Modellreduktionsverfahren / Model Order Reduction Techniques in SS 2019
|Vorlesung||Tuesday 8:30 - 10:00||Hauptgebäude 1010|224.3||Start: 03.04.2019|
|Friday 8:30 - 10:00||Hauptgebäude 1010|224.3||Start: 05.04.2019|
|Übung||Thursday 8:30 - 10:00||Hauptgebäude 1010|224.3||tbd|
First meeting: Wednesday, April 03.04.2019, at 10:30am in room 224.3 (main building).
Time of lectures to be discussed during the first lecture.
- The course is split into two parts, Model Order Reduction Techniques 1 and 2, each with 2SWS lecture and 1 SWS exercises. Each part accounts for 5 credit points (ECTS).
- The first part will be held during the first half of the term, the second part during the second half. Students can take only part 1 (5 ECTS) or both parts (10 ECTS).
- The problem sets will require some programming in Matlab.
The focus of this course is on model order reduction methods for large-scale systems. These systems typically arise when partial differential equations are solved using numerical methods such as the finite element method. Model order reduction techniques provide an efficient and reliable way of solving these problems in the many-query or real-time context, such as optimization, characterization, and control.
We first present reduced basis approximations and associated a posteriori error estimation procedures for rapid and reliable solution of parametrized partial differential equations (PDEs). The focus is on rapidly convergent Galerkin approximations on a space spanned by "snapshots" on the parametrically induced solution manifold; rigorous and sharp a posteriori error estimators for the outputs/quantities of interest; efficient selection of quasi-optimal samples in general parameter domains; and Offline-Online computational procedures for rapid calculation in the many-query and real-time contexts. We subsequently consider Proper Orthogonal Decomposition (POD) and Balanced Truncation for the approximation of systems which only depend on time.
Lecture notes and additional literature
The notes and slides from the lecture are the primary source for the class. The following references cover some of the topics or provide further reading:
- A.T. Patera and G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. Version 1.0, Copyright MIT 2006-2007, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering.
- A.C. Antoulas, Approximation of Large-Scale Dynamical Systems (Advances in Design and Control). SIAM, 2005.
- Additional reference for the general preliminaries: lecture notes on FEM from the class 16.920 Numerical Methods for PDEs . (links to Notes 1, Notes 2, Notes 3, Notes 4)
- Additional reference for SVD and POD: lecture notes by Prof. S. Volkwein, Uni Konstanz (link to Script)
- Further texts may be added during the term.
Lecture slides and notes (will be posted before each lecture)
- Lecture 1 (03.04.2019): Motivation and Introduction. (slides).
- Lecture 2 (09.04.2019): slides/notes of Lecture 1.
- Lecture 3 (11.04.2019): General Preliminaries, Linear Elliptic Problems (Definitions, Truth Approximation, Reduced Basis Approximation). (slides or article).
- Lecture 4 (12.04.2019): slides/notes of Lecture 2.
- Lecture 5 (30.04.2019): slides/notes of Lecture 2.
- Lecture 6 (02.05.2019): A Posteriori Error Estimation for Linear Elliptic Problems. (slides or article).
- Lecture 7 (03.05.2019): Convergence of RB Approximations. (slides or article).
- Lecture 8 (07.05.2019): Noncompliant Linear Elliptic Problems (slides or article).
- Lecture 9 (10.05.2019): slides/notes of Lecture 8. Coercivity Lower Bounds (slides: color or article).
- Lecture 10 (14.05.2019): Linear Parabolic Problems (slides or article).
- Lecture 11 (17.05.2019): slides/notes of Lecture 10.
- Lecture 12 (21.05.2019): slides/notes of Lecture 10.
- Lecture 13 (24.05.2019): slides/notes of Lecture 10.
- Lecture 14 (28.05.2019): slides/notes of Lecture 10. Linear Parabolic Problems, additional slides/notes on nonsymmetric parabolic problems (slides or article).
- Lecture 15 (31.05.2019): Nonaffine Problems, Empirical Interpolation Method (slides: color, or article).
- Lecture 16 (04.06.2019): slides/notes of Lecture 15.
- Lecture 17 (07.06.2019): slides/notes of Lecture 15.
- Lecture 18 (18.06.2019): slides/notes of Lecture 15, geometry parametrization (see notes from class).
- Lecture 19 (21.06.2019): ---
- Lecture 20 (25.06.2019): Parameter Estimation and Optimal Control (slides: color).
- Lecture 21 (28.06.2019): Parameter Estimation and Optimal Control
- Lecture 22 (02.07.2019): Noncoercive and saddle point problems (see notes from class).
- Lecture 23 (05.07.2019): ---
- Lecture 24 (09.07.2019): Noncoercive and saddle point problems (see notes from class).
- Lecture 25 (12.07.2019): ---