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## Über dieses Buch

The special volume offers a global guide to new concepts and approaches concerning the following topics: reduced basis methods, proper orthogonal decomposition, proper generalized decomposition, approximation theory related to model reduction, learning theory and compressed sensing, stochastic and high-dimensional problems, system-theoretic methods, nonlinear model reduction, reduction of coupled problems/multiphysics, optimization and optimal control, state estimation and control, reduced order models and domain decomposition methods, Krylov-subspace and interpolatory methods, and applications to real industrial and complex problems.

The book represents the state of the art in the development of reduced order methods. It contains contributions from internationally respected experts, guaranteeing a wide range of expertise and topics. Further, it reflects an important effor

t, carried out over the last 12 years, to build a growing research community in this field.

Though not a textbook, some of the chapters can be used as reference materials or lecture notes for classes and tutorials (doctoral schools, master classes).

## Inhaltsverzeichnis

### Chapter 1. Two Ways to Treat Time in Reduced Basis Methods

In this chapter, we compare two ways to treat the time within reduced basis methods (RBMs) for parabolic problems: Time-stepping and space-time variational based methods. We briefly recall both concepts and review well-posedness, error control and model reduction in both cases as well as the numerical realization. In particular, we highlight the conceptual differences of the two approaches.
We provide numerical investigations focussing on the performance of the RBM in both variants regarding approximation quality, efficiency and reliability of the error estimator. Pro’s and Con’s of both approaches are discussed.
Silke Glas, Antonia Mayerhofer, Karsten Urban

### Chapter 2. Simultaneous Empirical Interpolation and Reduced Basis Method: Application to Non-linear Multi-Physics Problem

This paper focuses on the reduced basis method in the case of non-linear and non-affinely parametrized partial differential equations where affine decomposition is not obtained. In this context, Empirical Interpolation Method (EIM) (Barrault et al. C R Acad Sci Paris Ser I 339(9):667–672, 2004) is commonly used to recover the affine decomposition necessary to deploy the Reduced Basis (RB) methodology. The build of each EIM approximation requires many finite element solves which increases significantly the computational cost hence making it inefficient on large problems (Daversin et al. ESAIM proceedings, EDP Sciences, Paris, vol. 43, pp. 225–254, 2013). We propose a Simultaneous EIM and RB method (ser) whose principle is based on the use of reduced basis approximations into the EIM building step. The number of finite element solves required by ser can drop to N + 1 where N is the dimension of the RB approximation space, thus providing a huge computational gain. The ser method has already been introduced in Daversin and Prud’homme (C R Acad Sci Paris Ser I 353:1105–1109, 2015) through which it is illustrated on a 2D benchmark itself introduced in Grepl et al. (Modél Math Anal Numér 41(03):575–605, 2007). This paper develops the ser method with some variants and in particular a multilevel ser, ser() which improves significantly ser at the cost of ℓN + 1 finite element solves. Finally we discuss these extensions on a 3D multi-physics problem.
Cécile Daversin, Christophe Prud’homme

### Chapter 3. A Certified Reduced Basis Approach for Parametrized Optimal Control Problems with Two-Sided Control Constraints

In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and two-sided control constraints, which play an important role in many industrial and economical applications. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using two slack variables, we then develop a reduced basis approximation for both slack problems by suitably restricting the solution space, and derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of both approaches and substantiate the comparison by presenting numerical results for a model problem.
Eduard Bader, Martin A. Grepl, Karen Veroy

### Chapter 4. A Reduced Basis Method with an Exact Solution Certificate and Spatio-Parameter Adaptivity: Application to Linear Elasticity

