State-Based Adjoint Method for Reduced Order Modeling

Abstract

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.

Keywords:

• initial conditions perturbation
• reduced order modeling
• subspace reduction
• adjoint methods

ACKNOWLEDGMENTS

This work is supported by the Consortium for Advanced Simulation of Light Water Reactors (www.casl.gov), an Energy Innovation Hub (http://www.energy.gov/hubs) for Modeling and Simulation of Nuclear Reactors under U.S. Department of Energy Contract No. DE-AC05-00OR22725.

Notes

Although this may appear as an unfortunate situation, it implies that the improvement in our understanding of the world still exceeds the startling growth in computer power, which needless to say is a fortunate situation. The opposite however would be alarming, to say the least.

Hybridization with forward methods will be reserved for future article(s).

Black-box identification techniques have also appeared in some engineering communities to perform the same objective as RSM methods.

In the one-shot Hessian-based approach, one can use either the forward or the adjoint model for small dimensional models. For real-world high dimensional models, however, both the forward and adjoint models must be used to avoid the explicit formation of the Hessian matrix. More details on this are forthcoming.

As discussed earlier, although the one-shot Hessian-based approach discussed in Bashir and colleagues (2008) and the goal-oriented approach of Bui-Thanh and colleagues (2006) have some differences in the way the optimization problem is solved, they are collectively referred to here as the go-ROM-LTI approaches.