TMA4310 Advanced Optimization (Spring 2015): Optimal Control of PDEs

• 16.01: Owing to too few participants this course will be run as a reading course. I will publish the program and exercises as we go. I am available for discussing the exercises on Wed 10-12.
• 13.01: Lectures will be held in R41.
• 06.01: The info-meeting will take place on 07.01, 12:15-13:00, in SBII/734.
• 03.01: Happy 2015 and welcome to the course! Next week (05.01-09.01) we have to meet and agree upon the schedule for the course. I will also tell you a little bit about the course's structure and syllabus. We will begin with the regular lectures/exercises starting from the week of 12.01. I propose the middle of the week: 07.01 at 12:15-13:00, most likely in the meeting room on the 7th floor of SBII. If this time does not suite you, please fill out the following doodle by the end of Monday, 06.01. I will book and communicate the room/place back to you as soon as we agree upon the time.

• 16.01: Exercise 1: introduction and concepts in finite-dimensions.
• 20.01: Reading list and exercises for this week are here. Please let me know if you need copies of Chapter 5 from Evans.
• 26.01: Reading list and exercises for this week are here.
• 03.02: Reading list and exercises for this week are here. From next week on we switch to Tröltzsch.
• 11.02: Reading list and exercises for this week are here.
• 24.02: Reading list and exercises for the last week (apologies - forgot to upload last week) are here.
• 24.02: Reading list and exercises for this week are here. You can download the FVM discretization of the Laplace eqn on polygonal grids here. Simply unzip the archive and run FVMLaplace in Matlab. The script is well commented and is easy to adapt for solving the control problem instead of the PDE. You do not need to bother with the details of the discretization.
• 10.03: Reading list and exercises are here.
• 18.03: Reading list/exercise set #9: link.
• 18.03: Reading list/exercise set #10: link.
• 18.03: Reading list/exercise set #11: link.
• 20.03: Reading list/exercise set #12: link.

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Linear and non-linear partial differential equations (PDEs) constitute one of the most widely used mathematical framework for modelling various physical or technological processes, such as fluid flow, structural deformations, propagation of acoustic and electromagnetic waves among countless other examples. Improvement in such processes therefore require 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 (and the textbook supports this, too) but of course some knowledge of PDEs, functional analysis, and optimization theory is beneficial.

After meeting the learning objectives of the course, the student will be able to:

• analyze control-to-state operators for model control problems
• derive necessary and sufficient optimality conditions for optimal control problems with or without state constraints
• assess existence of solutions to model optimal control problems
• implement optimization algorithms on a computer
• apply optimization algorithms to model problems
• explain the basic properties of the relevant functional spaces, in particular Sobolev spaces

The main textbook is Tröltzsch.

• 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. Amazon Springer
• L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, v. 19, 2010. Amazon AMS

The exam will test a selection of the Learning objectives.

Permitted examination support material: C: Specified, written and handwritten examination support materials are permitted. A specified, simple calculator is permitted. The permitted examination support materials are:

• F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, AMS Graduate Studies in Mathematics, v. 112, 2010.
• Your own lecture notes from the course