On estimation algorithms for ordinary differential equations

This article addresses the problem of estimating the parameters of a system of ordinary differential equations given data derived from noisy observations on the state variables. This problem is important in a range of applications in areas such as adaptive, real time control. There are two main classes of method for attacking this problem, and their equivalence and effectiveness (consistency) are discussed. Recent rate of convergence results for the major implementation techniques are summarized, and some matters requiring further consideration indicated. References H. G. Bock. Recent advances in parameter identification techniques in {ODE}. In P. Deuflhard and E. Hairer, editors, Numerical Treatment of Inverse Problems in Differential and Integral Equations, pages 95--121. Birkhauser, 1983. J. Nocedal and S. J. Wright. Numerical Optimization. Springer--Verlag, 1999. M. R. Osborne. Numerical questions in ODE boundary value problems. In Wayne Read, Jay W. Larson, and A. J. Roberts, editors, Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC-2006, volume 48 of ANZIAM J., pages C899--C926, February 2008. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/79 [February 11, 2008]. M. R. Osborne. Fisher's method of scoring. Int. Stat. Rev., 86:271--286, 1992. M. R. Osborne. The bock iteration for the ode estimation problem. 2008. in preparation. I. Tjoa and L. T. Biegler. Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic systems. Ind. Eng. Chem. Res., 30:376--385, 1991