Information about the lecture

here

Linear and non-linear partial differential equations (PDEs) constitute one of the most widely used mathematical frameworks for modelling various physical or technological processes, such as fluid flow, structural deformations, propagation of acoustic and electromagnetic waves amongst countless other examples. Improvement in such processes therefore requires modelling and solving optimization problems constrained with PDEs, and more generally convex and non-convex optimization problems in spaces of functions.

In this course you will learn the theory pertinent for analysing optimization problems of this type and also fundamental numerical methods for solving these problems. We will mostly concentrate on the optimal control of processes governed with linear and semilinear elliptic PDEs.

We will aim at a reasonably self-contained course but of course some knowledge of PDEs, functional analysis, and optimization theory is beneficial.

The lecture is held by Markus Grasmair, room 1040 SB2, markus.grasmair@math.ntnu.no, jointly with Dietmar Hömberg, hoemberg [at] wias [dash] berlin [dot] de.

According to the results of the doodle poll, the best time for the lecture and exercises is:

• Tuesday, 16:15–18:00, room 734 (Sentralbygg 2, 7th floor).
• Thursday, 16:15–18:00, room 656 (Sentralbygg 2, 6th floor).

The first lecture will be held on Tuesday, January 15.

In some of the lectures we will have to move to a different room - I will inform you about details later.

The exercises will be published here.

When and where:

• Monday, 12:15–14:00, room S22 (Sentralbygg 2).

The first exercise class will be held on Monday, January 21. Afterwards, we will have a class roughly every second week.

We will not follow a specific textbook during this class. However, if you want to study the topic of Optimisation with PDE controls in more details, I can recommend the following books:

• J.C. De los Reyes, Numerical PDE-Constrained Optimization, 2015. Available online via Springer link
• F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, AMS Graduate Studies in Mathematics, v. 112, 2010. Amazon AMS
• M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, v. 23, Springer, 2009. Available online via Springer link

De los Reyes' book is probably the simplest introduction to the topic of optimal control of PDEs, but the theoretical foundations are somehow lacking. Also, the more advanced parts are often somewhat superficial. In contrast, both Tröltzsch and Hinze et al. are much more focussed on a precise and detailed derivation of the theory, with Tröltzsch probably being quite a bit easier to follow (but not that general). Note that there is also a German (original) version of Tröltzsch's book available.

Additional material of interest:

• If you want to refresh (or build up) your knowledge on PDEs, there is the classic book by Evans:
  • L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, v. 19, 2010. Amazon AMS
• The following books contain several articles concerned with theory and applications of PDE constrained optimisation:
  • Eds.: Lorenz T. Biegler, M. Heinkenschloss, O. Ghattas, B. van Bloemen Waanders, Large-Scale PDE-Constrained Optimization, 2003. Available online via Springer link
  • Eds.: L.T. Biegler , O. Ghattas , M. Heinkenschloss , D. Keyes and B. van Bloemen Waanders, Real-Time PDE-Constrained Optimization, 2007. Available online via SIAM

There will be an oral exam at the end of the course, on May 31. Details about the time and place will be published soon.