Parameter Estimation and Optimum Experimental Design for Differential Equation Models
This article reviews state-of-the-art methods for parameter estimation and optimum experimental design in optimization based modeling. For the calibration of differential equation models for nonlinear processes, constrained parameter estimation problems a
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Abstract This article reviews state-of-the-art methods for parameter estimation and optimum experimental design in optimization based modeling. For the calibration of differential equation models for nonlinear processes, constrained parameter estimation problems are considered. For their solution, numerical methods based on the boundary value problem method optimization approach consisting of multiple shooting and a generalized Gauß–Newton method are discussed. To suggest experiments that deliver data to minimize the statistical uncertainty of parameter estimates, optimum experimental design problems are formulated, an intricate class of non-standard optimal control problems, and derivative-based methods for their solution are presented. Keywords Gauß–Newton method • Multiple shooting • Nonlinear differentialalgebraic equations • Optimum experimental design • Parameter estimation • Variance–covariance matrix
1 Introduction Dynamic processes in science, engineering or medicine often can be described by differential equation models. Solutions of the model equations, usually obtained by numerical methods, give simulations of the process behavior under various conditions. To obtain realistic model output results, the model has to be validated. An important task for this is the calibration of the model, i.e. to explain experimental data by appropriate simulation results. In order to fit the model to the data, unknown
H.G. Bock S. K¨orkel () J.P. Schl¨oder Interdisciplinary Center for Scientific Computing, Heidelberg University, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany e-mail: [email protected] H.G. Bock et al. (eds.), Model Based Parameter Estimation, Contributions in Mathematical and Computational Sciences 4, DOI 10.1007/978-3-642-30367-8 1, © Springer-Verlag Berlin Heidelberg 2013
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quantities in the model, called parameters, have to be estimated by minimizing a norm of the residuals between data and model response. A fit of given data yet does not mean a validated model. Due to the randomness of experimental errors, the parameter estimate is also a random variable. Only if the statistical uncertainty of the parameters is small, simulations can also predict the outcome of future experiments and explain the behavior of the real process in a qualitatively and quantitatively correct way. The statistical uncertainty of a parameter estimate depends on layout, setup, control and sampling of the experiments. Experimental design optimization problems minimize a function of the variance–covariance matrix of the parameter estimation problem. Optimal experiments can reduce the experimental effort for model validation drastically. In this article we review the formulation of parameter estimation and optimum experimental design problems and discuss methods for their numerical solution. The article is structured as follows. After this introduction, Sect. 2 introduces the class of differential equation models we consider in this article. In Sect. 3, parameter estimation probl
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