Adaptive Basis Construction and Improved Error Estimation for Parametric Nonlinear Dynamical Systems

TL;DR

This paper introduces an adaptive method for constructing reduced-order models of parametric nonlinear dynamical systems, enhancing stability, reliability, and efficiency through dynamic basis selection and improved error estimation.

Contribution

It develops an adaptive POD-Greedy algorithm with a novel error indicator, enabling automatic basis size determination and improved model accuracy.

Findings

The method produces stable, compact reduced models.

The new error indicator outperforms existing ones.

Validated on Burgers' equation and chemical engineering models.

Abstract

An adaptive scheme to generate reduced-order models for parametric nonlinear dynamical systems is proposed. It aims to automatize the 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 reduced basis and the interpolation basis vectors, making the generation of a stable, compact and reliable reduced-order model 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 reduced-order model 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.