[TMA4183 2017v] Optimization II: Optimal Control of PDEs

Messages

  • 14.06: Her er visualisering av iterasjoner av optimeringsalgoritmen (projected gradient) på prosjektoppgaven (distribuert kontroll av naturlig konveksjon):
  • 11.05: Her er noe informasjon om den muntlige eksamen - det er en liste med de viktigste konsepter som vi har diskutert. Jeg sender mer informasjon om timeplan snart.
  • 25.04: Jeg vil bare legge informasjonen om StudForsk program her.
  • 28.03:
    • Jeg er tilgjengelig på kontoret i dag hvis dere har spørsmål til prosjektet
    • Jeg har også laget en Google Group her. Den er public, så vi får å se, hvor mye spam vi får. Ellers bruk den vennligst til å stille spørsmål/diskutere prosjektet og alt annet relatert til emnet. Jeg skal prøve å svare hurtigst mulig.
  • 20.03: Prosjekt er online nå.
  • 22.02: Jeg har reservert Nullrommet til i morgen kl 14-16. Vi møtes der da.
  • 17.02: Nullrommet is booked Fri 12-14 for TMA4300. We can try Banachrommet, or simply BYO laptop to B2.
  • 16.02: Hvis Nullrommet ikke er opptatt i morgen kan vi møtes der; ellers blir jeg i B2. Spør gjerne hvis dere trenger mer hjelp med koden eller hva som helst relatert til emnet.
  • 14.02:
    • På datamaskiner i Nullrommet har man FENICS og Paraview installert. Der kan man bare kjøre Python/FENICS kode vha "python filename.py" i terminalen, og visualisere vha "paraview filename.pvd"). Husk å trykke på "apply" i Paraview!
    • FENICS i Nullrommet har versjon 2016.2, som er nyere enn den man får med Dokker (så vidt jeg vet i hver fall). PGA det "MixedFunctionSpace" som brukes i laplace05.py er ikke lengere med; istedenfor man må bruke "MixedFiniteElement", se det nye eksempel "laplace05_new.py"
  • 08.02: I will use Python interface to FENICs for implementing numerical methods. Download FENICs here. You may want to install Docker first. I will also use Paraview to visualize the solutions etc.
  • 31.01: Here (and here is a PDF) is an article about the optimization of a stealth bomber using only the first order optimality conditions.
  • 25.01: Here is a very brief summary of the main results we have discussed so far, with pointers to the book. I also wrote down a couple of proofs. I omit all the examples. Please let me know (email?) if something looks unfamiliar, needs to be elaborated, etc.
  • 17.01:
    • Note the change of rooms: Tue: R20→S23; Thu: R91→Kjl22; Fri: R41→B2
    • I shall keep office hours on Fri 14-15 (not 20.01 though). Otherwise send me an email, I try to reply them regularly.
  • 09.01: Happy 2017 and welcome to the course!

Synopsis

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.

Learning objectives

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

Reading material

The 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
  • J.C. De los Reyes, Numerical PDE-Constrained Optimization, 2015. Available online via Springer link
  • 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
  • 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
  • L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, v. 19, 2010. Amazon AMS

Schedule

MoTuWeThFr
08:15-09:00     
09:15-10:00     
10:15-11:00     
11:15-12:00     
12:15-13:00 Lecture (S23)  Exercises (B2)
13:15-14:00    
14:15-15:00   Lecture (KJL22) 
15:15-16:00    
16:15-17:00     

Lecture plan and log

[Tr]=Tröltzsch

Date Topic Reference
21.03 Presentation of the project.
16.03 Application: least squares FEM analysis article and computations article. Some FENICS examples laplace_dpg.py laplace_dpg_cg.py
14.03 Application of formal Lagrange method in the semi-linear case, adjoint problem (see example here laplace04nl.py). Second order derivatives and second-order sufficient conditions. 4.7, 4.9, 4.10.1, first example in 4.10.2
09.03 Necessary optimality conditions 4.5, 4.6
07.03 Existence of optimal controls 4.4
02.03 Dropping the boundedness assumption on the non-linearity (sketch). Nemytski operators 4.2.3, 4.3
28.02 Optimal control of semi-linear elliptic PDEs 4.1, 4.2
21.02 Higher order regularity 2.14, 2.15
16.02 Work on numerical exercises
14.02 Work on numerical exercises
09.02 Discretize or optimize first? Numerical methods: Conditional gradient method. Introduction to implementations in FENICs. laplace00.py, laplace01.py, laplace02.py, laplace03.py, laplace04.py, laplace05_new.py, laplace06.py Section 2.12 [Tr]
07.02 Special cases of first order necessary conditions. The formal Lagrangian method. Sections 2.8, 2.10 [Tr]
02.02 Differentiability, adjoint operators. First order necessary optimality conditions Sections 2.6-2.8 [Tr]
31.01 Existence of optimal controls. Differentiability in Banach spaces Sections 2.5-2.6 [Tr]
26.01 Weak solutions to elliptic PDEs. Existence of optimal controls Section 2.3, 2.5 [Tr]
14.01 Regular domains, Sobolev spaces. Weak solutions to elliptic PDEs Section 2.2-2.3 [Tr]
19.01 Weakly sequentially closed and compact sets, weakly sequentially lower semi-continuous functions. Existence of solutions (Weierstrass-type theorem). Section 2.4.2
17.01 Weak convergence in Banach spaces. Proof of Theorem 2.10 Section 2.4.2 [Tr]
12.01 Normed and inner product spaces, Banach and Hilbert spaces. Projection onto a closed and convex set in a Hilbert space. Riesz representation theorem as a consequence of this (exertise set 1). Dual and bidual, reflexive spaces. Section 2.1, 2.4 [Tr]
10.01 Introduction, examples, concepts in finite dimensions Chapter 1 [Tr]

Exercises

Exercises are not compulsory.

Project

Optimal control of forced convection: problem description: convection.pdf and solution of the state equations: convection.py

Deadline: end of semester

Exam information

The exam will test a selection of the Learning objectives. The final grade is based on the course project (10%) and oral exam (90%). The reserved date for the oral exam is 29.05.2017, from 09:00. We will discuss the exam schedule during the semester.

Contact

  • Lectures & exercises: Anton Evgrafov antone [at] math [dot] ntnu [dot] no; office hours: Fri 14:15-15:00
  • Reference group: TBD
  • Please put TMA4183 in the subject line when emailing
2017-06-14, Anton Evgrafov