Summary
A new algorithm is presented for computing vertices of a simplicial triangulation of thep-dimensional solution manifold of a parametrized equationF(x)=0, whereF is a nonlinear mapping fromR n toR m,p=n−m>1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of the tangent spaces that vary smoothly with their points of contact. The triangulation algorithm uses these bases, together with a chord form of the Gauss-Newton process as corrector, to compute the desired vertices. The Jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local “triangulation patches”. Several numerical examples show that the method is very efficient in computing triangulations, even around singularities such as limit points and bifurcation points. This opens up new possibilities for determining the form and special features of such solution manifolds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Allgower, E.L., Georg, K.: Generation of Triangulations by Reflections. Util. Math.16, 123–129 (1979)
Allgower, E.L., Schmidt, P.H.: An Algorithm for Piecewise-linear Approximation of an Implicitly Defined Manifold. SIAM J. Numer. Anal.22, 322–346 (1985)
Babuska, I., Rheinboldt, W.C.: Adaptive Finite Element Processes in Structural Mechanics. In: (Elliptic Problem Solvers II, Birkhoff G., Schoenstadt, A., eds.), pp. 345–378. New York: Academic Press
Bohl, E.: Finite Modelle gewöhnlicher Randwertaufgaben. Stuttgart: Teubner 1981
Berry, M.W., Heath, M.T., Kaneko, I., Law, M., Plemmons, R.J., Ward, R.C.: An Algorithm to Compute a Sparse Basis of the Null Space. Numer. Math.47, 483–504 (1985)
Coleman, T.F., Pothen, A.: The Null Space Problem II: Algorithms, Cornell University, Department of Computer Science, Technical Report TR 86-747, April 1986
Coleman, T.F., Sorensen, D.C.: A Note on the Computation of an Orthonormal Basis for the Null Space of a Matrix. Math. Program.29, 234–242 (1984)
Deuflhard, P., Heindl, G.: Affine Invariant Convergence Theorems for Newton's Method and Extensions to Related Methods. SIAM J. Numer. Anal.16, 1–10 (1979)
Fink, J.P., Rheinboldt, W.C.: Solution Manifolds of Parametrized Equations and their Discretization Error. Numer. Math.45, 323–343 (1984)
Fink, J.P., Rheinboldt, W.C.: A Geometric Framework for the Numerical Study of Singular Points. University of Pittsburgh, Institute for Computer Mathematics and Application. Technical Report ICMA-86-90, January 1986. SIAM J. Numer. Anal. (in press)
Griewank, A., Reddien, G.W.: Characterization and Computation of Generalized Turning Points. SIAM J. Numer. Anal.21, 176–185 (1984)
Golub, G.H., VanLoan, C.F.: Matrix Computations. Baltimore: The Johns Hopkins University Press 1983
Melhem, R.G., Rheinboldt, W.C.: A Comparison of Methods for Determining Turning Points of Nonlinear Equations. Computing29, 201–226 (1982)
Rheinboldt, W.C.: Numerical Methods for a Class of Finite-dimensional Bifurcation Problems. SIAM J. Numer. Anal.17, 221–237 (1980)
Rheinboldt, W.C.: Numerical Analysis of Parametrized Nonlinear Equations. New York: Wiley
Rheinboldt, W.C., Burkardt, J.V.: A Locally Parametrized Continuation Process. ACM Trans. Math. Software9, 236–246 (1983)
Spivak, M.: A Comprehensive Introduction to Differential Geometry. Five Volumes, Second Ed. Berkeley: Publish or Perish 1979
Todd, M.J.: The Computation of Fixed Points and Applications. New York, Berlin, Heidelberg: Springer 1976
Walker, A.C.: A Non-Linear Finite Element Analysis of Shallow Circular Arches. Int. J. Solids Struct.5, 97–107 (1969)
Author information
Authors and Affiliations
Additional information
Dedicated to Professor Ivo Babuška on the occasion of his sixtieth birthday
This work was supported in part by the National Science Foundation under Grant DCR-8309926, the Office of Naval Research under contract N-00014-80-C-9455, and the Air Force Office of Scientific Research under Grant 84-0131
Rights and permissions
About this article
Cite this article
Rheinboldt, W.C. On the computation of multi-dimensional solution manifolds of parametrized equations. Numer. Math. 53, 165–181 (1988). https://doi.org/10.1007/BF01395883
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01395883