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.