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

Published on Nov 5, 2015in Siam Review10.78
路 DOI :10.1137/130932715
Peter Benner54
Estimated H-index: 54
(MPG: Max Planck Society),
Serkan Gugercin31
Estimated H-index: 31
(VT: Virginia Tech),
Karen Willcox48
Estimated H-index: 48
(MIT: Massachusetts Institute of Technology)
Sources
Abstract
Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...
Download
馃摉 Papers frequently viewed together
2005
References222
Newest
Nonlinear parametric inverse problems appear in several prominent applications; one such application is diffuse optical tomography (DOT) in medical image reconstruction. Such inverse problems present huge computational challenges, mostly due to the need for solving a sequence of large-scale discretized, parametrized partial differential equations in the forward model. In this paper, we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in...
Source
In this paper, we focus on model reduction of large-scale bilinear systems. The main contributions are threefold. First, we introduce a new framework for interpolatory model reduction of bilinear systems. In contrast to the existing methods where interpolation is forced on some of the leading subsystem transfer functions, the new framework shows how to enforce multipoint interpolation of the underlying Volterra series. Then, we show that the first-order conditions for optimal \mathcal{H}_2mod...
Source
This paper proposes a data-driven strategy to assist online rapid decision making for an unmanned aerial vehicle that uses sensed data to estimate its structural state, uses this estimate to update its corresponding flight capabilities, and then dynamically replans its mission accordingly. The approach comprises offline and online computational phases constructed to address the sense鈥損lan鈥揳ct information flow while avoiding a costly online inference step. During the offline phase, high-fidelity ...
Source
#1Peter Benner (MPG: Max Planck Society)H-Index: 54
#2Tobias Breiten (University of Graz)H-Index: 15
In this paper, we investigate a recently introduced approach for nonlinear model order reduction based on generalized moment matching. Using basic tensor calculus, we propose a computationally efficient way of computing reduced-order models. We further extend the idea of two-sided interpolation methods to this more general setting by employing the tensor structure of the Hessian. We investigate the use of oblique projections in order to preserve important system properties such as stability. We ...
Source
The Bilinear Interpolatory Rational Krylov Algorithm (BIRKA; P. Benner and T. Breiten, Interpolation-based H2-model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl. 33 (2012), pp. 859鈥885. doi:10.1137/110836742) is a recently developed method for Model Order Reduction (MOR) of bilinear systems. Here, it is used and further developed for a certain class of parametric systems. As BIRKA does not preserve stability, two different approaches generating stable reduced models are prese...
Source
#1Peter Benner (MPG: Max Planck Society)H-Index: 54
#2Lihong Feng (MPG: Max Planck Society)H-Index: 16
Last. Yongjin Zhang (MPG: Max Planck Society)H-Index: 4
view all 4 authors...
A reduced basis method is applied to batch chromatography and the underlying optimization problem is solved efficiently based on the resulting reduced model. A technique of adaptive snapshot selection is proposed to reduce the complexity and runtime of generating the reduced basis. With the help of an output-oriented error bound, the construction of the reduced model is managed automatically. Numerical examples demonstrate the performance of the adaptive technique in reducing the offline time. T...
Source
#1Alberto Sartori (Polytechnic University of Milan)H-Index: 9
#2Davide Baroli (Polytechnic University of Milan)H-Index: 7
Last. Gianluigi Rozza (SISSA: International School for Advanced Studies)H-Index: 41
view all 5 authors...
In this work, a Reduced Order Model (ROM) for multi-group time-dependent parametrized reactor spatial kinetics is presented. The Reduced Basis method (built upon a high-fidelity 鈥渢ruth鈥 finite element approximation) has been applied to model the neutronics behavior of a parametrized system composed by a control rod surrounded by fissile material. The neutron kinetics has been described by means of a parametrized multi-group diffusion equation where the height of the control rod (i.e., how much t...
Source
#1Paul G. Constantine (Colorado School of Mines)H-Index: 24
#2David F. Gleich (Purdue University)H-Index: 33
Last. Jeremy Alan TempletonH-Index: 15
view all 4 authors...
We present a method for computing reduced-order models of parameterized partial differential equation solutions. The key analytical tool is the singular value expansion of the parameterized solution, which we approximate with a singular value decomposition of a parameter snapshot matrix. To evaluate the reduced-order model at a new parameter, we interpolate a subset of the right singular vectors to generate the reduced-order model's coefficients. We employ a novel method to select this subset th...
Source
#1Ulrike Baur (MPG: Max Planck Society)H-Index: 9
#2Peter Benner (MPG: Max Planck Society)H-Index: 54
Last. Lihong Feng (MPG: Max Planck Society)H-Index: 16
view all 3 authors...
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, mai...
