Adaptive reduced order modeling of dynamical systems through novel a posteriori error estimators : Application to uncertainty quantification

Abstract

Multi-query problems such as uncertainty quantification, optimization of a dynamical system require solving a governing differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. Replacing the original discretized higher dimensional model with a faster reduced order model (ROM) can alleviate this computationally prohibitive task significantly. However, a ROM incurs error in the solution due to approximation in a lower dimensional subspace. Moreover, ROMs lack robustness in terms of effectiveness over the entire parameter range. Accordingly, often they are classified as local and global, based on their construction in the parametric domain. Availability of an error bound or error estimator of a ROM helps in achieving this robustness, mainly by allowing adaptivity. The goal of this thesis is to propose such error estimators and use them to develop adaptive proper orthogonal decomposition-based ROM for uncertainty quantification.

Therefore, two a posteriori error estimators, one for linear and another for nonlinear dynamical system, respectively, are developed based on the residual in the differential equation. To develop an a posteriori error estimator for nonlinear systems, first, an upper bound on the norm of the state transition matrix is derived and then it is used to develop the error estimator. Numerically they are compared with the error estimators available in the current literature. This comparison revealed that the proposed estimators follow the trend of the exact error more closely, thus serving as an improvement over the state-of-the-art. These error estimators are used in conjunction with a greedy search to develop adaptive algorithms for the construction of robust ROM. The adaptively trained ROM is subsequently deployed for uncertainty quantification by invoking it in a statistical simulation. For the linear dynamical system, two algorithms are proposed for building robust ROMs --- one for local, and another for global. For the nonlinear dynamical system, an adaptive algorithm is developed for the global ROM. In the training stage of global ROM, a modification is proposed --- at each iteration of the greedy search, the ROM is trained at a few local maxima in addition to the global maxima of the error estimator --- leading to an accelerated convergence. For this purpose, a multi-frequency vibrational particle swarm optimization is employed. It is shown that the proposed algorithm for adaptive training of ROMs poses ample scope of parallelization.

Different numerical studies: (i) bladed disk assembly, (ii) Burgers' equation, and (iii) beam on nonlinear Winkler foundation, are performed to test the efficiency of the error estimators and the accuracy achieved by the modified greedy search. A speed-up of more than two orders of magnitude is achieved using the ROM, trained with the proposed algorithm, and error estimators.

However, adaptive training of ROM is also expensive due to multiple evaluations of HDM. To address this issue, a novel random excitation-based training is proposed in this thesis. Accordingly, depending upon the parameter range of interest, bandlimited random white noise excitations are chosen and the ROM is trained from the corresponding responses. This is applied to linear and nonlinear vibrating systems with spatial periodicity and imperfection. From the numerical studies, it is found that the proposed method reduces the cost of training significantly, and successfully captures the behavior such as alternate pass- and stop-bands of vibration propagation, peak response.