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Model Reduction SoSe 2019

General Information

The modeling of complex physical and technical processes often leads to systems of ordinary differential equations consisting of several million of variables and equations. The numerical simulation, control, and optimization of such systems is then often too expensive or even impossible due to the large requirement of computational resources. The aim of model reduction therefore consists of approximating the large-scale dynamical system by a system with much fewer variables and equations. In particular, we will discuss efficient numerical methods as well as statements concerning the approximation quality such as error bounds. In this course we will discuss the following topics:
  • examples of large-scale dynamical systems
  • basics of systems theory (controllability and observability, Gramians, state-space transformations, transfer functions and the Hardy spaces H2 and H)
  • eigenvalue based reductions (modal truncation)
  • singular value based approaches (balanced truncation)
  • Krylov subspace methods (moment matching, iterative rational Krylov algorithm)
  • data-driven methods (Loewner framework)
  • aspects of structure preservation (optional)
  • extension to DAEs and nonlinear problems (optional)

Lecturer

Schedule

  • Lecture: Wednesday, 12:15-1:45pm in Geom H2 and Thursday, 2:15-3:45pm in Geom H6 (every odd week)
  • Exercise: Thursday, 2:15-3:45pm in Geom H6 (every even week, starting at April 18, 2019)

Recommended Literature

In this course I will follow my lecture notes. You can also find the lecture notes on PaperHive, an online platform for discussing and annotating PDF documents. If you have questions or if something is unclear, you are welcome to leave a note there.

Further literature:
  • A. C. Antoulas. Approximation of Large-Scale Dynamical Systems, volume 6 of Adv. Des. Control. SIAM Publications, Philadelphia, PA, 2005. doi:10.1137/1.9780898718713.
  • K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ, 1996.
  • Slides on model reduction by Peter Benner, covering most aspects of this course.

Exams

Exams will be oral and of about 40 minutes length. They can be taken in English or German (your choice). The exact dates for the exam will be announced later.

Exercises

Every second week there will be an exercise session. It is highly recommended to work on the exercise sheets (and I would like that everybody presents some of his or her solutions on the blackboard).

 
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