1 Model order reduction: basic concepts and notation

This is the first chapter of a three-volume series dedicated to the theory and ap plication of Model Order Reduction (MOR) .We motivate and introduce the basic con- cepts and notation, with reference to the two main cultural approaches to MOR: the system-theoretic approach employing state-space models and transfer functionc con- cepts (Volume1), and the numerical analysis approach as applied to partial differen- tial operators (Volume2) ,for which project ion and approximation in suitable function spaces provide a rich set of tools for MOR.The set two approaches are complementary but share.Despite the sometimes different opted language and notation,they also share the main ideas andkey concepts, which are briefly summarized in this chapter. The material is presented so that all chapters in this three-volume series are put into context, by high lighting the specific problems that they address. An overview of all MOR applications in volume 3 is also provided.