Vorlesung im Sommersemester 2014:
V4E2 – Numerical Simulation
Requirements: Wissenschaftliches Rechnen I (V3E1/F4E1).
We will discuss optimization and inverse problems with PDEs. If we define the forward problem by computing the solution of a PDE given its right hand side and system coefficients, optimization and inverse problems refer to computing right hand sides or coefficients given a certain target solution. Such problems are particularly challenging for the following reasons:
1. 
Even for a linear forward problem, the inverse problem can be (strongly) nonlinear.

2. 
The inverse problem may be illposed: its solution may not depend continuously on the
input data, and there may be none or multiple exact solutions.

3. 
We may want to restrict the unknown coefficients to certain subspaces (for example, they
should be nonnegative if they enter the bilinear form of an elliptic operator).

4. 
Computationally, we solve a minimization problem where the PDE is a constraint. We
will introduce Lagrange multipliers in certain function spaces. Each step of the inverse
solver involves one or more solves of the PDE.

Date & time
Lectures:  Tue,  10:15–11:45 am,  Wegelerstraße 6, Room 6.020 
 Thu,  8:30–10:00 am,  Wegelerstraße 6, Room 6.020 
First lecture:  Tue,  8.4.2014 
Tutorial:  Wed,  4:30–6:00 pm,  Wegelerstraße 6, Room 6.020 
Exercise sheets
Exercises are handed out on tuesdays, and are to be handed in one week later.
Requirements for the exam
Students need to achieve 50% of all points, separately for theory and
programming exercises.
Date of the examinations will be 21.–23.7.
Literature
[1]  L. Biegler, G. Biros, O. Ghattas, Y. Marzouk, M. Heinkenschloss, D. Keyes,
B. Mallick, , L. Tenorio, B. van Bloemen Waanders, and K. Willcox, eds.,
Largescale Inverse Problems and Quantification of Uncertainty, Wiley, 2011.

[2] 
A. Borzì and V. Schulz, Computational Optimization of Systems Governed by Partial
Differential Equations, SIAM, 2012.

[3] 
F. Troltzsch, Optimale Steuerung partieller Differentialgleichungen, Vieweg, Wiesbaden,
Germany, 2005.

[4] 
F. Troltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and
Applications, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society,
2010.
