In the past decades, Model Order Reduction (MOR) has demonstrated its robustness and wide applicability for simulating large-scale mathematical models in engineering and the sciences. Recently, MOR has been intensively further developed for increasingly complex dynamical systems. Wide applications of MOR have been found not only in simulation, but also in optimization and control. In this survey paper, we review some popular MOR methods for linear and nonlinear large-scale dynamical systems, mainly used in electrical and control engineering, in computational electromagnetics, as well as in micro-and nanoelectro-mechanical systems design. This complements recent surveys on generating reduced-order models for parameterdependent problems (Benner et al. in 2013; Boyaval et al. in Arch Comput Methods Eng 17(4):435-454, 2010; Rozza et al. Arch Comput Methods Eng 15(3):229-275, 2008) which we do not consider here. Besides reviewing existing methods and the computational techniques needed to implement them, open issues are discussed, and some new results are proposed.

A robust algorithm for computing reduced-order models of parametric systems is proposed. Theoretical considerations suggest that our algorithm is more robust than previous algorithms based on explicit multi-moment matching. Moreover, numerical simulation results show that the proposed algorithm yields more accurate approximations than traditional non-parametric model reduction methods and parametric model reduction methods based on explicitly computing moments.

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.

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. Furthermore, we propose several variants of the error estimator, and compare those variants with the existing ones both theoretically and numerically. It has been 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 output error of reduced-order modeling for steady linear parametric systems. *

Abstract. We propose a posteriori error bounds for reduced-order models of non-parametrized linear time invariant (LTI) systems and parametrized LTI systems. The error bounds estimate the errors of the transfer functions of the reduced-order models, and are independent of the model reduction methods used. It is shown that for some special non-parametrized LTI systems, particularly efficiently computable error bounds can be derived. According to the error bounds, reduced-order models of both non-parametrized and parametrized systems, computed by Krylov subspace based model reduction methods, can be obtained automatically and reliably. Simulations for several examples from engineering applications have demonstrated the robustness of the error bounds.

Abstract. Krylov subspace recycling is a process for accelerating the convergence of sequences of linear systems. Based on this technique, the recycling BiCG algorithm has been developed recently. Here, we now generalize and extend this recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving sequences of dual linear systems, while the focus here is on efficiently solving sequences of single linear systems (assuming non-symmetric matrices for both recycling BiCG and recycling BiCGSTAB).As compared with other methods for solving sequences of single linear systems with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based recycling algorithms, like recycling Bi-CGSTAB, have the advantage that they involve a short-term recurrence, and hence, do not suffer from storage issues and are also cheaper with respect to the orthogonalizations.We modify the BiCGSTAB algorithm to use a recycle space, which is built from left and right approximate invariant subspaces. Using our algorithm for a parametric model order reduction example gives good results. We show about 40% savings in the number of matrix-vector products and about 35% savings in runtime.

In this work we present an a posteriori output error bound for model order reduction of parametrized evolution equations. With the help of the dual system and a simple representation of the relationship between the field variable error and the residual of the primal system, a sharp output error bound is derived. Such an error bound successfully avoids the accumulation of the residual over time, which is a common drawback in the existing error estimations for time-stepping schemes. An estimation needs to be performed for practical computation of the error bound, and as a result, the output error bound reduces to an output error estimation. The proposed error estimation is applied to four kinds of problems. The first one is a linear convection-diffusion equation, which is used to compare the performance of the new error estimation and an existing primal-dual error bound. The second one is the unsteady viscous Burgers' equation, an academic benchmark of nonlinear evolution equations in fluid dynamics often used as a first test case to validate nonlinear model order reduction methods. The other two problems arise from chromatographic separation processes. Numerical experiments demonstrate the performance and efficiency of the proposed error estimation. Furthermore, optimization based on the resulting reduced-order models is successful in terms of accuracy and runtime for obtaining the optimal solution. B911for linear or quadratically nonlinear parabolic problems [34,37,38,39]. Notably, these error estimations are all derived in the functional space in the framework of the finite element (FE) discretization except for [17]. In the FE discretization framework, the weak form of the partial differential equation (PDE) is used to derive the error bound, while the error bound in [17] is derived in the framework of the finite volume discretization for error estimation of the field variables.In this paper, we propose an efficient output error estimation for projection-based MOR methods applied to parametrized nonlinear evolution problems. For (nonlinear) evolution problems, time-stepping schemes are often used to solve them [24], and error estimations for projection-based MOR methods have been studied in recent years; see, e.g., [13,17,40]. The error estimator, however, may lose sharpness when a large number of time steps are needed, because the error estimator is actually a summation of the error over the previous time steps. To circumvent this problem, we introduce a suitable dual system at each time instance in the evolution process associated with the primal system, i.e., the original system. The output error for the primal system can thus be estimated sharply and efficiently with the help of the dual system and a simple representation of the relationship between the residual and the error of the field variable.Actually, an a posteriori output error bound for the RB method using the primaldual approach can be found in [16]. However, the derived output error bound is suitable only for linear evolution equations. From the ...

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