O. Roderick, M. Anitescu, Z. Wang, "Reduced Order Approximations in Uncertainty Analysis of Nuclear Engineering Applications," Preprint ANL/MCS-P2078-0412, April 2012. [pdf]
Uncertainty analysis of complex simulation models plays an important role in nuclear engineering, where better understanding of uncertainty leads to greater confidence in the models and in the improved safety and efficiency of engineering projects.
It is often the case that approximation of uncertainty induced variation in the outputs of a model requires extensive sampling. At the same time, running a computationally expensive simulation model more than a few times is impractical. The contradiction may be resolved if a complex model has a simplified, lower quality version that runs much faster. Order reduction of the model equations based on proper orthogonal decomposition is an attractive choice for this simplification, due to its applicability to nonlinear systems, straightforward implementation, and availability of an a posteriori error estimate . If the reduced model is very good, it replaces the full model for all purposes. In a more general case, the error can be described by a stochastic process with the covariance function fitted to the available training data .
The idea of uncertainty analysis on a combination of perfect and imperfect data is not completely new . Our work is distinguished by its emphasis on model order reduction, and also by its relationship with our ongoing work on the use of gradient information for uncertainty quantification [4,5]. Gradient-enhanced automatic learning used in combination with learning on imperfect data allows us to construct models of uncertainty in high dimension, using very little sampling.