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 effort, 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).

Peter Benner is Director of the Max Planck Institute for Dynamics of Complex Technical Systems and head of the department "Computational Methods in Systems and Control Theory". Moreover, he is a Professor at the TU Chemnitz and Adjunct Professor at the Otto-von-Guericke University Magdeburg. He serves on the editorial board of several scientific journals, including Advances in Computational Mathematics and the SIAM Journal on Matrix Analysis and Applications.

Mario Ohlberger is a Full Professor of Applied Mathematics and Managing Director of Applied Mathematics at the University of Münster's Institute of Analysis and Numerics. He is an Associate Editor of five mathematical journals, including SIAM Journal on Scientific Computing. He is a member of the Center for Nonlinear Science, the Center for Multiscale Theory and Computation, and the Cluster of Excellence "Cells in Motion".

Anthony T. Patera is the Ford Professor of Engineering and a Professor of Mechanical Engineering at MIT, and Co-Director of the MIT Center for Computational Engineering.  His research interests include partial differential equations, computational methods, model order reduction, a posteriori error estimation, and data assimilation. Professor Patera holds SB and SM degrees in Mechanical Engineering from MIT, and a PhD in Applied Mathematics, also from MIT. He served as Co-Editor-in-Chief of the journal Mathematical Modeling and Numerical Analysis from 2003 to 2012.

Gianluigi Rozza has been an Associate Professor of Numerical Analysis and Scientific Computing at SISSA, International School for Advanced Studies since 2014. He holds a degree in Aerospace Engineering from Politecnico di Milano (2002) and a PhD in Applied Mathematics at Ecole Polytechnique Federale de Lausanne (2005). He was a post-doctoral research associate at the Massachusetts Institute of Technology (MIT) Center for Computational Engineering (2006-08), t

hen a Researcher and Lecturer at EPFL (2008-2012). He is the author of over 100 scientific publications and recipient of the 2014 ECCOMAS young investigator Jacques Louis Lions Award in Computational Mathematics for researchers under the age of 40. Professor Rozza has been an associate editor of the SIAM/ASA Journal of Uncertainty Quantification since 2013, of the SIAM Journal of Numerical Analysis since 2015, and of Computing and Visualization in Science since 2016.

Karsten Urban is a Full Professor of Numerical Mathematics and Director of the Scientific Computing Centre at Ulm University. He is Managing Editor in Chief of Advances in Computational Mathematics and  Associate Editor of several mathematical journals, including SIAM Journal on Scientific  Computing. Further, he is currently directing several interdisciplinary research projects.

1 Two ways to treat time in Reduced Basis Methods.- 2 Simultaneous empirical interpolation and reduced basis method. Application to non-linear multi-physics problem.- 3 A Certified Reduced Basis Approach for Parametrized Optimal Control Problems with Two-sided Control Constraints.- 4 A reduced basis method with an exact solution certificate and spatio-parameter adaptivity: application to linear elasticity.- 5 A Reduced Basis Method for Parameter Functions using Wavelet Approximations.- 6 Reduced basis isogeometric mortar approximations for eigenvalue problems in vibroacoustics.- 7 Reduced Basis Approximations for Maxwell's Equations in Dispersive Media.- 8 Offline Error Bounds for the Reduced Basis Method.- 9 ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics.- 10 Reduced-order semi-implicit schemes for  fluid-structure interaction problems.- 11 True Error Control for the Localized Reduced Basis Method for Parabolic Problems.- 12 Automatic reduction of

PDEs defined on domains with variable shape.- 13 Localized Reduced Basis Approximation of a Nonlinear Finite Volume Battery Model with Resolved Electrode Geometry.- 14 A-posteriori error estimation of discrete POD models for PDE-constrained optimal control.- 15 Hi-POD solution of parametrized fluid dynamics problems: preliminary results.- 16 Adaptive sampling for nonlinear dimensionality reduction based on manifold learning.- 17 Cross-Gramian-Based Model Reduction: A Comparison.- 18 Truncated Gramians for Bilinear Systems and their Advantages in Model Order Reduction.- 19 Leveraging Sparsity and Compressive Sensing for Reduced Order Modeling.- 20 A HJB-POD approach to the control of the level set equation.- 21 Model order reduction approaches for infinite horizon optimal control problems via the HJB equation.- 22 Interpolatory methods for H model reduction of multi-input/multi-output systems.- 23 Model reduction of linear time-varying systems with applications for moving loads.- 24 Interpolation Strategy for BT-based Parametric MOR of Gas Pipeline-Networks.- 25 Energy stable model order reduction for the  Allen-Cahn equation.- 26 MOR-based Uncertainty Quantification in Transcranial Magnetic Stimulation.- 27 Model Order Reduction of Nonlinear Eddy Current Problems using Missing Point Estimation.- 28 On Efficient Approaches for Solving a Cake Filtration Model under Parameter Variation.- 29 Model reduction for coupled near-well and reservoir models using multiple space-time discretizations.- 30 Time-dependent Parametric Model Order Reduction for Material Removal Simulations