## Courses

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## IAM760 - Special Topics: Model Order Reduction

**Credit: **
3(3-0); **ECTS: **
8.0

**Instructor(s): **
Bülent Karasözen / Hamdullah Yücel

**Prerequisites: **
Consent of Instructor(s)

#### Course Catalogue Description

Reduced Order Modeling: proper orthogonal decomposition (POD), evolution problems; Active Subspaces: parametrized models in physics and engineering, discover the active subspaces, exploit the active subspaces, active subspaces in action; Dynamic Mode Decomposition: introduction, Koopman analysis; PDE-constrained optimization: elliptic and parabolic linear optimal control problems; equality and inequality constraints; numerical algorithms for PDE-constrained optimization; reduced order modeling.

#### Course Objectives

The aim of the course is to give the students an introduction to model order reduction techniques for solving problems in the engineering and the sciences described by systems of differential equations. The modeling of complex physical and technical processes often leads to systems of differential equations with several millions of equations and variables. The numerical simulation, real-time and optimal control of such large-scale systems is very time consuming and expensive due to high computing times and high memory requirements, and sometimes impossible. The goal of model order reduction is to approximate highly dimensional systems with systems of smaller dimension. Hereby, crucial physical properties of the system should be preserved in the reduced order model, while simultaneously the approximation error should be small, and the methods should be stable and efficient.

This course is designed for graduate students majoring in mathematics as well as mathematically inclined graduate engineering students. At the end of this course, the student will:

- learn the fundamental concepts of model order reduction techniques such as proper orthogonal decomposition, active subspaces method, and dynamic mode decomposition;
- learn the fundamental concepts of partial differential equation (PDE) constrained optimization;
- learn how to solve large-scale systems efficiently by preserving their crucial physical properties;
- develop proficiency in the applications of model order reduction (modeling, analysis, and interpretation of results) to realistic engineering problems through the use of major commercial general purpose MATLAB codes.

#### Course Learning Outcomes

Upon successful completion of this course, the student will be able to

- formulate and solve (with a computer) partial differential equations efficiently by using various model order reduction methods;
- formulate and solve linear-quadratic optimal control problems;
- solve large-scale systems efficiently by preserving their crucial physical properties;
- apply model order reduction techniques in their thesis and understand the current research in this area;
- tackle real-life applications in science, engineering, and finance using state of art model order reduction techniques.

#### Tentative (Weekly) Outline

- Reduced Order Modeling: proper orthogonal decomposition (POD)
- Reduced Order Modeling: evolution problems
- Active Subspaces: parametrized models in physics and engineering
- Active Subspaces: discover the active subspaces
- Active Subspaces: exploit the active subspaces
- Active Subspaces: active subspaces in action
- Dynamic Mode Decomposition: introduction to DMD
- Dynamic Mode Decomposition: Koopman analysis
- PDE-constrained optimization: motivation and introduction
- PDE-constrained optimization: unconstrained problems
- PDE-constrained optimization: control constraints
- PDE-constrained optimization: boundary control problems
- PDE-constrained optimization: parabolic control problems
- Reduced Order Modeling: optimal control of linear quadratic problems

#### Course Textbook(s)

- P. Benner, M. Ohlberger, A. Cohen and K. Willcox, Model Reduction and Approximation Theory and Algorithms, SIAM 2017.
- P. G. Constantine, Active Subspaces Emerging Ideas for Dimension Reduction in Parameter Studies, SIAM, 2015.
- J. N. Kutz, S. L. Brunton, B. W. Brunton and J. L. Proctor, Dynamic Mode Decomposition: Data- Driven Modeling of Complex Systems, SIAM, 2016.
- A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations, Springer, 2015.
- F. Tröltzsch, Optimal Control of Partial Differential Equations-Theory, Methods and Applications, AMS, 2010.
- S. Volkwein, Proper Orthogonal Decomposition: Theory and Reduced Order Modeling, Uni Konstanz, 2013.

#### Supplementary Materials and Resources

- Resources:
- MATLAB
- htps://odtuclass.metu.edu.tr

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