Numerical parameter estimation in differential equations

• Main Author: Benson, Maurice W.
• Language: English
• Published: 2010
• Online Access: http://hdl.handle.net/2429/21231

The problem of numerical least squares parameter estimation in differential equations is considered. Several new algorithms that pay particular attention to the differential equation aspect of the problem are presented. These reduce some of the difficulties encountered when the problem is treated solely as a question of nonlinear optimization. The extremely powerful interactive approach is considered and an interactive package incorporating standard techniques using sensitivity equations along with a selection of our special algorithms is presented. We consider methods involving the fitting of integrals and derivatives using piecewise polynomial approximations to the observations. Continuation methods with a quasi multiple shooting technique to bridge the gap between these coarse but well behaved methods and the full least squares problem are explored. Special methods are developed for the important case of two state variables with observations available on only one of them. In particular we consider algorithms which use an initial guess at the behavior of the unobserved state variable and then iteratively improve this guess. The need for effective algorithms for fitting population growth models in ecology is one motivation for this thesis. We devote a chapter to an important predator-prey model of population dynamics and extensive experiments are presented which demonstrate some of the typical difficulties which can arise and which illustrate the ability of our algorithms to overcome some of these difficulties. Some special problems involving jumps from one equilibrium to another (loosely referred to as catastrophes) are examined. This type of model has important applications in ecology. Models involving stiff differential equations are also considered. A short chapter is devoted to the use of sequential reestimation techniques. Experiments indicate that such methods can be useful for improving a crude initial guess at the parameters and this improvement can be crucial for the successful solution of the problem. Finally a chapter is devoted to a selection of "real world" problems. It is on such problems that the true value of an algorithm is determined. === Science, Faculty of === Computer Science, Department of === Graduate