General Information about the course

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

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

This course is held by André Massing, room 1340 SB2, andre.massing@ntnu.no, jointly with Dietmar Hömberg, dietmar [dot] homberg [at] ntnu [dot] no

The detailed timeplan can be found on the official course description page.

The first lecture will take place on Tuesday, 10th of January at 12:15 in R91 Realfagbygget. There will be no exercise session in the first week.

This course also includes a project, where you will be asked to analyze both theoretically and numerically some optimal control problem arising from real world applications. Detailed information will follow shortly and be published on the Project wiki page

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, we can recommend the following books:

• A. Manzoni, A. Quarteroni, S. Salsa, Optimal Control of Partial Differential Equations: Analysis, Approximation, and Applications, Springer, 2021. Individual chapters are available at Springer link.
• 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

The book by Manzoni, Quarteroni and Salsa provides a very nice balanced approach to theory, algorithms and applications. 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.
• Also, Brezis has written a nice book covering the most important aspects from functional analysis, Sobolev spaces, and PDEs: H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer Springer Link

Throughout this course we will use the Julia-based finite element package Gridap.jl to showcase the techniques we learn in this course. Gridap.jl provides a very high-level user interface to implement and solve Partial Differential Equations (PDEs). The Gridap developers have also written a number of nice tutorials on how to use Gridap.jl, and we will at suitable time use run a few tutorial sessions to get acquainted with the package.

To install Julia and Gridap locally on your machine, please take at look at download and the platform-specific installation instructions at https://julialang.org if you want to have a local copy on your computer. You can use the Jupyter environment from your Anaconda installation with Julia, see this tutorial.

There will be an oral exam at the end of the course; the dates for the exam are will be announced towards the end of the semester.