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Chemische Prozesse können durch nichtlineare Differentialgleichungen beschrieben werden, typischerweise mit unbekannten Modellparametern. Durch Parameterschätzung können die Modelle an experimentelle Daten angepasst werden. Die geschätzten Parameter sind mit statistischen Unsicherheiten behaftet, die von der Art der Experimente abhängen und bereits vor deren Durchführung berechnet und durch optimale Versuchsplanung minimiert werden können. In diesem Beitrag werden diese Optimierungsprobleme formuliert, Methoden zur Lösung diskutiert und ihre Wirksamkeit durch Anwendung auf ein chemisches Reaktionssystem demonstriert.
Chemische Prozesse können durch nichtlineare Differentialgleichungen beschrieben werden, typischerweise mit unbekannten Modellparametern. Durch Parameterschätzung können die Modelle a\n experimentelle Daten angepasst werden. Die geschätzten Parameter sind mit statistischen Unsicherheiten behaftet, die von der Art der Experimente abhängen und bereits vor deren Durchführung berechnet und durch optimale Versuchsplanung minimiert werden können. In diesem Beitrag werden diese Optimierungsprobleme formuliert, Methoden zur Lösung diskutiert und ihre Wirksamkeit durch Anwendung auf ein chemisches Reaktionssystem demonstriert.
Computational models of cognition provide an interface to connect advanced mathematical tools and methods to empirically supported theories of behavior in psychology, cognitive science, and neuroscience. In this article, we consider a computational model of instance-based learning, implemented in the ACT-R cognitive architecture. We propose an approach for obtaining mathematical reformulations of such cognitive models that improve their computational tractability. For the well-established Sugar Factory dynamic decision making task, we conduct a simulation study to analyze central model parameters. We show how mathematical optimization techniques can be applied to efficiently identify optimal parameter values with respect to different optimization goals. Beyond these methodological contributions, our analysis reveals the sensitivity of this particular task with respect to initial settings and yields new insights into how average human performance deviates from potential optimal performance. We conclude by discussing possible extensions of our approach as well as future steps towards applying more powerful derivative-based optimization methods.
The performance of cognitive models often depends on the settings of specific model parameters, such as the rate of memory decay or the speed of motor responses. The systematic exploration of a model’s parameter space can yield relevant insights into model behavior and can also be used to improve the fit of a model to human data. However, exhaustive parameter space searches quickly run into a combinatorial explosion as the number of parameters investigated increases. Taking an established instance-based learning task as example, we show
how simulation using parallel computing and derivative-free optimization methods can be applied to investigate the effects
of different parameter settings. We find that both global optimization methods involving genetic algorithms as well as local methods yield satisfactory results in this case. Furthermore, we show how a model implemented in a specific cognitive architecture (ACT-R) can be mathematically reformulated to prepare the application of derivative-based optimization methods which promise further efficiency gains for quantitative analysis.
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.
In this work, a method for reducing the number of degrees of freedom in online optimal dynamic experiment design problems for systems described by differential equations is proposed. The online problems are posed such that only the inputs which extend an operation policy resulting from an experiment designed offline are optimized. This is done by formulating them as multiple experiment designs, considering explicitly the information of the experiment designed offline and possible time delays unknown a priori. The performance of the method is shown for the case of the separation of isopropanolol isomers in a Simulated Moving Bed plant.
We propose a method for an efficient optimization of experimental designs, using a combination of discrete adjoint computations, Taylor arithmetic and matrix calculus. Compared to the state of the art of using finite differences or the forward mode of automatic differentiation, our proposed approach leads to a reduction of the relative temporal complexity from linear to constant time in the number of control variables and measurement weights. We demonstrate that the advantageous complexity results are not only of theoretical nature, but lead to significant speedups in practice as well. With our implementation we are very close to the theoretical bound of the cheap gradient principle. We present one academic (spatially discretized heat equation) and two industrial application examples (biochemical process/Diesel-oxidation catalysis process) where we achieve speedups that range between 10 and 100. In addition to our core results, we also describe an efficient adjoint approach for the treatment of differential algebraic equations and present adjoint formulas for constrained least-squares problems.
Optimum experimental design for extended Gaussian disorder modeled organic semiconductor devices
(2013)
We apply optimum experimental design (OED) to organic semiconductors modeled by the extended Gaussian disorder model (EGDM) which was developed by Pasveer et al. [Phys. Rev. Lett. 94, 206601 (2005)]. We present an extended Gummel method to decouple the corresponding system of equations and use automatic differentiation to get derivatives with the required accuracy for OED. We show in two examples, whose parameters are taken from Pasveer et al. [Phys. Rev. Lett. 94, 206601 (2005)] and Mensfoort and Coehoorn [Phys. Rev. B 78, 085207 (2008)] that the linearized confidence regions of the parameters can be reduced significantly by applying OED resulting in new experiments with a different setup.
The adaptive input design (also called online redesign of experiments) for parameter estimation is very effective for the compensation of uncertainties in nonlinear processes. Moreover, it enables substantial savings in experimental effort and greater reliability in modeling.
We present theoretical details and experimental results from the real-time adaptive optimal input design for parameter estimation. The case study considers separation of three benzoate by reverse phase liquid chromatography. Following a receding horizon scheme, adaptive D-optimal input designs are generated for a precise determination of competitive adsorption isotherm parameters. Moreover, numerical techniques for the regularization of arising ill-posed problems, e.g. due to scarce measurements, lack of prior information about parameters, low sensitivities and parameter correlations are discussed. The estimated parameter values are successfully validated by Frontal Analysis and the benefits of optimal input designs are highlighted when compared to various standard/heuristic input designs in terms of parameter accuracy and precision.
Nonlinear ill-posed problem analysis in model-based parameter estimation and experimental design
(2015)
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.