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.
Abstract. We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reduced-order model. The parameter dependence may be linear or nonlinear and is retained in the reduced-order model. Moreover, we are able to give conditions under which the gradient and Hessian of the system response with respect to the system parameters is matched in the reduced-order model. We provide a systematic approach built on established interpolatory H 2 optimal model reduction methods that will produce parameterized reduced-order models having high fidelity throughout a parameter range of interest. For single input/single output systems with parameters in the input/output maps, we provide reduced-order models that are optimal with respect to an H 2 ⊗ L 2 joint error measure. The capabilities of these approaches are illustrated by several numerical examples from technical applications. 1. Introduction. Numerical simulation has steadily increased in importance across virtually all scientific and engineering disciplines. In many application areas, experiments have been largely replaced by numerical simulation in order to save costs in design and development. High accuracy simulation requires high fidelity mathematical models which in turn induce dynamical systems of very large dimension. The ensuing demands on computational resources can be overwhelming and efficient model utilization becomes a necessity. It often is both possible and prudent to produce a lower dimension model that approximates the response of the original one to high accuracy. There are many model reduction strategies in use that are remarkably effective in the creation of compact, efficient, and high fidelity dynamical system models. Such a reduced model can then be used reliably as an efficient surrogate to the original system, replacing it as a component in larger simulations, for example, or in allied contexts that involve design optimization or the development of low-order, fast controllers suitable for real time applications.Typically, a reduced-order model will represent a specific instance of the physical system under study and as a consequence will have high fidelity only for small variations around that base system instance. Significant modifications to the physical model such as geometric variations, changes in material properties, or alterations in
Model reduction is a common theme within the simulation, control and optimization of complex dynamical systems. For instance, in control problems for partial differential equations, the associated large-scale systems have to be solved very often. To attack these problems in reasonable time it is absolutely necessary to reduce the dimension of the underlying system. We focus on model reduction by balanced truncation where a system theoretical background provides some desirable properties of the reduced-order system. The major computational task in balanced truncation is the solution of large-scale Lyapunov equations, thus the method is of limited use for really large-scale applications. We develop an effective implementation of balancing-related model reduction methods in exploiting the structure of the underlying problem. This is done by a data-sparse approximation of the large-scale state matrix A using the hierarchical matrix format. Furthermore, we integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the Lyapunov equations. This approach is well-suited for a class of practical relevant problems and allows the application of balanced truncation and related methods to systems coming from 2D and 3D FEM and BEM discretizations.
We investigate the numerical solution of large-scale Lyapunov equations with the sign function method. Replacing the usual matrix inversion, addition, and multiplication by formatted arithmetic for hierarchical matrices, we obtain an implementation that has linearpolylogarithmic complexity and memory requirements. The method is well suited for Lyapunov operators arising from FEM and BEM approximations to elliptic differential operators. With the sign function method it is possible to obtain a low-rank approximation to a full-rank factor of the solution directly. The task of computing such a factored solution arises, e.g., in model reduction based on balanced truncation. The basis of our method is a partitioned Newton iteration for computing the sign function of a suitable matrix, where one part of the iteration uses formatted arithmetic while the other part directly yields approximations to the full-rank factor of the solution. We discuss some variations of our method and its application to generalized Lyapunov equations. Numerical experiments show that the method can be applied to problems of order up to O(10 5 ) on workstations.
1 Complexity reduction is an important issue in the mathematical modelling of cells. The use of small effective models can speed up numerical simulations considerably, and on top of this, focusing on the most important modes of dynamics can help us to understand the design of biological systems. In this article, we concentrate on small biochemical systems (e.g. a single metabolic pathway) that are embedded in a complex environment. For the sake of modelling, reactions in the environment are often ignored, while external metabolite concentrations are held at fixed values. To justify this, it is typically assumed that these metabolite concentrations are either very high or efficiently buffered, which is not always the case. If the buffering is incomplete, then the system will influence its environment and create perturbations that can act back on the system. If this feedback loop is neglected, then the model is possibly not suited to describe the data, and fitted model parameters may be wrong even if the fit looks satisfactory. Hence, we are looking for a more faithful representation of the environment that can provide realistic boundary conditions. For the modelling of steady states, this has been accomplished by using phenomenological relations between different external metabolite concentrations [1]. For dynamic simulations, however, the problem becomes harder: the environment has to be modelled dynamically, which can increase the simulation time. As a remedy, we propose to replace it by a small linear model that is supposed to mimic the dynamic responses of the original environment. Reduction of linear models has been studied for a long time, and various methods have been proposed. We use balanced truncation [2], which is numerically demanding but yields a stable reduced system with a bounded approximation error (the Matlab code for balanced truncation can be found at http://www.tu-chemnitz.de/mathematik/ industrie_technik/software/software.php). Moreover, by tuning the dimensionality, one can choose a compromise between approximation accuracy and numerical efficiency. Balanced truncation has successfully been Modelling of biochemical systems usually focuses on certain pathways, while the concentrations of so-called external metabolites are considered fixed. This approximation ignores feedback loops mediated by the environment, that is, via external metabolites and reactions. To achieve a more realistic, dynamic description that is still numerically efficient, we propose a new methodology: the basic idea is to describe the environment by a linear effective model of adjustable dimensionality. In particular, we (a) split the entire model into a subsystem and its environment, (b) linearize the environment model around a steady state, and (c) reduce its dimensionality by balanced truncation, an established method for large-scale model reduction. The reduced variables describe the dynamic modes in the environment that dominate its interaction with the subsystem. We compute metabolic response coefficients that accou...
In this paper, a method for solving large-scale Sylvester equations is presented. The method is based on the sign function iteration and is particularly effective for Sylvester equations with factorized right-hand side. In this case, the solution will be computed in factored form as it is for instance required in model reduction. The hierarchical matrix format and the corresponding formatted arithmetic are integrated in the iteration scheme to make the method feasible for large-scale computations.
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