Towards Optimum Experimental Design for Partial Differential Equations

• Antragstellerin: Professorin Dr. Ekaterina Kostina
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2009 bis 2016
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 131726439

Parameter estimation and optimal design of experiments are important steps in establishing models that reproduce a given process quantitatively correctly. The aim of parameter estimation is to reliably and accurately identify model parameters from sets of noisy experimental data. The “accuracy” of the parameters, i.e. their statistical distribution depending on data noise, can be estimated up to first order by means of a covariance matrix approximation and corresponding confidence regions. In practical applications, however, one often finds that the experiments performed to obtain the required measurements are expensive, but nevertheless do not guarantee satisfactory parameter accuracy or even well-posedness of the parameter estimation problem. In order to maximize the accuracy of the parameter estimates, additional experiments can be designed with optimal experimental settings or controls (e.g. initial conditions, measurement devices, sampling times, temperature profiles, feed streams etc.) subject to constraints. As an objective functional a suitable function of the covariance matrix can be used. The possible constraints in this problem describe costs, feasibility of experiments, domain of models etc.The methods for optimum experimental design that have been developed over the last few years can handle processes governed by DAE, but methods for processes governed by PDE are still in their infancy because of the extreme complexity of models and the optimization problem. The aim of this project is to provide a framework for efficient numerical methods for optimum experimental design for processes described by systems of non-stationary PDEs. The PDE system under consideration is motivated by an actual production process in chemical engineering. The aim of the optimal experiments is to maximize information to gain accuracy of a small number of (physical) parameters and/or predicted accuracy of some functions of the states and parameters.The main results of this project will be: generally applicable methods for design of optimal experiments for parameter estimation of a small number of quantities of interest in PDE systems based on a simultaneous optimization approach; reduced approximate SQP (or Gauss-Newton) methods for solving the nonlinear programming problem resulting from discretization of model equations and controls that allow handling of inexact constraint Jacobians; fast iterative solvers and preconditioning, e.g. multigrid methods, for solving quadratic subproblems and for computing covariance matrices and their derivatives.

• DFG-Verfahren: Schwerpunktprogramme
• Teilprojekt zu: SPP 1253: Optimierung mit partiellen Differentialgleichungen