We present a reduced basis method for parametrized linear elasticity equations with two objectives: providing an error bound with respect to the exact weak solution of the PDE, as opposed to the typical finite-element “truth”, in the online stage; providing automatic adaptivity in both physical and parameter spaces in the offline stage. Our error bound builds on two ingredients: a minimum-residual mixed formulation with a built-in bound for the dual norm of the residual with respect to an infinite-dimensional function space; a combination of a minimum eigenvalue bound technique and the successive constraint method which provides a lower bound of the stability constant with respect to the infinite-dimensional function space. The automatic adaptivity combines spatial mesh adaptation and greedy parameter sampling for reduced bases and successive constraint method to yield a reliable online system in an efficient manner. We demonstrate the effectiveness of the approach for a parametrized linear elasticity problem with geometry transformations and parameter-dependent singularities induced by cracks.
Masayuki Yano

### Chapter 5. A Reduced Basis Method for Parameter Functions Using Wavelet Approximations

We consider parameterized parabolic partial differential equations (PDEs) with variable initial conditions, which are interpreted as a parameter function within the Reduced Basis Method (RBM). This means that we are facing an infinite-dimensional parameter space. We propose to use the space-time variational formulation of the parabolic PDE and show that this allows us to derive a two-step greedy method to determine offline separately the reduced basis for the initial value and the evolution. For the approximation of the initial value, we suggest to use an adaptive wavelet approximation. Online, for a given new parameter function, the reduced basis approximation depends on its (quasi-)best N-term approximation in terms of the wavelet basis. A corresponding offline-online decomposable error estimator is provided. Numerical experiments show the flexibility and the efficiency of the method.
Antonia Mayerhofer, Karsten Urban

### Chapter 6. Reduced Basis Isogeometric Mortar Approximations for Eigenvalue Problems in Vibroacoustics

We simulate the vibration of a violin bridge in a multi-query context using reduced basis techniques. The mathematical model is based on an eigenvalue problem for the orthotropic linear elasticity equation. In addition to the nine material parameters, a geometrical thickness parameter is considered. This parameter enters as a 10th material parameter into the system by a mapping onto a parameter independent reference domain. The detailed simulation is carried out by isogeometric mortar methods. Weakly coupled patch-wise tensorial structured isogeometric elements are of special interest for complex geometries with piecewise smooth but curvilinear boundaries. To obtain locality in the detailed system, we use the saddle point approach and do not apply static condensation techniques. However within the reduced basis context, it is natural to eliminate the Lagrange multiplier and formulate a reduced eigenvalue problem for a symmetric positive definite matrix. The selection of the snapshots is controlled by a multi-query greedy strategy taking into account an error indicator allowing for multiple eigenvalues.
Thomas Horger, Barbara Wohlmuth, Linus Wunderlich

### Chapter 7. Reduced Basis Approximations for Maxwell’s Equations in Dispersive Media

Simulation of electromagnetic and optical wave propagation in, e.g. water, fog or dielectric waveguides requires modeling of linear, temporally dispersive media. Using a POD-greedy and ID-greedy sampling driven by an error indicator, we seek to generate a reduced model which accurately captures the dynamics over a wide range of parameters, modeling the dispersion. The reduced basis model reduction reduces the model order by a factor of more than 20, while maintaining an approximation error of significantly less than 1% over the whole parameter range.
Peter Benner, Martin Hess

### Chapter 8. Offline Error Bounds for the Reduced Basis Method

The reduced basis method is a model order reduction technique that is specifically designed for parameter-dependent systems. Due to an offline-online computational decomposition, the method is particularly suitable for the many-query or real-time contexts. Furthermore, it provides rigorous and efficiently evaluable a posteriori error bounds, which are used offline in the greedy algorithm to construct the reduced basis spaces and may be used online to certify the accuracy of the reduced basis approximation. Unfortunately, in real-time applications a posteriori error bounds are of limited use. First, if the reduced basis approximation is not accurate enough, it is generally impossible to go back to the offline stage and refine the reduced model; and second, the greedy algorithm guarantees a desired accuracy only over the finite parameter training set and not over all points in the admissible parameter domain. Here, we propose an extension or “add-on” to the standard greedy algorithm that allows us to evaluate bounds over the entire domain, given information for only a finite number of points. Our approach employs sensitivity information at a finite number of points to bound the error and may thus be used to guarantee a certain error tolerance over the entire parameter domain during the offline stage. We focus on an elliptic problem and provide numerical results for a thermal block model problem to validate our approach.
Robert O’Connor, Martin Grepl

