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This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.

Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects.

This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.



1. A Novel Approach to Model Order Reduction for Coupled Multiphysics Problems

Model order reduction (MOR) has become an important tool in the design of complex high-tech systems. It can be used to find a low-order model that approximates the behavior of the original high-order model, where this low-order approximation facilitates both the computationally efficient analysis and controller design for the system to induce desired behavior. This chapter introduces MOR techniques that are designed especially for coupled problems, meaning that different physical phenomena are simulated in conjunction with each other. The method developed makes use of the reduction of the individual systems, and low rank approximations of the coupling blocks. This is done in such a way that existing software for industrial problems can be adapted in a straightforwward way. An industrial test case is described in detail, so as to demonstrate the effectiveness of the reduction technique.
Wil H. A. Schilders, Agnieszka Lutowska

2. Case Study: Parametrized Reduction Using Reduced-Basis and the Loewner Framework

In this case study, we compare two methods for model reduction of parametrized systems, namely, Reduced-Basis and Loewner rational interpolation.
While having the same goal of constructing reduced-order models for large-scale parameter-dependent systems, the two methods follow fundamentally different approaches. On the one hand, the well known Reduced-Basis method takes a time domain approach, using offline snapshots of the full-order system combined with a rigorous error bound. On the other hand, the recently introduced Loewner matrix framework takes a frequency-domain approach that constructs rational interpolants of transfer function measurements, and has the flexibility of allowing different reduced-orders for each of the frequency and parameter variables.
We apply the two methods to a parametrized partial differential equation modeling the transient temperature evolution near the surface of a cylinder immersed in fluid. Then, we compare the resulting reduced-order models with the full-order finite element system by running both time- and frequency-domain simulations.
Antonio C. Ionita, Athanasios C. Antoulas

3. Comparison of Some Reduced Representation Approximations

In the field of numerical approximation, specialists considering highly complex problems have recently proposed various ways to simplify their underlying problems. In this field, depending on the problem they were tackling and the community that are at work, different approaches have been developed with some success and have even gained some maturity, the applications can now be applied to information analysis or for numerical simulation of PDE’s. At this point, a crossed analysis and effort for understanding the similarities and the differences between these approaches that found their starting points in different backgrounds is of interest. It is the purpose of this paper to contribute to this effort by comparing some constructive reduced representations of complex functions. We present here in full details the Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM) together with other approaches that enter in the same category.
Mario Bebendorf, Yvon Maday, Benjamin Stamm

4. Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear and Parametric Systems

Projection based methods lead to reduced order models (ROMs) with dramatically reduced numbers of equations and unknowns. However, for nonlinear or parametrically varying problems the cost of evaluating these ROMs still depends on the size of the full order model and therefore is still expensive. The Discrete Empirical Interpolation Method (DEIM) further approximates the nonlinearity in the projection based ROM. The resulting DEIM ROM nonlinearity depends only on a few components of the original nonlinearity. If each component of the original nonlinearity depends only on a few components of the argument, the resulting DEIM ROM can be evaluated efficiently at a cost that is independent of the size of the original problem. For systems obtained from finite difference approximations, the ith component of the original nonlinearity often depends only on the ith component of the argument. This is different for systems obtained using finite element methods, where the dependence is determined by the mesh and by the polynomial degree of the finite element subspaces. This paper describes two approaches of applying DEIM in the finite element context, one applied to the assembled and the other to the unassembled form of the nonlinearity. We carefully examine how the DEIM is applied in each case, and the substantial efficiency gains obtained by the DEIM. In addition, we demonstrate how to apply DEIM to obtain ROMs for a class of parameterized system that arises, e.g., in shape optimization. The evaluations of the DEIM ROMs are substantially faster than those of the standard projection based ROMs. Additional gains are obtained with the DEIM ROMs when one has to compute derivatives of the model with respect to the parameter.
Harbir Antil, Matthias Heinkenschloss, Danny C. Sorensen

5. Greedy Sampling Using Nonlinear Optimization

We consider the reduced basis generation in the offline stage. As an alternative for standard Greedy-training methods based upon a-posteriori error estimates on a training subset of the parameter set, we consider a nonlinear optimization combined with a Greedy method. We define an optimization problem for selecting a new parameter value on a given reduced space. This new parameter is then used -in a Greedy fashion- to determine the corresponding snapshot and to update the reduced basis. We show the well-posedness of this nonlinear optimization problem and derive first- and second-order optimality conditions. Numerical comparisons with the standard Greedy-training method are shown.
Karsten Urban, Stefan Volkwein, Oliver Zeeb

