Numerische Mathematik

, Volume 62, Issue 1, pp 269–294 | Cite as

Adaptive data distribution for concurrent continuation

  • Eric F. Van de Velde
  • Jens Lorenz


Continuation methods compute paths of solutions of nonlinear equations that depend on a parameter. This paper examines some aspects of the multicomputer implementation of such methods. The computations are done on a mesh connected multicomputer with 64 nodes.

One of the main issues in the development of concurrent programs is load balancing, achieved here by using appropriate data distributions. In the continuation process, many linear systems have to be solved. For nearby points along the solution path, the corresponding system matrices are closely related to each other. Therefore, pivots which are good for theLU-decomposition of one matrix are likely to be acceptable for a whole segment of the solution path. This suggests to choose certain data distributions that achieve good load balancing. In addition, if these distributions are used, the resulting code is easily vectorized.

To test this technique, the invariant manifold of a system of two identical nonlinear oscillators is computed as a function of the coupling between them. This invariant manifold is determined by the solution of a system of nonlinear partial differential equations that depends on the coupling parameter. A symmetry in the problem reduces this system to one single equation, which is discretized by finite differences. The solution of the discrete nonlinear system is followed as the coupling parameter is changed.

Mathematics Subject Classification (1991)

15-04, 15A05 15A12 15A23 34C30 35L50 65-04 65F05 65F35 65Y05 65Y20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aronson, D.G., Doedel, E.J., Othmer, H.G. (1987): An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators. Physica,25D, 20–104Google Scholar
  2. Chan, T.F. (1984): Deflation techniques and block-elimination algorithms for solving bordered singular systems. SIAM J. Sci. Stat. Comput.5, 121–134Google Scholar
  3. Deuflhard, B., Fiedler, B., Kunkel, P. (1987): Efficient numerical pathfollowing beyond critical points. SIAM J. Numer. Anal.24, 912–927Google Scholar
  4. Dieci, L., Lorenz, J., Russel, R.D. (1989): Numerical calculation of invariant tori. SIAM J. Sci. Stat. Comput.12, 607–647Google Scholar
  5. Fenichel, N. (1971): Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J.21, 193–226Google Scholar
  6. Fox, G.C., Johnson, M.A., Lyzenga, G.A., Otto, S.W., Salmon, J.K., Walker, D.W. (1988): Solving Problems on Concurrent Processors. Prentice Hall, Englewood Cliffs, NJGoogle Scholar
  7. Higham, N.J., Higham, D.J. (1989): Large growth factors in Gaussian elimination with pivoting. SIAM J. Matrix Anal.10, 155–164Google Scholar
  8. Johnson, R.A. (1989): Hopf bifurcation from non-periodic solutions of differential equations i: Linear theory. Preprint, University of Southern CaliforniaGoogle Scholar
  9. Keller, H.B. (1982): Practical procedures in path following near limit points. In: R. Glowinski, J.L. Lions, eds., Computing Methods in Applied Sciences and Engineering. North-Holland, AmsterdamGoogle Scholar
  10. Keller, H.B. (1987): Numerical Methods in Bifurcation Problems. Tata Institute of Fundamental Research, BombayGoogle Scholar
  11. Rheinboldt, W.C. (1986): Numerical Analysis of Parametrized Nonlinear Equations. Wiley, New York, NYGoogle Scholar
  12. Sacker, R. (1969): A perturbation theorem for invariant manifolds and Hölder continuity. J. Math. Mech.18, 705–762Google Scholar
  13. Seydel, R. (1988): From Equilibrium to Chaos. Elsevier, New York Amsterdam LondonGoogle Scholar
  14. Toy, B.: LINPACK benchmark available through netlib@ornlGoogle Scholar
  15. Van de Velde, E.F. (1990): Experiments with multicomputer lu-decomposition. Concurrency: Practice and Experience2, 1–26Google Scholar
  16. Wilkinson, J.H. (1961): Error analysis of direct methods of matrix inversion. J. Assoc. Comput. Mach.8, 281–330Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Eric F. Van de Velde
    • 1
  • Jens Lorenz
    • 2
  1. 1.Applied Mathematics 217-50CaltechPasadenaUSA
  2. 2.Department of Mathematics and StatisticsThe University of New MexicoAlbuquerqueUSA

Personalised recommendations