### Chapter 9. ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics

The simulation method ArbiLoMod (Buhr et al., SIAM J. Sci. Comput. 2017, accepted) has the goal of providing users of Finite Element based simulation software with quick re-simulation after localized changes to the model under consideration. It generates a Reduced Order Model (ROM) for the full model without ever solving the full model. To this end, a localized variant of the Reduced Basis method is employed, solving only small localized problems in the generation of the reduced basis. The key to quick re-simulation lies in recycling most of the localized basis vectors after a localized model change. In this publication, ArbiLoMod’s local training algorithm is analyzed numerically for the non-coercive problem of time harmonic Maxwell’s equations in 2D, formulated in H(curl).
Andreas Buhr, Christian Engwer, Mario Ohlberger, Stephan Rave

### Chapter 10. Reduced-Order Semi-Implicit Schemes for Fluid-Structure Interaction Problems

POD–Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.
Francesco Ballarin, Gianluigi Rozza, Yvon Maday

### Chapter 11. True Error Control for the Localized Reduced Basis Method for Parabolic Problems

We present an abstract framework for a posteriori error estimation for approximations of scalar parabolic evolution equations, based on elliptic reconstruction techniques (Makridakis and Nochetto, SIAM J. Numer. Anal. 41(4):1585–1594, 2003. doi:10.1137/S0036142902406314; Lakkis and Makridakis, Math. Comput. 75(256):1627–1658, 2006. doi:10.1090/S0025-5718-06-01858-8; Demlow et al., SIAM J. Numer. Anal. 47(3):2157–2176, 2009. doi:10.1137/070708792; Georgoulis et al., SIAM J. Numer. Anal. 49(2):427–458, 2011. doi:10.1137/080722461). In addition to its original application (to derive error estimates on the discretization error), we extend the scope of this framework to derive offline/online decomposable a posteriori estimates on the model reduction error in the context of Reduced Basis (RB) methods. In addition, we present offline/online decomposable a posteriori error estimates on the full approximation error (including discretization as well as model reduction error) in the context of the localized RB method (Ohlberger and Schindler, SIAM J. Sci. Comput. 37(6):A2865–A2895, 2015. doi:10.1137/151003660). Hence, this work generalizes the localized RB method with true error certification to parabolic problems. Numerical experiments are given to demonstrate the applicability of the approach.
Mario Ohlberger, Stephan Rave, Felix Schindler

### Chapter 12. Efficient Reduction of PDEs Defined on Domains with Variable Shape

In this work we propose a new, general and computationally cheap way to tackle parametrized PDEs defined on domains with variable shape when relying on the reduced basis method. We easily describe a domain by boundary parametrizations, and generate domain (and mesh) deformations by means of a solid extension, obtained by solving a linear elasticity problem. The proposed procedure is built over a two-stages reduction: (1) first, we construct a reduced basis approximation for the mesh motion problem; (2) then, we generate a reduced basis approximation of the state problem, relying on finite element snapshots evaluated over a set of reduced deformed configurations. A Galerkin-POD method is employed to construct both reduced problems, although this choice is not restrictive. To deal with unavoidable nonaffine parametric dependencies arising in both the mesh motion and the state problem, we apply a matrix version of the discrete empirical interpolation method, allowing to treat geometrical deformations in a non-intrusive, efficient and purely algebraic way. In order to assess the numerical performances of the proposed technique, we address the solution of a parametrized (direct) Helmholtz scattering problem where the parameters describe both the shape of the obstacle and other relevant physical features. Thanks to its easiness and efficiency, the methodology described in this work looks promising also in view of reducing more complex problems.
Andrea Manzoni, Federico Negri

### Chapter 13. Localized Reduced Basis Approximation of a Nonlinear Finite Volume Battery Model with Resolved Electrode Geometry

