Reduced order modeling of random linear dynamical systems based on a new a posteriori error bound

Reduced order models (ROMs) are becoming increasingly useful for saving computational cost in response prediction of vibrating systems. In a number of applications such as uncertainty quantification, ROMs require robustness over a wide variation of parameters. Accordingly, often they are classified as local and global, based on their performance in the parametric domain. Availability of an error bound of a ROM helps in achieving this robustness, mainly by allowing adaptivity. In this work, for a linear random dynamical system, first, an a posteriori error bound is developed based on the residual in the governing differential equation. Next, based on this error bound, two adaptive methods are proposed for building robust ROMs, that is, one for local, and another for global. While both methods are based on a greedy search approach, a modification is proposed in the training stage of the global ROM for accelerated convergence. These methods are then applied to an uncertainty quantification problem in a statistical simulation framework, and accordingly, two algorithms are developed. A detailed numerical study on vibration of a bladed disk assembly is conducted to study the accuracy and efficiency of the proposed error bound and adaptive ROMs. It is found that these adaptive ROMs provide a considerable speed-up in estimating the probability of failure