Information about the lecture

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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 optimisation problems in spaces of functions.

In this course you will learn the theory pertinent for analysing optimisation 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 optimisation theory is beneficial. Depending on the background of the students attending the class, we will spend the first few weeks with an overview of these topics, though.

The lecture is held by Markus Grasmair, room 1052 SB2, markus.grasmair@ntnu.no, jointly with Dietmar Hömberg, hoemberg [at] wias [dash] berlin [dot] de.

• Tuesday, 10:15–12:00, B2 (Berg).
• Friday, 10:15–12:00, F4 (Gamle Fysikk).

The first lecture will take place on Tuesday, January 12.

The lectures will be held on campus with live streaming via zoom on campus with live streaming via zoom on campus with live streaming via zoom.

• Thursday, 16:15-18:00, room F4 (Gamle Fysikk) in zoom room F4 (Gamle Fysikk) in zoom.

The exercises start in calendar week 4, that is, on Thursday, January 27. Note the change in exercise times.

This course also includes a project, where you will be asked to analyse both theoretically and numerically some optimal control problem arising from real world applications. The project will be organised by Dietmar Hömberg; calendar weeks 8-10 are (tentatively) reserved for work on the project.

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 being quite a bit easier to follow. 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; the dates for the exam are May 12 and June 2.