Index-aware model order reduction methods. Applications to differential-algebraic equations (Q2792600)

This short monograph, which is essentially based on three recent publications of the authors and \textit{C. Tischendorf} [Math. Comput. Model. Dyn. Syst. 20, No. 4, 345--373 (2014; Zbl 1298.93096); SIAM J. Sci. Comput. 35, No. 3, A1487-A1510 (2013; Zbl 1272.78022); Implicit-IMOR method for index-1 and index-2 linear constant DAEs. External Report, CASA Report, No. 14--10. Eindhoven: Technische Universiteit Eindhoven], is devoted to model order reduction methods for differential-algebraic equations (DAEs). In the main focus are linear constant coefficient DAEs, but the methods in this book are extendable to linear time-varying DAEs as well as to nonlinear DAEs. The book is divided into six chapters. The first two chapters aim to introduce DAEs and several real-life applications such as problems arising in electronical network, computational fluid dynamics, and constrained mechanics. The motivation to consider model order reduction for DAEs is also discussed. Chapter 3 deals with the decoupling technique for linear constant DAEs by using the projector-based approach. Matrix pencils associated with the systems are classified by the existence of a finite eigenvalue and the corresponding cases are investigated. A detailed description of the decoupled systems for DAEs up to index 3 is given. Some practical issues such as decoupling of DAEs with special structures and fast construction of projector and basis chains are discussed as well. Chapter 4 contains the main part of the book, where index-aware model order reduction (IMOR) methods are introduced. The main idea is as follows. First, the system is decoupled into a differential subsystem and an algebraic one. Then, well-known model order reduction methods are applied to the differential part and the algebraic part. Finally, by combining the results, reduced order DAEs are obtained. An implicit version of the IMOR method which is called implicit index-aware model reduction (IIMOR) method is also derived, where matrix inversions are avoided. The IIMOR method is cheaper but numerical experiments show that it is less accurate than the IMOR method. Chapter 5 contains numerical experiments illustrating the efficiency and the robustness of the IMOR methods on large scale problems from real-life applications. In the last chapter, some conclusions are given. There are a few typos in the reference list, for example, the author names of \textit{R. Lamour} and \textit{V. Mehrmann} [Differential-algebraic equations. Zürich: EMS (2006)] are not correct and references [\textit{T. Stykel}, Analysis and numerical solution of generalized Lyapunov equations. Berlin: TU Berlin, Fakultät II -- Mathematik und Naturwissenschaften (2002; Zbl 1097.65074)] and [\textit{T. Stykel}, Anaylsis and numerical solution of generalized Lyapunov equations. Berlin: Technical University (PhD Thesis) (2002)] coincide.