# V4E2 – Numerical Simulation

### Prof. Dr. Carsten Burstedde

#### 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.

#### 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.