Efficient reduced models anda posteriorierror estimation for parametrized dynamical systems by offline/online decomposition

Abstract

We address the problem of model order reduction (MOR) of parametrized dynamical systems. Motivated by reduced basis (RB) methods for partial differential equations, we show that some characteristic components can be transferred to model reduction of parametrized linear dynamical systems. We assume an affine parameter dependence of the system components, which allows an offline/online decomposition and is the basis for efficient reduced simulation. Additionally, error control is possible by a posteriori error estimators for the state vector and output vector, based on residual analysis and primaldual techniques. Experiments demonstrate the applicability of the reduced parametrized systems, the reliability of the error estimators and the runtime gain by the reduction technique. The a posteriori error estimation technique can straightforwardly be applied to all traditional projection-based reduction techniques of non-parametric and parametric linear systems, such as model reduction, balanced truncation, moment matching, proper orthogonal decomposition (POD) and so on.

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

“…The reduced basis community in particular has promoted a strong emphasis on the derivation of error estimates for parametric model reduction [112,123,189,197,217,218]. This work has created new methods that certify the reduced model through error estimates that can be computed without recourse to the full model [189].…”

“…In other words, the basis behind the model reduction step can come from rational interpolation, balanced truncation, or POD. This can be seen by analyzing these error estimates in state-space form as recently presented in [123]. Recall that projection-based model reduction as in (4.1) corresponds to approximating x(t; p) by Vx r (t; p).…”

“…Recall that projection-based model reduction as in (4.1) corresponds to approximating x(t; p) by Vx r (t; p). As in [123], take E = I and let e(0;p) denote the error in the state x(t; p) 5 The concepts of passivity and contractivity appear frequently in circuit and microwave theory. The output error at time t and the parameterp can be bounded similarly using Although the error estimates apply to a general basis, a key remaining question is the computability of the constant ξ(p).…”

Solving Parameter-Dependent Lyapunov Equations Using the Reduced Basis Method with Application to Parametric Model Order Reduction

“…Remark 6.1. Note that the parametric reduced-order system (6.2) can also be determined by applying the reduced basis method directly to (6.1), see [11]. However, the model reduction approach presented here has several advantages over that method.…”

Greedy‐based approximation of frequency‐weighted Gramian matrices for model reduction in multibody dynamics

“…During an iterative reduction process in every step a new projection matrix is calculated. The aim is to split the calculation of the error estimator into parts which need to be calculated only once, and into parts which need to be calculated in every iteration step, similar to an offline/online decomposition known from the Reduced Basis (RB) community [14,23] to save computation time.…”