Abstract
The present article is a short summary or a more extensive presentation — see [9]. Multiple shooting (MS) techniques as developed in [4,16,19,5] are one of the popular approaches for the numerical solution of (in general nonlinear) boundary value problems (BVP’s) for ordinary differential equations (ODE’s). For alternative approaches see e.g. [1,14]. For a recent survey on MS techniques see [6], where also further references and historical remarks can be found. The application of MS techniques to parameter identification has been nicely developed in [2]. In fact, the efficient and economic implementation of a parameter identification algorithm has motivated the present study. Even though, for the purpose of simplification, most of the presentation is confined to the standard BVP case (without explicit dependence on parameters), the results also apply — mutatis mutandis — to the parameter identification case.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
U. Ascher, I. Christiansen, R.D. Russell: A Collocation Solver for Mixed Order Systems of Boundary Value Problems. Math. Comp. 33, 659–679 (1979)
H.G. Bock: Numerical Treatment of Inverse Problems in Chemical Reaction Kinetics. Springer Series Chemical Physics 18, section 8, p.102–125 (1981).
C.G. Broyden: A class of methods for solving non-linear simultaneous ecuations. Math. Comp. 19, 577–583 (1965).
R. Bulirsch: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Carl-Cranz-Gesellschaft: Tech.Rep. (Oct. 1971).
P. Deuflhard: Non-linear Equation Solvers in Boundary Value Problem Codes. Sprinqer Lecture Notes Computer Science 76, (Childs et al., ed.), p. 40–66 (1979).
P. Deuflhard: Recent Advances in Multiple Shooting Techniques. In: Computational Techniques for Ordinary Differential Equations (Gladwell/ Sayers, ed.), Section 10, p.217–272. London, New York: Academic Press 1980).
P. Deuflhard: Order and Stepsize Control in Extrapolation Methods. Universität Heidelberg, SFB 123: Tech. Rep. 93 (1980). To appear in Numer. Math. (1983).
P. Deuflhard, V. Apostolescu: An Underrelaxed Gauss-Newton Method for Equality Constrained Nonlinear Least Squares Problems. Springer Lecture Notes Control Inf.Sci. 7, 22–32 (1978).
P. Deuflhard, G. Bader: Multiple Shooting Techniques Revisited. University of Heidelberg, SFB 123: Tech. Rep. 163 (June 1982).
P. Deuflhard, W. Sautter: On Rank-Deficient Pseudo-Inverses. J. Lin. Alg. Appl. 29, 91–111 (1980).
H.J. Diekhoff, P. Lory, H.J. Oberle, H.J. Pesch, P. Rentrop, R. Seydel: Comparing Routines for the Numerical Solution of Initial Value Problems for Ordinary Differential Equations in Multiple Shooting. Numer.Math. 27, 449–469 (1977).
M. Hermann, H. Berndt: RWPM: A Multiple Shooting Code for Nonlinear Two-Point Boundary Value Problems. Forschungsergebnisse der Friedrich-Schi lier-Universität Jena: Nr. N/81/27 (May 1981).
M. Jankowski, H. Wozniakowski: Iterative Refinement Implies Numerical Stability. BIT 17, 303–311 (1977).
M. Lentini, V. Pereyra: An adaptive finite difference solver for nonlinear two-point boundary value problems with mild boundary layers. SIAM J. Numer. Anal. 14, 91–111 (1977).
R.M.M. Mattheij: The Conditioning of Linear Boundary Value Problems. To appear in SIAM J. Numer. Anal. (1982).
M.R. Osborne: On shooting methods for boundary value problems. J.Math. Anal. Appl. 27, 417–433 (1969).
L.K. Schubert: Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian. Math. Comp. 24, 27–30 (1970).
R.D. Skeel: Iterative Refinement Implies Numerical Stability for Gaussian Elimination. Math.Comp. 35, 817–832 (1980).
J. Stoer, R. Bulirsch: Einführung in die Numerische Mathematik II. Berlin, Heidelberg, New York: Springer (1st ed., 1973).
J.M. Varah: Alternate row and column elimination for solving certain linear systems. SIAM J. Numer. Anal. 13, 71–75 (1976).
R. Weiss: The Convergence of Shooting Methods. BIT 13, 470–475 (1973).
J.H. Wilkinson: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press (1965).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Birkhäuser Boston
About this chapter
Cite this chapter
Deuflhard, P., Bader, G. (1983). Multiple Shooting Techniques Revisited. In: Deuflhard, P., Hairer, E. (eds) Numerical Treatment of Inverse Problems in Differential and Integral Equations. Progress in Scientific Computing, vol 2. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-7324-7_6
Download citation
DOI: https://doi.org/10.1007/978-1-4684-7324-7_6
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3125-3
Online ISBN: 978-1-4684-7324-7
eBook Packages: Springer Book Archive