Adaptive basis construction and improved error estimation for parametric nonlinear dynamical systems

Abstract

Summary An adaptive scheme to generate reduced‐order models for parametric nonlinear dynamical systems is proposed. It aims to automatize the proper orthogonal decomposition (POD)‐Greedy algorithm combined with empirical interpolation. At each iteration, it is able to adaptively determine the number of the reduced basis vectors and the number of the interpolation basis vectors for basis construction. The proposed technique is able to derive a suitable match between the RB and the interpolation basis vectors, making the generation of a stable, compact and reliable ROM possible. This is achieved by adaptively adding new basis vectors or removing unnecessary ones, at each iteration of the greedy algorithm. An efficient output error indicator plays a key role in the adaptive scheme. We also propose an improved output error indicator based on previous work. Upon convergence of the POD‐Greedy algorithm, the new error indicator is shown to be sharper than the existing ones, implicating that a more reliable ROM can be constructed. The proposed method is tested on several nonlinear dynamical systems, namely, the viscous Burgers' equation and two other models from chemical engineering.

A Training Set Subsampling Strategy for the Reduced Basis Method

“…The number of POD modes r POD , r EI to enrich the RB and DEIM bases is determined at each iteration based on the adaptive approach proposed in [16]. We have also used the primal-dual error estimator proposed by the authors of [16] for our implementation of Algorithms 2.1, 3.1 and 3.2. The dual RB basis required for the error estimator is generated separately.…”

“…Several enhancements in the form of primal-dual error estimation, hyper-reduction, adaptive basis construction, etc. exist [6,16,26,28,58].…”

“…For both algorithms, we do not reset the value of iter at the end of Stage 1, so the final value of iter upon convergence for Algorithms 3.1 and 3.2 is the total number of iterations required by either algorithms to converge to the desired tolerance. It is worth pointing out that an a posteriori error estimator [16] is used in both stages of the proposed algorithms so that the parameter sample picked at each iteration is the one at which an estimated output error is the largest. Furthermore, the adaptive basis enrichment technique proposed in [16] is implemented and serves as a possible solution to the issue of repeated parameter selection.…”

“…It is worth pointing out that an a posteriori error estimator [16] is used in both stages of the proposed algorithms so that the parameter sample picked at each iteration is the one at which an estimated output error is the largest. Furthermore, the adaptive basis enrichment technique proposed in [16] is implemented and serves as a possible solution to the issue of repeated parameter selection. The number of POD modes corresponding to a selected parameter sample is adaptively decided: when the estimated error is large, a higher number of POD modes (r POD ) for the selected parameter are added; otherwise fewer POD modes are added.…”

Special Issue on Credible High‐Fidelity and Low‐Cost Simulations in Computational Engineering

“…The 12 manuscripts in this special issue cover a wide range of techniques related to the above mentioned topics, including: accurate high‐order 1,2 and fast lowest‐order discontinuous Galerkin discretisations; 3,4 mesh, 1‐3,5 degree, 1 and modal basis 6,7 adaptivity; a priori error estimates, 8 a posteriori error indicators, 1,3 error bounds, 5,7,9,10 and error estimates in quantities of interest; 2,6,9,10 a priori 5,10,11 and a posteriori 2,6‐9,12 reduced order models.…”