In this contribution we present first results towards localized model order reduction for spatially resolved, three-dimensional lithium-ion battery models. We introduce a localized reduced basis scheme based on non-conforming local approximation spaces stemming from a finite volume discretization of the analytical model and localized empirical operator interpolation for the approximation of the model’s nonlinearities. Numerical examples are provided indicating the feasibility of our approach.
Mario Ohlberger, Stephan Rave

### Chapter 14. A-Posteriori Error Estimation of Discrete POD Models for PDE-Constrained Optimal Control

In this work a-posteriori error estimates for linear-quadratic optimal control problems governed by parabolic equations are considered. Different error estimation techniques for finite element discretizations and model-order reduction are combined to validate suboptimal control solutions from low-order models which are constructed by a Galerkin discretization and the application of proper orthogonal decomposition. The theoretical findings are used to design an updating algorithm for the reduced-order models; the efficiency and accuracy are illustrated by numerical tests.
Martin Gubisch, Ira Neitzel, Stefan Volkwein

### Chapter 15. Hi-POD Solution of Parametrized Fluid Dynamics Problems: Preliminary Results

Numerical modeling of fluids in pipes or network of pipes (like in the circulatory system) has been recently faced with new methods that exploit the specific nature of the dynamics, so that a one dimensional axial mainstream is enriched by local secondary transverse components (Ern et al., Numerical Mathematics and Advanced Applications, pp 703–710. Springer, Heidelberg, 2008; Perotto et al., Multiscale Model Simul 8(4):1102–1127, 2010; Perotto and Veneziani, J Sci Comput 60(3):505–536, 2014). These methods—under the name of Hierarchical Model (Hi-Mod) reduction—construct a solution as a finite element axial discretization, completed by a spectral approximation of the transverse dynamics. It has been demonstrated that Hi-Mod reduction significantly accelerates the computations without compromising the accuracy. In view of variational data assimilation procedures (or, more in general, control problems), it is crucial to have efficient model reduction techniques to rapidly solve, for instance, a parametrized problem for several choices of the parameters of interest. In this work, we present some preliminary results merging Hi-Mod techniques with a classical Proper Orthogonal Decomposition (POD) strategy. We name this new approach as Hi-POD model reduction. We demonstrate the efficiency and the reliability of Hi-POD on multiparameter advection-diffusion-reaction problems as well as on the incompressible Navier-Stokes equations, both in a steady and in an unsteady setting.
Davide Baroli, Cristina Maria Cova, Simona Perotto, Lorenzo Sala, Alessandro Veneziani

### Chapter 16. Adaptive Sampling for Nonlinear Dimensionality Reduction Based on Manifold Learning

We make use of the non-intrusive dimensionality reduction method Isomap in order to emulate nonlinear parametric flow problems that are governed by the Reynolds-averaged Navier-Stokes equations. Isomap is a manifold learning approach that provides a low-dimensional embedding space that is approximately isometric to the manifold that is assumed to be formed by the high-fidelity Navier-Stokes flow solutions under smooth variations of the inflow conditions. The focus of the work at hand is the adaptive construction and refinement of the Isomap emulator: We exploit the non-Euclidean Isomap metric to detect and fill up gaps in the sampling in the embedding space. The performance of the proposed manifold filling method will be illustrated by numerical experiments, where we consider nonlinear parameter-dependent steady-state Navier-Stokes flows in the transonic regime.
Thomas Franz, Ralf Zimmermann, Stefan Görtz

### Chapter 17. Cross-Gramian-Based Model Reduction: A Comparison

As an alternative to the popular balanced truncation method, the cross Gramian matrix induces a class of balancing model reduction techniques. Besides the classical computation of the cross Gramian by a Sylvester matrix equation, an empirical cross Gramian can be computed based on simulated trajectories. This work assesses the cross Gramian and its empirical Gramian variant for state-space reduction on a procedural benchmark based on the cross Gramian itself.
Christian Himpe, Mario Ohlberger

