On error estimation for reduced-order modeling of linear non-parametric and parametric systems

Abstract

Motivated by a recently proposed error estimator for the transfer function of the reduced-order model of a given linear dynamical system, we further develop more theoretical results in this work. Moreover, we propose several variants of the error estimator, and compare those variants with the existing ones both theoretically and numerically. It is shown that some of the proposed error estimators perform better than or equally well as the existing ones. All the error estimators considered can be easily extended to estimate the output error of reduced-order modeling for steady linear parametric systems.

Multi-fidelity error estimation accelerates greedy model reduction of complex dynamical systems

“…Then it was extended to POD-greedy for dynamical systems, which is used to construct the ROM using snapshots in the time domain. Later the greedy algorithm found its capability in adaptively choosing interpolation points for frequency-domain MOR methods [15,16]. The greedy algorithm for steady systems and frequency-domain MOR has the same formulation, whereas POD-greedy for time-domain MOR of time-dependent systems needs an SVD step at each greedy iteration.…”

“…The multi-fidelity error estimation we are going to introduce depends on the formulation of the highfidelity error estimator ∆(µ). To illustrate the basic concept, we use an error estimator proposed in [16] as the high-fidelity error estimator and discuss how to further reduce the computational load by using multi-fidelity error estimation.…”

“…∆(µ) defined in (11) is one of the error estimators proposed in [16], where the proposed error estimators were shown to outperform other existing error estimators in the literature [34,15] in terms of both accuracy and computational efficiency. Furthermore, it has been discussed in [16] that ∆(µ) is even more accurate but has less computational complexity than other proposed estimators, including the one used in [14]. Even with this error estimator, the greedy algorithm could take several hours to converge for some complex systems, for example, the time-delay systems we consider in this work.…”

“…It is known that an efficient error estimator makes the greedy algorithm successful in producing an accurate ROM without running many iterations. Therefore, many efforts have been made in this direction to develop computable error estimators for different problems [15,16,18,19,20,21,22,23,24,25,28,33,34,35]. However, more attention has been paid to improve the effectivity or accuracy of the error estimator than to develop more efficient strategies to accelerate the greedy process [13,25,33,34,36].…”

Automatic model order reduction for systems with frequency-dependent material properties

“…Reduced basis methods have also been successfully applied to vibro-acoustic systems in, for example, [42]. Error estimators required for adaptive algorithms are often based on residual expressions or the comparison of multiple reduced models having different orders or with different expansion points [35,[43][44][45]. Algorithms employing such techniques have been found suitable to reduce systems with frequencydependent material properties: The residuals of the intermediate reduced models of systems with general nonlinear damping effects are used as an error estimate in [46]; two independent reduced models of vibro-acoustic systems with poroelastic material damping are computed based on different expansion points and compared to obtain an estimate for the approximation error in [47].…”