6. A Robust Algorithm for Parametric Model Order Reduction Based on Implicit Moment Matching

Parametric model order reduction (PMOR) has received a tremendous amount of attention in recent years. Among the first approaches considered, mainly in system and control theory as well as computational electromagnetics and nanoelectronics, are methods based on multi-moment matching. Despite numerous other successful methods, including the reduced-basis method (RBM), other methods based on (rational, matrix, manifold) interpolation, or Kriging techniques, multi-moment matching methods remain a reliable, robust, and flexible method for model reduction of linear parametric systems. Here we propose a numerically stable algorithm for PMOR based on multi-moment matching. Given any number of parameters and any number of moments of the parametric system, the algorithm generates a projection matrix for model reduction by implicit moment matching. The implementation of the method based on a repeated modified Gram-Schmidt-like process renders the method numerically stable. The proposed method is simple yet efficient. Numerical experiments show that the proposed algorithm is very accurate.
Peter Benner, Lihong Feng

7. On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

We propose a modified parallel-in-time — parareal — multi-level time integration method that, in contrast to previously proposed methods, employs a coarse solver based on a reduced model, built from the information obtained from the fine solver at each iteration. This approach is demonstrated to offer two substantial advantages: it accelerates convergence of the original parareal method for similar problems and the reduced basis stabilizes the parareal method for purely advective problems where instabilities are known to arise. When combined with empirical interpolation methods (EIM), we develop this approach to solve both linear and nonlinear problems and highlight the minimal changes required to utilize this algorithm to accelerate existing implementations. We illustrate the advantages through algorithmic design, through analysis of stability, convergence, and computational complexity, and through several numerical examples.
Feng Chen, Jan S. Hesthaven, Xueyu Zhu

8. On the Stability of Reduced-Order Linearized Computational Fluid Dynamics Models Based on POD and Galerkin Projection: Descriptor vs Non-Descriptor Forms

The Galerkin projection method based on modes generated by the Proper Orthogonal Decomposition (POD) technique is very popular for the dimensional reduction of linearized Computational Fluid Dynamics (CFD) models, among many other typically high-dimensional models in computational engineering. This, despite the fact that it cannot guarantee neither the optimality nor the stability of the Reduced- Order Models (ROMs) it constructs. Short of proposing any variant of this model order reduction method that guarantees the stability of its outcome, this paper contributes a best practice to its application to the construction of linearized CFD ROMs. It begins by establishing that whereas the solution snapshots computed using the descriptor and non-descriptor forms of the discretized Euler or Navier-Stokes equations are identical, the ROMs obtained by reducing these two alternative forms of the governing equations of interest are different. Focusing next on compressible fluid-structure interaction problems associated with computational aeroelasticity, this paper shows numerically that the POD-based fluid ROMs originating from the non-descriptor form of the governing linearized CFD equations tend to be unstable, but their counterparts originating from the descriptor form of these equations are typically stable and reliable for aeroelastic applications. Therefore, this paper argues that whereas many computations are performed in CFD codes using the non-descriptor form of discretized Euler and/or Navier-Stokes equations, POD-based model reduction in these codes should be performed using the descriptor form of these equations.
David Amsallem, Charbel Farhat

9. Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.
Toni Lassila, Andrea Manzoni, Alfio Quarteroni, Gianluigi Rozza

10. Window Proper Orthogonal Decomposition: Application to Continuum and Atomistic Data

Proper Orthogonal Decomposition (POD) is a powerful tool for analyzing multidimensional data, especially of vector fields in large-scale simulations. In this article we review the Window Proper Orthogonal Decomposition (WPOD) proposed in [7] for analysis of continuum data and in [5] for analysis of atomistic fields.
Leopold Grinberg, Mingge Deng, George Em Karniadakis, Alexander Yakhot

11. Reduced Order Models at Work in Aeronautics and Medicine

We review a few applications of reduced-order modeling in aeronautics and medicine. The common idea is to determine an empirical approximation space for a model described by partial differential equations. The empirical approximation space is usually spanned by a small number of global modes. In case of time-periodic or mainly diffusive phenomena it is shown that this approach can lead to accurate fast simulations of complex problems. In other cases, models based on definition of transport modes significantly improve the accuracy of the reduced model.
Michel Bergmann, Thierry Colin, Angelo Iollo, Damiano Lombardi, Olivier Saut, Haysam Telib


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