### Chapter 18. Truncated Gramians for Bilinear Systems and Their Advantages in Model Order Reduction

In this paper, we discuss truncated Gramians (TGrams) for bilinear control systems and their relations to Lyapunov equations. We show how TGrams relate to input and output energy functionals, and we also present interpretations of controllability and observability of the bilinear systems in terms of these TGrams. These studies allow us to determine those states that are less important for the system dynamics via an appropriate transformation based on the TGrams. Furthermore, we discuss advantages of the TGrams over the Gramians for bilinear systems as proposed in Al-baiyat and Bettayeb (Proceedings of 32nd IEEE CDC, pp. 22–27, 1993). We illustrate the efficiency of the TGrams in the framework of model order reduction via a couple of examples, and compare to the approach based on the full Gramians for bilinear systems.
Peter Benner, Pawan Goyal, Martin Redmann

### Chapter 19. Leveraging Sparsity and Compressive Sensing for Reduced Order Modeling

Sparsity can be leveraged with dimensionality-reduction techniques to characterize and model parametrized nonlinear dynamical systems. Sparsity is used for both sparse representation, via proper orthogonal decomposition (POD) modes in different dynamical regimes, and by compressive sensing, which provides the mathematical architecture for robust classification of POD subspaces. The method relies on constructing POD libraries in order to characterize the dominant, low-rank coherent structures. Using a greedy sampling algorithm, such as gappy POD and one of its many variants, an accurate Galerkin-POD projection approximating the nonlinear terms from a sparse number of grid points can be constructed. The selected grid points for sampling, if chosen well, can be shown to be effective sensing/measurement locations for classifying the underlying dynamics and reconstruction of the nonlinear dynamical system. The use of sparse sampling for interpolating nonlinearities and classification of appropriate POD modes facilitates a family of local reduced-order models for each physical regime, rather than a higher-order global model. We demonstrate the sparse sampling and classification method on the canonical problem of flow around a cylinder. The method allows for a robust mathematical framework for robustly selecting POD modes from a library, accurately constructing the full state space, and generating a Galerkin-POD projection for simulating the nonlinear dynamical system.
J. Nathan Kutz, Syuzanna Sargsyan, Steven L. Brunton

### Chapter 20. A HJB-POD Approach to the Control of the Level Set Equation

We consider an optimal control problem where the dynamics is given by the propagation of a one-dimensional graph controlled by its normal speed. A target corresponding to the final configuration of the front is given and we want to minimize the cost to reach the target. We want to solve this optimal control problem via the dynamic programming approach but it is well known that these methods suffer from the “curse of dimensionality” so that we can not apply the method to the semi-discrete version of the dynamical system. However, this is made possible by a reduced-order model for the level set equation which is based on Proper Orthogonal Decomposition. This results in a new low-dimensional dynamical system which is sufficient to track the dynamics. By the numerical solution of the Hamilton-Jacobi-Bellman equation related to the POD approximation we can compute the feedback law and the corresponding optimal trajectory for the nonlinear front propagation problem. We discuss some numerical issues of this approach and present a couple of numerical examples.
Alessandro Alla, Giulia Fabrini, Maurizio Falcone

### Chapter 21. Model Order Reduction Approaches for Infinite Horizon Optimal Control Problems via the HJB Equation

We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this report we focus on the infinite horizon optimal control problem with quadratic cost functionals. We compare several model reduction methods such as Proper Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati equation based approach. Finally, we present numerical examples and discuss several features of the different methods analyzing advantages and disadvantages of the reduction methods.
Alessandro Alla, Andreas Schmidt, Bernard Haasdonk

