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Optimization with Partial Differential Equations (4V, 2Ü)

Prof. HerzogWS2013/14

Partial differential equations (PDEs) describe a countless number of phenomena in the natural sciences, such as heat conduction, the propagation of sound and electromagnetic waves the motion of fluids and the behavior of quantum physical particles. Besides the numerical simulation of such processes, one is often interested in their optimization. This includes optimal control problems, parameter identification as well as shape optimization problems, all of which will be treated in this class.

Goals of this class

In this class you will

get to know some basic examples of optimization problems, mainly with elliptic PDEs,
learn about necessary and sufficient optimality conditions (as a starting point for numerical solution schemes),
learn to use numerical methods for the solution of optimal control problems.

This class complements well the following other classes:

Numerical Methods for Partial Differential Equations,
Infinite-Dimensional Optimization,
Nonlinear Optimization,
Functional Analysis,
Analysis of Partial Differential Equations,
Inverse Problems.

This class can serve as a preparatory step towards a thesis in the work group Numerical Mathematics (Partial Differential Equations).

You may also consider the list of all classes for additional information.

This class can serve as a research module in numerical mathematics (medium) or as a research module in optimization (medium).

News

24.10.2013	Aufgrund des Ausfalls der Übung durch den Reformationstag findet am 07.11.2013 von 10:45 Uhr bis 12:15 Uhr im Anschluss an die reguläre Übung eine Zusatzübung im Raum Rh 41/702 statt.
15.10.2013	Für diese Lehrveranstaltung steht ein Vorlesungsskript zur Verfügung.
01.10.2013	Office hours of Roland Herzog are Tuesdays, 10:30 - 11:15 and by appointment.

Dates

Lecture

Additional lecture material

Skript zur Vorlesung (Stand 06.02.2014)
Companion notes An Introduction to Sobolev Spaces (Version of May 21, 2013)
Matlab-Package: Floor Heating (constrained)

Supplementary References

Matlab Tutorials:

We highly recommend to familiarize yourself (by participating in the labs, and by independent studies) with the basics in Matlab. This will be useful not only for this class. Find additional material on Matlab here.

Documentation of the Matlab Toolboxes:

PDE Toolbox Online Documentation, MathWorks
PDE Toolbox User's Guide (pdf-Version), MathWorks
Optimization Toolbox Online Documentation, MathWorks
Optimization Toolbox User's Guide (pdf-Version), MathWorks

Optimal Control Problems:

Fredi Tröltzsch: Optimale Steuerung partieller Differentialgleichungen, Vieweg, 2005 (Neuauflage 2009)
list of errata in the book (1st edition 2005)
Fredi Tröltzsch: Optimal Control of Partial Differential Equations, AMS, 2010
Michael Hinze, Michael Ulbrich, Stefan Ulbrich, Rene Pinnau: Optimization with PDE Constraints, Springer, 2009

Parameter Identification:

Chapter 10 in Jorge Nocedal, Stephen J. Wright: Numerical Optimization, Springer, 2006; is (at the Chemnitz UT) available online here
George A. F. Seber, Alan J. L. Wild, Nonlinear Regression, John Wiley & Sons, 2005
Bangti Jin, Peter Maass, Sparsity Regularization for Parameter Identification Problems, Inverse Problems 28, 123001, 2012

Shape Optimization:

Jan Sokolowski, Jean-Paul Zolesio: Introduction to Shape Optimization, Springer, 1992
Karsten Eppler: On Hadamard Representations of Shape Gradients - A Computational Guide, Preprint SPP1253-079, 2009
Stephan Schmidt, Volker Schulz: Shape Derivatives for General Objective Functions and the Incompressible Navier-Stokes Equations, Control and Cybernetics 39(3), p.677-713, 2010

Theory and Numerical Methods of Finite Dimensional Optimization:

Jorge Nocedal, Stephen J. Wright: Numerical Optimization, Springer, 2006; is (at the Chemnitz UT) available online here
Carl Geiger, Christian Kanzow: Theorie und Numerik restringierter Optimierungsaufgaben, Springer, 2002

In addition, you may want to consider a book about functional analysis, e.g.,

Hans Wilhelm Alt: Lineare Funktionalanalysis, Springer, 2012
Manfred Dobrowolski: Angewandte Funktionalanalysis, Springer, 2010
Dirk Werner: Funktionalanalysis, Springer, 2011

Additional tutorial material

Exercise	Additional material
1. Exercise
2. Exercise	setup_problem.m plot_state.m plot_control.m plot_subdomains.m plot_boundary_data.m pdemodel_Fussbodenheizung.m setup_matrices.m
3. Exercise
4. Exercise	steepest_descent_QP.m cg_QP.m setup_matrices_full.m
5. Exercise	bc_engine.m geometry_engine.m pdebound_engine.m pdemodel_engine.m plot_state.m setup_problem.m
6. Exercise
7. Exercise	Engine_Cooling_Exercise.zip
8. Exercise	Heat_Sources_Exercise.zip
9. Exercise
10. Exercise
11. Exercise	EIT_Exercise.zip
12. Exercise
13. Exercise

Use of Matlab

There are various possibilities to launch Matlab at the computers in the MRZ pool (where the labs are taught):

Windows: Start / Alle Programme / Mathematik / Matlab / Matlab2013b
Linux: On the desktop, in the applications:/Mathe/ folder, click on Mathematical programs / Matlab R2013b
Linux: Enter matlab in a linux shell

The graphical user interface (GUI) of the PDE toolbox can be reached within Matlab by entering pdetool, or via the tab Apps / PDE.

Exam

There will be oral examinations for participants who would like to acquire a "Fachprüfung" (subject examination) or a "Schein mit Note" (certificate with mark). For participants who would like a "Schein ohne Note" (certificate without mark) it is sufficient to hand in at least 10 successfully processed homework problems (out of a total of 14 problem sheets).