteration Methods. Linear Algebra and its Applications, 103: 175–192.
Einsestat S.C., Walker S.C. (1994). Choosing the Forcing Terms in an Inexact Newton Method. Center for Research on Parallel computation. Technical Report CRPRPC-TR94463. Rice University.
Frayssé V., Giraud L., Gratton S. (1997). A Set of GMRES routines for Real and Complex Arithmetics. CERFACS Technical Report TR/PA/97/49.
Frayssé V., Giraud L., Kharraz-Aroussi. (1998). On the influence of the othogonalization scheme on the parallel performance of GMRES. EUROPAR 98’. Parallel Processing. pp. 751–762, Vol 1470. Springer Verlag. Eds. D. Pritchard, J. Reeve.
Golub G. H., Van Loan C. F. (1996). Matrix Computations (third edition). John Hopkins University Press.
Kelley C.T. (1995). Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics, SIAM Publications, Philadelphia.
Kumar R., Grama A., Gupta A., Karypis G. (1994). Introduction to Parallel Computing: Design and Analysis of Algorithms. The Benjamin Cumimngs Publishing Company.
Myrinet.Myrinet overview. (Online). http://www.myri.com/myrinet/overview/index. Html.
Moré J.J., Garbow B.S., Hillstrom K.E, (1980). User Guide for MINPACK-1. Technical Report. ANL-80-74. Argonne National Laboratory.
Moré J.J, Garbow B.S and Hillstrom K.E. (1981). Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software, Vol 7, No.1, March 1981.
Moré J.J. (1990). A Collection of Nonlinear Model problems. Lectures in Applied Mathematics, Volume 26.,pp 723–762.
Peinado J, Vidal A.M. (2001). A new Parallel approach to the Toeplitz Inverse Eigenproblem using the Newton’s Method and Finite Difference techniques. VecPar’ 2000. 4th International Meeting on Vector and Parallel Processing. Selected Papers and Invited Talks. Lecture Notes in Computer Science. Ed. Springer.
Peinado J, Vidal A.M. (2001). Aproximación Paralela al Problema Inverso de los Valores Propios de Matrices Toeplitz mediante métodos tipo Newton. XII Jornadas de Paralelismo. Septiembre 2001. Valencia. pp 235–240.
Peinado J, Vidal A.M. (2001). Aproximación paralela al Método Newton-GMRES. T.Report DSIC-II/23/01. Departamento de Sistemas Informáticos y Computación Universidad Politécnica de Valencia. December 2001. Valencia.
Peinado J, Vidal A.M. (2002). Un marco Computacional Paralelo para la Resolución de Sistemas de gran Dimension. V Congreso de Métodos Numéricos en Ingenieria. Madrid, 3-6 de Junio de 2002. Accepted.
Saad Y., Schultz M. (1986). GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat, Comput. 7, 856–869.
Skjellum A. (1994). Using MPI: Portable Programming with Message-Passing Interface. MIT Press Group.
Whaley R.C. (1994). Basic Linear Algebra Communication Subprograms (BLACS): Analysis and Implementation Across Multiple Parallel Architectures. Computer Science Dept. Technical Report CS 94-234, University of Tennesee, Knoxville.
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Peinado, J., Vidal, A.M. (2003). A Parallel Newton-GMRES Algorithm for Solving Large Scale Nonlinear Systems. In: Palma, J.M.L.M., Sousa, A.A., Dongarra, J., Hernández, V. (eds) High Performance Computing for Computational Science — VECPAR 2002. VECPAR 2002. Lecture Notes in Computer Science, vol 2565. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36569-9_21
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