### Chapter 22. Interpolatory Methods for Model Reduction of Multi-Input/Multi-Output Systems

We develop here a computationally effective approach for producing high-quality $$\mathcal{H}_{\infty }$$-approximations to large scale linear dynamical systems having multiple inputs and multiple outputs (MIMO). We extend an approach for $$\mathcal{H}_{\infty }$$ model reduction introduced by Flagg et al. (Syst Control Lett 62(7):567–574, 2013) for the single-input/single-output (SISO) setting, which combined ideas originating in interpolatory $$\mathcal{H}_{2}$$-optimal model reduction with complex Chebyshev approximation. Retaining this framework, our approach to the MIMO problem has its principal computational cost dominated by (sparse) linear solves, and so it can remain an effective strategy in many large-scale settings. We are able to avoid computationally demanding $$\mathcal{H}_{\infty }$$ norm calculations that are normally required to monitor progress within each optimization cycle through the use of “data-driven” rational approximations that are built upon previously computed function samples. Numerical examples are included that illustrate our approach. We produce high fidelity reduced models having consistently better $$\mathcal{H}_{\infty }$$ performance than models produced via balanced truncation; these models often are as good as (and occasionally better than) models produced using optimal Hankel norm approximation as well. In all cases considered, the method described here produces reduced models at far lower cost than is possible with either balanced truncation or optimal Hankel norm approximation.
Alessandro Castagnotto, Christopher Beattie, Serkan Gugercin

### Chapter 23. Model Reduction of Linear Time-Varying Systems with Applications for Moving Loads

In this paper we consider different model reduction techniques for systems with moving loads. Due to the time-dependency of the input and output matrices, the application of time-varying projection matrices for the reduction offers new degrees of freedom, which also come along with some challenges. This paper deals with both, simple methods for the reduction of particular linear time-varying systems, as well as with a more advanced technique considering the emerging time derivatives.
M. Cruz Varona, B. Lohmann

### Chapter 24. Interpolation Strategy for BT-Based Parametric MOR of Gas Pipeline-Networks

Proceeding from balanced truncation-based parametric reduced order models (BT-pROM) a matrix interpolation strategy is presented that allows the cheap evaluation of reduced order models at new parameter sets. The method extends the framework of model order reduction (MOR) for high-order parameter-dependent linear time invariant systems in descriptor form by Geuss (2013) by treating not only permutations and rotations but also distortions of reduced order basis vectors. The applicability of the interpolation strategy and different variants is shown on BT-pROMs for gas transport in pipeline-networks.
Y. Lu, N. Marheineke, J. Mohring

### Chapter 25. Energy Stable Model Order Reduction for the Allen-Cahn Equation

The Allen-Cahn equation is a gradient system, where the free-energy functional decreases monotonically in time. We develop an energy stable reduced order model (ROM) for a gradient system, which inherits the energy decreasing property of the full order model (FOM). For the space discretization we apply a discontinuous Galerkin (dG) method and for time discretization the energy stable average vector field (AVF) method. We construct ROMs with proper orthogonal decomposition (POD)-greedy adaptive sampling of the snapshots in time and evaluating the nonlinear function with greedy discrete empirical interpolation method (DEIM). The computational efficiency and accuracy of the reduced solutions are demonstrated numerically for the parametrized Allen-Cahn equation with Neumann and periodic boundary conditions.
Murat Uzunca, Bülent Karasözen

### Chapter 26. MOR-Based Uncertainty Quantification in Transcranial Magnetic Stimulation

Field computation for Transcranial Magnetic Stimulation requires the knowledge of the electrical conductivity profiles in the human head. Unfortunately, the conductivities of the different tissue types are not exactly known and vary from person to person. Consequently, the computation of the electric field in the human brain should incorporate the uncertainty in the conductivity values. In this paper, we compare a non-intrusive polynomial chaos expansion and a new intrusive parametric Model Order Reduction approach for the sensitivity analysis in Transcranial Magnetic Stimulation computations. Our results show that compared to the non-intrusive method, the new intrusive method provides similar results but shows two orders of magnitude reduced computation time. We find monotonically decreasing errors for increasing state-space dimensions, indicating convergence of the new method. For the sensitivity analysis, both Sobol coefficients and sensitivity coefficients indicate that the uncertainty of the white matter conductivity has the largest influence on the uncertainty in the field computation, followed by gray matter and cerebrospinal fluid. Consequently, individual white matter conductivity values should be used in Transcranial Magnetic Stimulation field computations.
Lorenzo Codecasa, Konstantin Weise, Luca Di Rienzo, Jens Haueisen

