Parameter estimation of ordinary differential equations

• Published in: IMA journal of numerical analysis Vol. 25; no. 2; pp. 264 - 285
• Main Authors: Li, Zhengfeng, Osborne, Michael R., Prvan, Tania
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01.04.2005; Oxford Publishing Limited (England)
• Subjects: Calculus of variations and optimal control; constrained optimization; data fitting; Exact sciences and technology; Gauss–Newton approximation; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Ordinary differential equations; orthogonal cyclic reduction; parameter estimation; Sciences and techniques of general use; SQP methods; Normal approximation; data fitting; Gauss approximation; Parameter estimation; Differential equation; Optimization method; Large sample; Embedding; Quadratic programming; Algorithm; Gauss-Newton approximation; ordinary differential equations; Numerical analysis; Sequential method; Equality constraint; orthogonal cyclic reduction; SQP methods; Lagrangian function; constrained optimization
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/drh016

This paper addresses the development of a new algorithm for parameter estimation of ordinary differential equations. Here, we show that (1) the simultaneous approach combined with orthogonal cyclic reduction can be used to reduce the estimation problem to an optimization problem subject to a fixed number of equality constraints without the need for structural information to devise a stable embedding in the case of non-trivial dichotomy and (2) the Newton approximation of the Hessian information of the Lagrangian function of the estimation problem should be used in cases where hypothesized models are incorrect or only a limited amount of sample data is available. A new algorithm is proposed which includes the use of the sequential quadratic programming (SQP) Gauss–Newton approximation but also encompasses the SQP Newton approximation along with tests of when to use this approximation. This composite approach relaxes the restrictions on the SQP Gauss–Newton approximation that the hypothesized model should be correct and the sample data set large enough. This new algorithm has been tested on two standard problems.