Model Reduction and Approximation

Many physical, chemical, biomedical, and technical processes can be described by partial differential equations or dynamical systems. In spite of increasing computational capacities, many problems are of such high complexity that they are solvable only with severe simplifications, and the design of efficient numerical schemes remains a central research challenge. This book presents a tutorial introduction to recent developments in mathematical methods for model reduction and approximation of complex systems.

Model Reduction and Approximation: Theory and Algorithms contains three parts that cover (I) sampling-based methods, such as the reduced basis method and proper orthogonal decomposition, (II) approximation of high-dimensional problems by low-rank tensor techniques, and (III) system-theoretic methods, such as balanced truncation, interpolatory methods, and the Loewner framework. The book is tutorial in nature, giving an accessible introduction to state-of-the-art model reduction and approximation methods; it covers a wide range of methods drawn from typically distinct communities (sampling based, tensor based, system-theoretic).

Audience: This book is intended for researchers interested in model reduction and approximation, particularly graduate students and young researchers.

Contents: Part I: Sampling-Based Methods; Chapter 1: Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control; Chapter 2: A Tutorial on Reduced Basis Methods; Chapter 3: The Theoretical Foundation of Reduced Basis Methods; Part II: Tensor-Based Methods; Chapter 4: Low-Rank Methods for High-Dimensional Approximation and Model Order Reduction; Chapter 5: Model Reduction for High-Dimensional Parametric Problems by Tensor Techniques; Part III: System-Theoretic Methods; Chapter 6: Model Order Reduction Based on Systems Balancing; Chapter 7: Model Reduction by Rational Interpolation; Chapter 8: The Loewner Framework for Model Reduction; Chapter 9: Comparison of Methods for Parametric Model Order Reduction.

Peter Benner is director at the Max Planck Institute for Dynamics of Complex Technical Systems and head of the Computational Methods in Systems and Control Theory department. He is also a professor at TU Chemnitz and adjunct professor at Otto-von-Guericke University Magdeburg, and he is a member of the Research Center Dynamic Systems: Systems Engineering in Magdeburg. He serves on the editorial board of several scientific journals, including SIAM Journal on Matrix Analysis and Applications.

Mario Ohlberger is a full professor of applied mathematics and managing director of Applied Mathematics: Institute of Analysis and Numerics at the University of Munster. He is 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.

Albert Cohen is a professor at Laboratoire Jacques Louis Lions, Universite Pierre et Marie Curie, Paris, France. He was awarded the Vasil Popov Prize (1995), the Jacques Herbrant Prize (2000), and the Blaise Pascal Prize (2004), and he has been the PI of the ERC Advanced Grant BREAD since 2014. He has been an invited speaker at ICM 2002 (Numerical Analysis section) and plenary speaker at ICIAM 2007. He is the managing editor of Foundations of Computational Mathematics. He has been a senior member of Institut Universitaire de France since 2013.

Karen E. Willcox is Professor of Aeronautics and Astronautics at the Massachusetts Institute of Technology and Co-Director of the MIT Center for Computational Engineering. Prior to joining the faculty at MIT, she worked at Boeing Phantom Works with the Blended-Wing-Body aircraft design group. She has served in multiple leadership positions within AIAA and SIAM, including on the SIAM Activity Group on Computational Science and Engineering. She is Section Editor of SIAM Journal on Scientific Computing and Associate Editor of AIAA Journal.