(179g) Bayesian Approach for Optimal Experimental Design of Thermo Gravimetric Analysis of CO2 on Amine Sorbents

Experimental design, in general, aims at determining the values of factors or process variables, at which the experiments when performed would provide maximum information of the system under study. In particular, model based Optimal Experimental Design (OED) ensures that the accurate estimation of model parameters can be achieved with the minimal number of experiments performed to gather the data. OED is very essential in situations where it is expensive and/or time consuming to conduct the experiments.
Conventional methods of optimal experimental design are based on the well-known
‘alphabetical optimality’ criterion to determine the value or the information gain of an experiment [1]. For e.g. ‘A optimality’ minimizes the variance of the parameter estimates, ‘D optimality’ minimizes the volume of the confidence region of the parameter estimates, and ‘G optimality’ minimizes the variance of the model predictions. On the other hand, under Bayesian methodology, a decision theoretic framework [2] is used to formulate the objective function for the optimal design criteria. The expectation of the objective or the utility function is maximized over the design space to determine the optimal experimental condition. A comprehensive review of the Bayesian experimental design is presented by Chaloner and Verdinelli [3]. There have been numerous literature studies on Bayesian optimal design, the majority of which have been restricted to linear models [3]. In case of non-linear models, the utility functions' integral becomes analytically intractable and is approximated in several ways, such as assuming a normal posterior distribution to parameters. Simulation based Bayesian optimal design offers a framework which does not require the use of limiting assumptions for nonlinear models [4]. The recent work of Terejanu et al [5] evaluates the design criteria on every finite element of design space dimension (on the order of ten designs in demonstrated examples) rather than optimizing over the design space. However, they reported that the computational expense can be prohibitive for larger search design space. With physically realistic complex models (nonlinear), it still remains an open challenge to determine the optimal experimental design condition using information theoretic design criteria.
In this work, we propose to demonstrate the use of a simulation based Bayesian design to determine the optimal experimental design condition such that the prediction uncertainty of a complex physiochemical process is reduced. It is worthwhile to note that the design criteria based on reducing prediction uncertainty is much more computationally intensive than the one based on reducing the parametric uncertainty. The involved computational complexity is handled in two parts. First, we propose to use the Shannon information, a measure of parametric uncertainty, as the design criteria and apply it only for those parameters which are highly sensitive to the prediction variable. Global Sensitivity Analysis (GSA) [6] is used to determine the sensitivity of the parameters with respect to the prediction variable. In the proposed simulation based
optimal design, the design criteria is evaluated for every finite element in the design space of factors to obtain the utility function surface of the design space. The computational expense of the design criteria evaluation is handled by parallelizing the computation across the design points resulting in reduction of the total computation time.
The physiochemical process considered in this work is the adsorption of CO2 on amine sorbents in a fixed bed adsorber, which is described by a complex model of Partial Differential Algebraic Equations (PDAE). The parameters which need to be estimated are the adsorption isotherm parameters determined using Thermo Gravimetric Analysis (TGA) experiments of CO2 adsorption on amine sorbents. The objective of the proposed paper is to determine the optimal experimental design condition which would reduce the prediction uncertainty of CO2 concentration at the exit of the fixed bed adsorber.
The utility function ??(??) based on Shannon information design criteria is of the
following form

??(??|??, ??)

??(??) = ? ? ??(??, ??, ??)??(??, ??|??)???????? = ? ? ln (

) ??(??|??, ??)??(??|??)????????

?? ??

?? ??

??(??)

where ??(??) is the expected utility at the design condition ??, ??(??, ??, ??) is the utility or loss
function defined based on the criterion chosen to determine the information gain of an
experiment at given value of parameters ?? and outcome ??. The utility function is
numerically evaluated as follows.

1

??(??) =

1 ??????

?=1

???????

?=1

ln (??(??? |??? , ??)) - ln (??(??? |??)) )

?????? ??????

where ?????? is the number of Markov Chain Monte Carlo (MCMC) simulations needed
to determine the posterior distribution of parameters and ?????? is the number of likely
observations of the outcome variable ??, to evaluate the likelihood distribution of ??(??|??)
at the design condition ??.
References
[1]. A. C. Atkinson, A. N. Donev, Optimum Experimental Designs, with SAS, Oxford
Statistical Science Series, Oxford University Press, 2007.
[2]. D. V. Lindley, On a measure of the information provided by an experiment, The
Annals of Mathematical Statistics 27 (1956) 986–1005.
[3]. K. Chaloner, I. Verdinelli, Bayesian experimental design: A review, Statistical
Science 10 (1995) 273–304.
[4].P. Müller, Simulation based optimal design, in: Bayesian Statistics 6: Proceedings of the Sixth Valencia International Meeting, Oxford University Press, 1998, pp. 459–474.
[5]. G. Terejanu, R. R. Upadhyay, K. Miki, J. Marschall, Bayesian experimental design for the active nitridation of graphite by atomic nitrogen, Experimental Thermal and Fluid Science 36 (2012) 178–193.
[6]. He. F., Yue. H., and Brown, M., Investigating Bayesian Robust Experimental Design with Principles of Global Sensitivity Analysis, Proc. of the 9th Int. Symposium on Dynamics and Control of Process Systems (DYCOPS 2010), Belgium, July 2010