The advancement in computer processing power; computationally and graphically, presents a hallmark in human development that help enclave their curiosity and tentatively fuel the need to comprehend the functioning of various systems. Against this backdrop, the real potential behind mathematical modeling and simulations, particularly relating to the dynamic behavior of complex systems is starting to be realized. Regrettably, models often involve uncertainty, which reduces their accuracy because of assumptions and simplifications made in a model’s formulation. Therefore, in order to improve the accuracy of a simulation in the presence of uncertainty, a probabilistic description of model prediction is often desired. So far, the uncertainty quantification (UQ) algorithm has delivered promising results. For instance, the UQ based on the intrusive generalized polynomial chaos (gPC) expansion has been shown to be an efficient tool in different applications, including dynamic modelling, control, and optimization problems in chemical engineering. It is well known that the successful application of intrusive gPC-based UQ is associated with the stochastic Galerkin projection, which yields a family of models described by several coupled equations of gPC coefficients. Using these coefficients, the evolution of uncertainty in a dynamic system can be quickly determined when there is probabilistic uncertainty in the system.
In essence, stochastic Galerkin projection is elegant when dealing with models that involve complex functions and larger numbers of uncertainties; however, it becomes computationally intractable and cannot be applied directly to solve gPC coefficients in real-time. To address this issue, researchers from the Department of Chemical and Biomolecular Engineering at Clarkson University in New York: Jeongeun Son (Graduate candidate) and Professor Yuncheng Du, proposed to use the generalized dimension reduction method (gDRM) to convert a high-dimensional integral involved in the stochastic Galerkin projection into several lower-dimensional integrals that can be easily solved. Their work is currently published in the research journal, Computers and Chemical Engineering.
The objective of their work was to show the accuracy of the gDRM-based gPC approach when dealing with larger numbers of uncertainties in a nonlinear dynamic system, which existing literature reports to be challenging based on existing methods. As such, they presented a UQ method that integrates the gDRM with the gPC and reported the detailed comparison between intrusive and nonintrusive UQ methods. Specifically, in order to prove their approach, three cases: a nonlinear algebraic benchmark, a penicillin manufacturing process, and autocrine signaling networks of living cells, were presented for algorithm verification.
The focus of their study was the parametric uncertainty, and the main objective was to quantify accurately just how such an uncertainty impacts model prediction. Overall, the two researchers established that the overall performance of the intrusive gDRM-based gPC in terms of UQ accuracy and computational time was superior to nonintrusive SC-SP. This demonstrated its capability to deal with more complicated problems.
In summary, the study presented an intrusive UQ algorithm that provides an accurate quantification of how parametric uncertainty affects model predictions in nonlinear and complex systems. Remarkably, the obtained results showed that the gDRM-based gPC could provide accurate UQ results and was computationally efficient when dealing with a large numbers of uncertainties, thus laying the foundation to pursue more complicated problems in future work. For Du, this is also the start of a broader and more comprehensive research in UQ. “We want to probabilistically predict how complex systems work, and capitalize on the prediction to tweak the systems to make them work better”.
Jeongeun Son, Yuncheng Du. Comparison of intrusive and nonintrusive polynomial chaos expansion-based approaches for high dimensional parametric uncertainty quantification and propagation. Computers and Chemical Engineering, volume 134 (2020) 106685.