@article{abdel-khalik_bang_wang_2013, title={Overview of hybrid subspace methods for uncertainty quantification, sensitivity analysis}, volume={52}, ISSN={["1873-2100"]}, DOI={10.1016/j.anucene.2012.07.020}, abstractNote={The role of modeling and simulation has been heavily promoted in recent years to improve understanding of complex engineering systems. To realize the benefits of modeling and simulation, concerted efforts in the areas of uncertainty quantification and sensitivity analysis are required. The manuscript intends to serve as a pedagogical presentation of the material to young researchers and practitioners with little background on the subjects. We believe this is important as the role of these subjects is expected to be integral to the design, safety, and operation of existing as well as next generation reactors. In addition to covering the basics, an overview of the current state-of-the-art will be given with particular emphasis on the challenges pertaining to nuclear reactor modeling. The second objective will focus on presenting our own development of hybrid subspace methods intended to address the explosion in the computational overhead required when handling real-world complex engineering systems.}, journal={ANNALS OF NUCLEAR ENERGY}, author={Abdel-Khalik, Hany S. and Bang, Youngsuk and Wang, Congjian}, year={2013}, month={Feb}, pages={28–46} }
 @article{bang_abdel-khalik_2013, title={Projection-based second order perturbation theory}, volume={52}, ISSN={["0306-4549"]}, DOI={10.1016/j.anucene.2012.07.009}, abstractNote={Abstract Reactor analysis represents a typical example of a complex engineering system that is described by multi-scale and multi-physics nonlinear models with many input parameters and output responses. Obtaining reference solutions to these models is computationally expensive which renders impractical their repeated executions for engineering-oriented studies such as design optimization, uncertainty quantification, and safety analysis. To overcome this challenge, sensitivity analysis based on first-order perturbation theory has been widely used in the reactor analysis community to estimate changes in responses of interest due to input parameter variations. Although perturbation theory has been rigorously developed over the past four decades in order to extend its applicability to estimate higher order variations, engineering applications have primarily focused on first-order perturbation theory only. This is because the computational overhead of higher order perturbation theory are often overwhelming and do not justify the development effort required for their implementation. This manuscript further develops a recently introduced higher order approach to estimate second order variations. The objective is to demonstrate that first-order perturbation theory can be employed in practical engineering calculations to estimate higher order variations. The applicability of the introduced approach is analyzed with TSUNAMI-2D for typical lattice physics calculations.}, journal={ANNALS OF NUCLEAR ENERGY}, author={Bang, Youngsuk and Abdel-Khalik, Hany S.}, year={2013}, month={Feb}, pages={80–85} }
 @article{bang_abdel-khalik_hite_2012, title={Hybrid reduced order modeling applied to nonlinear models}, volume={91}, ISSN={["1097-0207"]}, DOI={10.1002/nme.4298}, abstractNote={SUMMARY Reduced order modeling plays an indispensible role for most real‐world complex models. The objective of this manuscript is to hybridize local and global sensitivity analysis methods to enable the application of reduced order modeling to complex nonlinear models, often encountered in real system design and analysis calculations, for example, nuclear reactors. This is achieved by first employing local variational methods to identify important nonlinear features of the original model that are required to reach a user‐defined accuracy for the reduced model. This information is obtained by sampling local first‐order derivatives of a pseudoresponse utilizing a modified representation of an infinite series expansion around some reference point. The resulting derivative information is aggregated in a subspace of dimension much less than the dimension of the input parameter space. The accuracy of the reduced model can be mathematically quantified using a bounding norm. Next, global sensitivity methods are employed to exhaustively search the reduced subspace for sensitivity information. The theory and implementation details of the proposed method are exposed in this manuscript. Numerical tests based on prototype nonlinear functions and radiation transport models with many input parameters and many responses are conducted as proof of principle. Copyright © 2012 John Wiley & Sons, Ltd.}, number={9}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING}, author={Bang, Youngsuk and Abdel-Khalik, Hany S. and Hite, Jason M.}, year={2012}, month={Aug}, pages={929–949} }
 @article{bang_wang_abdel-khalik_2012, title={State-Based Adjoint Method for Reduced Order Modeling}, volume={41}, ISSN={["1532-2424"]}, DOI={10.1080/00411450.2012.672359}, abstractNote={Introduced here is an adjoint state-based method for model reduction, which provides a single solution to two classes of reduction methods that are currently in the literature. The first class, which represents the main subject of this manuscript, is concerned with linear time invariant problems where one is interested in calculating linear responses variations resulting from initial conditions perturbations. The other class focuses on perturbations introduced in the operator, which result in nonlinear responses variations. Unlike existing adjoint-based methods where an adjoint function is calculated based on a given response, the state-based method employs the state variations to set up a number of adjoint problems, each corresponding to a pseudoresponse. This manuscript extends the applicability of state-based method to generate reduced order models for linear time invariant problems. Previous developments focusing on operator perturbations are reviewed briefly to highlight the common features of the state-based algorithm as applied to these two different classes of problems. Similar to previous developments, the state-based reduction is shown to set an upper-bound on the maximum discrepancy between the reduced and original model predictions. The methodology is applied and compared to other state-of-the-art methods employing several nuclear reactor diffusion and transport models.}, number={1-2}, journal={TRANSPORT THEORY AND STATISTICAL PHYSICS}, author={Bang, Youngsuk and Wang, Congjian and Abdel-Khalik, Hany S.}, year={2012}, pages={101–132} }