Numerical questions in ODE boundary value problems

Abstract

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di-stability) in problem discretizations. However, computational categories are not precise, and qualitative descriptors such as `of moderate size' cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive Newton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate boundary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood. References U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical solution of boundary value problems for ordinary differential equations, SIAM, Philadelphia, 1995. J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley and Sons, 2003. G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33--53. G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), 27--43. F. R. de Hoog and R. M. M. Mattheij, On dichotomy and well-conditioning in BVP, SIAM J. Numer. Anal. 24 (1987), 89--105. P. Deuflhard, Newton methods for nonlinear problems, Springer--Verlag, Berlin Heidelberg, 2004. R. England and R. M. M. Mattheij, Boundary value problems and dichotomic stability, SIAM J. Numer. Anal. 25 (1988), 1037--1054. Z. Li, M. R. Osborne, and T. Prvan, Parameter estimation of ordinary differential equations, IMA J. Numer. Anal. 25 (2005), 264--285. J. Nocedal and S. J. Wright, Numerical optimization, Springer--Verlag, 1999. M. R. Osborne, On shooting methods for boundary value problems, J. Math. Analysis and Applic. 27 (1969), 417--433. M. R. Osborne, Cyclic reduction, dichotomy, and the estimation of differential equations, J. Comp. and Appl. Math. 86 (1997), 271--286. B. J. Quinn and E. J. Hannan, The estimation and tracking of frequency, Cambridge University Press, Cambridge, United Kingdom, 2001. J. O. Ramsay, G. Hooker, C. Cao, and C. Campbell, Estimating differential equations, preprint, Department of Psychology, McGill University, Montreal, Canada, 2005, p. 40.