Benner P., Cohen A., Ohlberger M., Willcox K. (Eds.) Model Reduction and Approximation: Theory and Algorithms

Philadelphia: SIAM, 2017. — 433 p.Many physical, chemical, biomedical, and technical processes can be described by means of partial differential equations (PDEs) or dynamical systems. If the underlying processes exhibit nonlinear dynamics, analysis and prediction of the complex behavior is often only possible by solving the (partial) differential equations numerically. For this reason, the design of efficient numerical schemes is a central research challenge. In spite of increasing computational capacity, many problems are of such high complexity that they are still only solvable with severe simplifications. In recent years, large-scale problems—often involving multiphysics, multiscale, or stochastic behavior—have become a particular focus of applied mathematical and engineering research. A numerical treatment of such problems is usually very time-consuming and thus requires the development of efficient discretization schemes that are often realized in large parallel computing environments. In addition, these problems often need to be solved repeatedly for many varying parameters, introducing a curse of dimensionality when the solution is also viewed as a function of these parameters. With this book we aim to introduce recent developments on complexity reduction of such problems, both from a theoretical and an algorithmic perspective.
The main purpose of the book resulting from the workshop is to extend the keynote lectures of the workshop to tutorials accessible to developers and users of mathematical methods for model reduction and approximation of complex systems. In addition to the keynote lecturers of the workshop, we also invited other experts to contribute chapters on methods not represented in the keynotes. The book thus contains tutorial-style introductions to several promising emerging fields in model reduction and approximation. It focuses in particular on sampling-based methods (Part I), such as the RB method and POD, approximation of high-dimensional problems by low-rank tensor techniques (Part II), and system-theoretic methods (Part III), such as balanced truncation (BT), interpolatory methods, and the Loewner framework.
Both application-driven aspects and fundamental points of view from approximation theory and information-based complexity are discussed This reveals the great success of the proposed techniques for certain classes of applications but at the same
time also shows their limitations. Real-life problems often pose major challenges that are currently covered by neither the mathematical theory nor the presented methods and thus constitute a driving force for future research.