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Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

Nurtaj Hossain, M and Ghosh, D (2020) Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification. In: International Journal for Numerical Methods in Engineering, 121 (15). pp. 5417-3441.

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Official URL: https://dx.doi.org/10.1002/nme.6365


Multiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12-13): 741-758) to adaptively construct a robust proper orthogonal decomposition-based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed-up in UQ is achieved.

Item Type: Journal Article
Publication: International Journal for Numerical Methods in Engineering
Publisher: John Wiley and Sons Ltd
Additional Information: The copyright of this article belongs to John Wiley and Sons Ltd
Keywords: Differential equations; Dynamical systems; Errors; Nonlinear dynamical systems; Nonlinear equations; Principal component analysis, Dimensional subspace; Greedy search algorithms; Posteriori error estimator; Proper orthogonal decompositions; Reduced order models; State transition Matrix; Statistical simulation; Uncertainty quantifications, Uncertainty analysis
Department/Centre: Division of Mechanical Sciences > Civil Engineering
Date Deposited: 13 Aug 2020 07:38
Last Modified: 13 Aug 2020 07:38
URI: http://eprints.iisc.ac.in/id/eprint/65512

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