Numerical methods and application strategies for optimum experimental design for nonlinear differential equation models
Seminar Room 1, Newton Institute
We consider dynamic processes which are modeled by systems of nonlinear differential equations. Usually the models contain parameters of unknown quantity. To calibrate the models, the parameters have to be estimated from experimental data. Due to the uncertainty of data, the resulting parameter estimate is random. Its uncertainty can be described by confidence regions and the relevant variance-covariance matrix. The statistical significance of the parameter estimation can be maximized by minimizing design criteria defined on the variance-covariance matrix with respect to controls describing layout and processing of experiments and subject to constraints on experimental costs and operability. The resulting optimum experimental design problems are constrained non-standard optimal control problems whose objective depends implicitly on the derivatives of the model states with respect to the parameters. For a numerical solution we have developed methods based on the direct approach of optimal control, on quasi-Newton methods for nonlinear optimization, and on the efficient integration and differentiation of differential equations. To use experimental design for practical problems, we have developed strategies including robustification, multiple experiment formulations, a sequential strategy and an on-line approach. Application examples show that optimally designed experiments yield information about processes much more reliable, much faster and at a significantly lower cost than trial-and-error or black-box approaches. We have implemented our methods in the software package VPLAN which is applied to practical problems from several partners from different fields like chemistry, chemical engineering, systems biology, epidemiology and robotics. In this talk we formulate experimental design problems, present numerical methods for the solution, discuss application strategies and give application examples from practice.