Model Reduction for Dynamical Systems (Summer Semester 2023)

This course deals with Model Order Reduction (MOR) for the efficient simulation, analysis, control, and optimization of large-scale dynamical systems. A large variety of MOR methods will be introduced and discussed, including both frequency-domain and time-domain methods. MOR methods applied to real-world problems will be presented in order to highlight the necessity and usefulness of model reduction.

Contents -  Schedule -  Certification -  Lectures -  Exercises -  Homework -  References

Contents

1. Basics of Linear Systems and Control Theory
   • Singular Value Decomposition
   • Input and Output Dynamical Systems
   • Frequency Domain Behavior and Transfer Functions
   • Gramians and Energy Functionals
2. System-theoretic Model Reduction
   • Balanced Truncation
   • Interpolation-based Model Reduction
3. Model Reduction of Nonlinear Systems
   • Proper Orthogonal Decomposition (POD)
   • Discrete Empirical Interpolation Method (DEIM)
4. Model Reduction of Parametric Systems
   • Multi-moment Matching
   • Reduced Basis (RB) Method
5. Data-Driven Methods for Dynamical Systems
   • Eigenvalue Realization Algorithm (ERA)
   • Loewner Framework
   • Dynamic Mode Decomposition
   • Operator Inference

The content of this lecture is summarized in the follwoing slides.

 

Schedule

Lecture:		Tuesdays 17:00 - 19:00 and Thursdays 13:00 - 15:00 (The dates will be discussed during the first week a might be changed according to the availability of the students) in room G02-20
Exercise:		Thursday 13:00 - 15:00 (every even week)

Certificate for successful participation

Will be announced in the lecture

Lectures

• Week 1 (11/04 and 13/04): Slides (Introduction to Model Reduction and SVD)
• Week 2 (18/04): Slides (System theory I)
• Week 3 (25/04 and 27/04): Slides (System theory II)
• Week 4 (02/05): Slides (Modal Truncation)  
• Week 5 (09/05, 11/05 and 16/05): Slides (Balanced Truncation)
• Week 6 (25/05): Slides Playlist with Videos (End of Balanced truncation and start of Interpolation based MOR)

Videos

• Singular Value Decomposition (SVD)
• Linear Dynamical Systems I: Matrix exponential, analytical solution and frequency domain representation
• Linear Dynamical Systems II: Reachability, observability, minimal realizations and system norms
• Modal Truncation
• Balanced Truncation
• Complementary Topics on Balanced Truncation (Not mandatory)
• Model Reduction via Rational Interpolation

Exercises

• Exercise 1 (Hand in on 04/05)
• Exercise 2  (Hand in on 23/05) 
• Exercise 3 (Hand in on 01/06)

Homework

Supplementary Material

• Matlab intro file

Recomended Literature

 

• A.C. Antoulas: Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, 2005.
• A. C. Athanasios, C. A. Beattie, S. Gugercin: Interpolatory Methods for Model Reduction, SIAM, 2020.
• P. Benner, M. Ohlberger, A. Cohen, K. Willcox: Model Reduction and Approximation: Theory and Algorithms, SIAM, 2017.
• Lecture notes from Dr. Matthias Voigt