Source
Purpose 鈥 The Reduced Basis Method (RBM) generates low-order models of parametrized PDEs to allow for efficient evaluation of parametrized models in many-query and real-time contexts. The purpose of this paper is to investigate the performance of the RBM in microwave semiconductor devices, governed by Maxwell's equations. Design/methodology/approach 鈥 The paper shows the theoretical framework in which the RBM is applied to Maxwell's equations and present numerical results for model reduction und...
Source
Cited By779
Newest
Last. Eleni ChatziH-Index: 29
view all 6 authors...
Source
#1Michael Sleeman (U of T: University of Toronto)H-Index: 1
#2Masayuki Yano (U of T: University of Toronto)H-Index: 14
Abstract null null We present a projection-based model reduction formulation for parametrized time-dependent nonlinear partial differential equations (PDEs). Our approach builds on the following ingredients: reduced bases (RB), which provide rapidly convergent approximations of the parameter-temporal solution manifold; reduced quadrature (RQ) rules, which provide hyperreduction of the nonlinear residual; and the dual-weighted residual (DWR) method, which provides an error representation formula ...
Source
#1Yukiko S. Shimizu (SNL: Sandia National Laboratories)
#2Eric J. Parish (SNL: Sandia National Laboratories)H-Index: 9
Abstract null null Projection-based reduced-order models (PROMs) restrict the full-order model (FOM) to a low-dimensional subspace. Space鈥搕ime PROMs in particular restrict the FOM to a temporally space鈥搕ime trial subspace, and compute approximate solutions through a residual orthogonalization or minimization process. One such technique of interest is the space鈥搕ime least-squares Petrov鈥揋alerkin method (ST-LSPG), which reduces both the spatial and temporal dimensions. However, ST-LSPG is computat...
Source
A data-driven approach for the identification of local turbulent-flow states and of their dynamics is proposed. After subdividing a flow domain in smaller regions, the -medoids clustering algorithm is used to learn from the data the different flow states and to identify the dynamics of the transition process. The clustering procedure is carried out on a two-dimensional (2-D) reduced-order space constructed by the multidimensional scaling (MDS) technique. The MDS technique is able to provide mean...
Source
#1Kai Li (NPU: Northwestern Polytechnical University)H-Index: 1
#2Jiaqing Kou (NPU: Northwestern Polytechnical University)H-Index: 15
Last. Weiwei Zhang (NPU: Northwestern Polytechnical University)H-Index: 19
view all 3 authors...
Abstract null null Computational-fluid-dynamics-based prediction of unsteady aerodynamics is an essential research topic in the design of aircraft, which usually requires very high computational cost. Therefore, developing efficient and accurate reduced-order models (ROMs) for unsteady aerodynamics is important. With recent progress in modeling unsteady aerodynamics for a fixed configuration, there is a need of developing reduced-order models across multiple geometrical shapes. In this work, the...
Source
#1Nirav Vasant Shah (SISSA: International School for Advanced Studies)
#2Michele GirfoglioH-Index: 8
Last. P. BarralH-Index: 6
view all 7 authors...
This contribution focuses on the development of Model Order Reduction (MOR) for one-way coupled steady state linear thermomechanical problems in a finite element setting. We apply Proper Orthogonal Decomposition (POD) for the computation of reduced basis space. On the other hand, for the evaluation of the modal coefficients, we use two different methodologies: the one based on the Galerkin projection (G) and the other one based on Artificial Neural Network (ANN). We aim at comparing POD-G and PO...
#1Alan A. KaptanogluH-Index: 4
#2Brian M. de SilvaH-Index: 4
view all 10 authors...
Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy, we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. T...
#2Kai Jiang (XTU: Xiangtan University)H-Index: 9
Last. Shi Shu (XTU: Xiangtan University)H-Index: 19
view all 3 authors...
In this paper, we propose a network model, the multiclass classification-based ROM (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying the deep learning-based reduced order model (DL-ROM) to solve diffusion-dominant PPDEs. We find that the DL-ROM has a good approximation for some special model parameters, but it cannot approximate the drastic changes of the solution as time evolves. Based on this fact, we cl...
#2Youssef MrouehH-Index: 19
Last. Steven G. JohnsonH-Index: 85
view all 5 authors...
We present a "physics-enhanced deep-surrogate ("PEDS") approach towards developing fast surrogate models for complex physical systems described by partial differential equations (PDEs) and similar models: we show how to combine a low-fidelity "coarse" solver with a neural network that generates "coarsified'' inputs, trained end-to-end to globally match the output of an expensive high-fidelity numerical solver. In this way, by incorporating limited physical knowledge in the form of the low-fideli...
Source
This website uses cookies.
We use cookies to improve your online experience. By continuing to use our website we assume you agree to the placement of these cookies.
To learn more, you can find in our Privacy Policy.