Vorlesung im Sommersemester 2014:

V4E2 – Numerical Simulation

Prof. Dr. Carsten Burstedde

Assistent: Philipp Morgenstern

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 ill-posed: 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 non-negative 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.

• Exercise Sheet 1
• Exercise Sheet 2
• Exercise Sheet 3
• Exercise Sheet 4
• Exercise Sheet 5
• Exercise Sheet 6
• Exercise Sheet 7
• Exercise Sheet 8
• Exercise Sheet 9
• Exercise Sheet 10

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., Large-scale 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.