### Chapter 27. Model Order Reduction of Nonlinear Eddy Current Problems Using Missing Point Estimation

In electromagnetics, the finite element method has become the most used tool to study several applications from transformers and rotating machines in low frequencies to antennas and photonic devices in high frequencies. Unfortunately, this approach usually leads to (very) large systems of equations and is thus very computationally demanding. This contribution compares three model order reduction techniques for the solution of nonlinear low frequency electromagnetic applications (in the so-called magnetoquasistatic regime) to efficiently reduce the number of equations—leading to smaller and faster systems to solve.
Y. Paquay, O. Brüls, C. Geuzaine

### Chapter 28. On Efficient Approaches for Solving a Cake Filtration Model Under Parameter Variation

In this work, we are considering a mathematical model for an industrial cake filtration process. The model is of moving boundary type and involves a set of parameters, which vary in a given range. We are interested in the case when the model has to be solved for thousands of different parameter values, and therefore model order reduction (MOR) is desirable, so that from full order solutions with one or several sets of parameters we derive a reduced model, which is used further to perform the simulations with new parameters. We study and compare the performance of several MOR techniques known from the literature. We start with standard MOR based on proper orthogonal decomposition (POD) and consider also several more advanced techniques based on combination of MOR and reduced basis techniques, including approaches relying on computation of sensitivities. The transformation from a moving to a fixed domain introduces time varying coefficients into the equations, which makes it reasonable to use an offline/online decomposition. Several test cases involving different simulation time horizons and short time training are considered. Numerical tests show that the discussed methods can approximate the full model solution accurately and work efficiently for new parameters belonging to a given parameter range.
S. Osterroth, O. Iliev, R. Pinnau

### Chapter 29. Model Reduction for Coupled Near-Well and Reservoir Models Using Multiple Space-Time Discretizations

In reservoir simulations, fine fully-resolved grids deliver accurate model representations, but lead to large systems of nonlinear equations to solve every time step. Numerous techniques are applied in porous media flow simulations to reduce the computational effort associated with solving the underlying coupled nonlinear partial differential equations. Many models treat the reservoir as a whole. In other cases, the near-well accuracy is important as it controls the production rate. Near-well modeling requires finer space and time resolution compared with the remaining of the reservoir domain. To address these needs, we combine Model Order Reduction (MOR) with local grid refinement and local time stepping for reservoir simulations in highly heterogeneous porous media. We present a domain decomposition algorithm for a gas flow model in porous media coupling near-well regions, which are locally well-resolved in space and time with a coarser reservoir discretization. We use a full resolution for the near-well regions and apply MOR in the remainder of the domain. We illustrate our findings with numerical results on a gas flow model through porous media in a heterogeneous reservoir.
Walid Kheriji, Yalchin Efendiev, Victor Manuel Calo, Eduardo Gildin

### Chapter 30. Time-Dependent Parametric Model Order Reduction for Material Removal Simulations

Machining of thin and lightweight structures is a crucial manufacturing step in industries ranging from aerospace to power engineering. In order to enable efficient simulations of elastic workpieces and solve typical tasks like the prediction of process stability, reduced elastic models have to be determined by model order reduction. Thereby, the system matrices need to be constant, which cannot be assumed for elastic bodies with varying geometry due to material removal. In this contribution we propose a technique to generate reduced elastic bodies for systems with time-varying geometries and their application in time-domain simulations. Therefore, the model is described as a parameter-dependent system. Due to the fact that the considered parameter varies in time-domain simulations, time-dependent parametric model order reduction techniques for elastic bodies are presented.
Michael Baumann, Dominik Hamann, Peter Eberhard
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