Stefan Körkel

Numerical methods for optimum experimental design in DAE systems

Subject of this paper is the design of optimal experiments for chemical processes described by nonlinear DAE models. The optimization aims at maximizing the statistical quality of a parameter estimate from experimental data. This leads to optimal control problems with an unusual and intricate objective function which depends implicitly on first derivatives of the solution of the underlying DAE. We treat these problems by the direct approach and solve them using a structured SQP method. The required first and second derivatives of the solution of the DAE are computed very efficiently by a special coupling of the techniques of internal numerical differentiation and automatic differentiation. The performance of our approach is demonstrated for an application to chemical reaction kinetics.

Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes

Optimization of experiments for nonlinear dynamic processes to maximize the reliability of parameter estimates subject to cost and other inequality-constraints leads to very complex optimal control problems. First, the objective function already depends on a generalized inverse of the Jacobian of the underlying parameter estimation problem. Second, optimization results depend on the assumed parameter values which are only known to lie in a confidence region. Hence robust optimal experiments are required. New efficient methods and numerical results are presented. E-mail: stefan@koerkel.de

Numerische Isotropieoptimierung von FIR-Filtern mittels Querglättung

Richtungsunabhangige Filter fur Bilder mit mehreren Dimensionen werden in der digitalen Bildverarbeitung, z.B. in Textur- oder Bewegungsanalyse, benotigt. In diesem Paper werden Finite-Impulse-Response-Filter fur die erste und zweite Ableitung in zwei und drei Dimensionen optimiert. Dafur wird eine Norm einer Winkelfehlerfunktion, die den ganzen Ortsfrequenzraum einnimmt, mit numerischen Optimierungsverfahren minimiert. Durch eine Glattung der eindimensionalen Filter in allen Querrichtungen werden optimierbare Freiheitsgrade eingefuhrt. Dieser Ansatz fuhrt auf effiziente, d.h. kompakte und separable Filter. Da die Filter vom verwendeten Bildmaterial weitgehend unabhangig sind, konnen sie universell eingesetzt werden.

Nonlinear ill-posed problem analysis in model-based parameter estimation and experimental design

Abstract Discrete ill-posed problems are often encountered in engineering applications. Still, their sound analysis is not yet common practice and difficulties arising in the determination of uncertain parameters are typically not assigned properly. This contribution provides a tutorial review on methods for identifiability analysis, regularization techniques and optimal experimental design. A guideline for the analysis and classification of nonlinear ill-posed problems to detect practical identifiability problems is given. Techniques for the regularization of experimental design problems resulting from ill-posed parameter estimations are discussed. Applications are presented for three different case studies of increasing complexity.

Robustness Aspects in Parameter Estimation, Optimal Design of Experiments and Optimal Control

Estimating model parameters from experimental data is crucial to reliably simulate dynamic processes. In practical applications, however, it often appears that the data contains outliers. Thus, a reliable parameter estimation procedure is necessary that delivers parameter estimates insensitive (robust) to errors in measurements.

Numerical Methods for Nonlinear Experimental Design

Nonlinear experimental design leads to a challenging class of optimization problems which occur in the procedure of the validation of process models. This paper discusses the formulation of such problems for a general class of underlying process models, presents numerical methods for the solution and shows their successful application to industrial processes.

Derivative-Extended POD Reduced-Order Modeling for Parameter Estimation

In this article we consider model reduction via proper orthogonal decomposition (POD) and its application to parameter estimation problems constrained by parabolic PDEs. We use a first discretize then optimize approach to solve the parameter estimation problem and show that the use of derivative information in the reduced-order model is important. We include directional derivatives directly in the POD snapshot matrix and show that, equivalently to the stationary case, this extension yields a more robust model with respect to changes in the parameters. Moreover, we propose an algorithm that uses derivative-extended POD models together with a Gauss--Newton method. We give an a posteriori error estimate that indicates how far a suboptimal solution obtained with the reduced problem deviates from the solution of the high dimensional problem. Finally we present numerical examples that showcase the efficiency of the proposed approach.

Optimum experimental design for key performance indicators

Abstract In this paper the methods of experimental design are used to minimize the uncertainty of the prediction of specific process output quantities, the so called key performance indicators. This is achieved by experimental design for constrained parameter estimation problems. We formulate these problems and apply our methods to an example from chemical reaction kinetics.

Optimization of Reactive Flows in a Single Channel of a Catalytic Monolith: Conversion of Ethane to Ethylene

We discuss the modeling, simulation, and, for the first time, optimization of the reactive flow in a channel of a catalytic monolith with detailed chemistry. We use boundary layer approximation to model the process and obtain a high dimensional PDE. We discuss numerical methods based on the efficient solution of high dimensional stiff DAEs arising from spatial semi-discretization and SQP method for the optimal control problem parameterized by the direct approach. We have investigated the application of conversion of ethane to ethylene which involves a complex reaction scheme for gas phase and surface chemistry. Our optimization results show that the maximum yield, an improvement of a factor of two, is achieved for temperatures around 1300 K.

Parameter Estimation for High-Dimensional PDE Models Using a Reduced Approach

Partial differential equations (PDE) are indispensable to describe complex processes. PDE constrained parameter estimation is still a prevailing topic of research. The increase in computation time with increasing complexity of the problem is one of the main problems. With the application of multiple shooting, the number of required derivatives for the generalized Gauss–Newton method rises rapidly. We introduce a method to overcome this challenge. By using directional derivatives the computational effort can be reduced to the minimal number. We demonstrate our methods with help of the heat equation.