The idea for this book originated during the workshop "Model order reduction, coupled problems and optimization" held at the Lorentz Center in Leiden from S- tember 19-23, 2005. During one of the discussion sessions, it became clear that a book describing the state of the art in model order reduction, starting from the very basics and containing an overview of all relevant techniques, would be of great use for students, young researchers starting in the ?eld, and experienced researchers. The observation that most of the theory on model order reduction is scattered over many good papers, making it dif?cult to ?nd a good starting point, was supported by most of the participants. Moreover, most of the speakers at the workshop were willing to contribute to the book that is now in front of you. The goal of this book, as de?ned during the discussion sessions at the workshop, is three-fold: ?rst, it should describe the basics of model order reduction. Second, both general and more specialized model order reduction techniques for linear and nonlinear systems should be covered, including the use of several related numerical techniques. Third, the use of model order reduction techniques in practical appli- tions and current research aspects should be discussed. We have organized the book according to these goals. In Part I, the rationale behind model order reduction is explained, and an overview of the most common methods is described.

The fourth international conference on Scientific Computing in Electrical En- gineering (SCEE) was held at the Eindhoven University of Technology, from 23rd to 28th June, 2002. It was sponsored by Philips Research Laborato- ries Eindhoven, the Eindhoven University of Technology, Computer Simula- tion Technology (CST) from Darmstadt, ABB Corporate Research, Thales Netherlands,the European Consortium for Mathematics in Industry (ECMI), the University of Rostock (organiser of SCEE-2000), the European network for Mathematics, Computing and Simulation for Industry (MACSI-net), the Royal Netherlands Academy of Arts and Sciences (KNAW), and the Scien- tific Computing Group of the Eindhoven University of Technology. The Program Committee consisted of: Dr. Alain Bossavit, Electricite de France, Clamart, France. Dr. Uwe Feldmann, Infineon Technologies A.G., Munich, Germany. Prof.Dr. Leszek Demkowicz, University of Texas at Austin, USA. Dr. Michael Gunther, Universitat Karlsruhe, Germany. Prof.Dr. Ulrich Langer, Johannes Kepler Universitat, Linz, Austria. Dr. Jan ter Maten,Philips Research Laboratories Eindhoven, The Nether- lands. Prof.Dr. Ursula van Rienen, Universitat Rostock, Germany. Prof.Dr. Jaijeet Roychowdhury, University of Minnesota, USA. - Prof.Dr. Wil Schilders, Technische Universiteit Eindhoven and Philips Research Laboratories Eindhoven, The Netherlands. - Prof.Dr. Thomas Weiland, Technische Universitat Darmstadt, Germany.

The idea for this book originated during the workshop "Model order reduction, coupled problems and optimization" held at the Lorentz Center in Leiden from S- tember 19-23, 2005. During one of the discussion sessions, it became clear that a book describing the state of the art in model order reduction, starting from the very basics and containing an overview of all relevant techniques, would be of great use for students, young researchers starting in the ?eld, and experienced researchers. The observation that most of the theory on model order reduction is scattered over many good papers, making it dif?cult to ?nd a good starting point, was supported by most of the participants. Moreover, most of the speakers at the workshop were willing to contribute to the book that is now in front of you. The goal of this book, as de?ned during the discussion sessions at the workshop, is three-fold: ?rst, it should describe the basics of model order reduction. Second, both general and more specialized model order reduction techniques for linear and nonlinear systems should be covered, including the use of several related numerical techniques. Third, the use of model order reduction techniques in practical appli- tions and current research aspects should be discussed. We have organized the book according to these goals. In Part I, the rationale behind model order reduction is explained, and an overview of the most common methods is described.

The fourth international conference on Scientific Computing in Electrical En- gineering (SCEE) was held at the Eindhoven University of Technology, from 23rd to 28th June, 2002. It was sponsored by Philips Research Laborato- ries Eindhoven, the Eindhoven University of Technology, Computer Simula- tion Technology (CST) from Darmstadt, ABB Corporate Research, Thales Netherlands,the European Consortium for Mathematics in Industry (ECMI), the University of Rostock (organiser of SCEE-2000), the European network for Mathematics, Computing and Simulation for Industry (MACSI-net), the Royal Netherlands Academy of Arts and Sciences (KNAW), and the Scien- tific Computing Group of the Eindhoven University of Technology. The Program Committee consisted of: Dr. Alain Bossavit, Electricite de France, Clamart, France. Dr. Uwe Feldmann, Infineon Technologies A.G., Munich, Germany. Prof.Dr. Leszek Demkowicz, University of Texas at Austin, USA. Dr. Michael Gunther, Universitat Karlsruhe, Germany. Prof.Dr. Ulrich Langer, Johannes Kepler Universitat, Linz, Austria. Dr. Jan ter Maten,Philips Research Laboratories Eindhoven, The Nether- lands. Prof.Dr. Ursula van Rienen, Universitat Rostock, Germany. Prof.Dr. Jaijeet Roychowdhury, University of Minnesota, USA. - Prof.Dr. Wil Schilders, Technische Universiteit Eindhoven and Philips Research Laboratories Eindhoven, The Netherlands. - Prof.Dr. Thomas Weiland, Technische Universitat Darmstadt, Germany.