Department of Information Technology

Numerical Functional Analysis (7.5hp)

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This course is given during the spring 2019.

Schedule

Seminar Date Time Place Hand in/Prepare Key concepts
S1 0503 13--15 2344 Read Chap. 1 (Metric spaces) Continuity, Separability, Convergence
S2 0510 13--15 2344 Inlupp #1, Read Chap. 2 (Banach spaces) Completion, Compactness, Bounded linear operators
S3 0524 13--15 2344 Inlupp #2, Read Chap. 3 (Hilbert spaces) Orthogonal and Riesz's representations, Adjoints
S4 0604 13--15 2344 Inlupp #3, Read Chap. 4+5+6 (Theorems and Applications), group presentations "Big" Theorems and Applications
S5 0610 13--15 2344 Group presentations "Big" Theorems and Applications
Deadline 0819 Draft mini-essay (2--6 pages) ready/sent to 2 referees (With CC to Stefan Engblom)
Deadline 0826 2 reviews (1/2--1 A4-page), sent to 2 authors (With CC to Stefan Engblom)
Deadline 0902 Final version of mini-essay ready

Notes: Date "0604" means "June 4th". Room 2344 means "house 2, floor 3, room (23)44". The seminars are scheduled for 2+epsilon hours.

The Student's book is available here: NFAStudentBook.pdf. Feel free to let me know what you think.

5hp or 7.5hp: To get 5hp you should hand in and pass inlupp #1--#3, you should participate in presenting one "Big" Theorem, and you must be present at least on 4 of the 5 seminars S1 through S5. Any possible other arrangements must be agreed upon beforehand. To get 7.5hp you also should participate in the mini-essay writing and reviewing.

Instructions regarding the mini-essay: The final part of the course is a mini-essay where you write in a free format approximately 2--6 pages (not counting the title page) on a topic of your own choice. Let me know at your earliest convenience what topic you have chosen; I prefer that all students write about a subject of their own, but some minor collisions are acceptable. Use the distributed LaTeX-template.

Author Topic Reviewers
Vidar Stiernström Adjoint-based a posteriori error estimation Ivo Dravins, Fredrik Laurén
Ivo Dravins Optimal control problems with state constraints for systems governed by differential inclusions Vidar Stiernström, Fredrik Laurén
Fredrik Laurén The universal approximation theorem Vidar Stiernström, Ivo Dravins
Robin Eriksson Bayesian inversion of the model problem Joar Bagge, Anton Artemov
Samuel Bronstein About the Fredholm Alternative Fredrik Fryklund, Anton Artemov
Joar Bagge Integral equations for Stokes flow Robin Eriksson, Samuel Bronstein
Fredrik Fryklund Condition number for compact operator equations Robin Eriksson, Samuel Bronstein
Anton Artemov The Schauder-Tikhonov theorem and its applications Joar Bagge, Fredrik Fryklund

When reviewing, your sole task is to suggest changes that improve the mini-essay. Say what your impressions were, what you appreciated, what you missed, ...

About the written mandatory assignments: Work together if you like, but hand in your own solutions. Practise a formal style, clarity, and non-ambiguous constructs. Hand in your solutions (on paper) no later than at the communicated deadline. If you cannot make it I prefer that you hand in whatever material you have at that occasion, let me correct it, and then hand in a final version at a later occasion. Note: hand-written solutions are NOT accepted.

Prepare the lectures thoroughly by reading what has been indicated, and by attempting the exercises. Bring sketches of your own solutions and be prepared to discuss and explain them to others. Your active participation is of vital importance for the quality of each meeting. Please read these instructions once more. Thanks!

Fourth and Fifth meeting

Before the meetings: the material for these two meetings are found in Chap. 4+5+6, so read through these and attempt some exercises of your own choice. Instead of an inlupp #4 you will participate in a group presentation and present a result according to the table below.

Result Group Presents
Hahn-Banach theorem Anton Artemov,Robin Eriksson 190610
Uniform boundedness (Banach-Steinhaus) theorem Joar Bagge,Fredrik Fryklund 190604
(Banach) Open mapping theorem Vidar Stiernström,Igor Tominec,Fredrik Laurén 190610
(Banach) Closed graph theorem Isaac Enyogoi, Samuel Bronstein 190610
 Banach fixed point theorem Ivo Dravins,Ylva Rydin 190604

It is mandatory to participate in a group presentation. If you cannot come to one of the meetings, take the responsibility and make sure that your whole group can show up at the same meeting. Or, take the responsibility and change group with someone!

The task is to present "your" theorem in a way you would have liked to hear about it. What is the result? What is the required background? Are there any interesting historical details? What are the key concepts of the proof? Solve some of the exercises in the book to show your fellow students how the result is used. What are the main applications? Ensure that all students in the group take a definite responsibility for parts of the presentation.

Practically, the presentation time limit is 20 minutes and this will be strictly enforced. Additionally, 5 minutes or so of questions are welcome. In addition to a serious attempt at answering to these matters and to these constraints, the group is also required to mail me 3 (preferably, but not less than 2) suitable True/False quiz questions on the material (the resulting "global" quiz will be assembled and given at the fifth meeting). Deadline 190607 @ 2400hrs.

Third meeting

Material:
Fast Quiz: No 3
Slow Quiz: No 3

Before the meeting: (as before) Read Chap. 3 and try to solve the exercises (finishing about half of them before class should be reasonable). It is recommended to try as many exercises as you can. The following is a selection which connects in various ways to Numerical Analysis (all from Chap. 3):

  • 3.1.5, 3.1.7, 3.1.8, 3.1.11, 3.1.14
  • 3.2.3, 3.2.5, 3.2.9
  • 3.3.1, 3.3.2, 3.3.3, !3.3.4!, 3.3.9, 3.3.10
  • 3.4.3, 3.4.4, 3.4.5, !3.4.6!, 3.4.7, 3.4.8, 3.4.10
  • 3.5.1, 3.5.3, 3.5.4, 3.5.5, 3.5.6, 3.5.7, 3.5.8
  • 3.6.4, 3.6.6, 3.6.8, 3.6.10
  • 3.7.2, 3.7.8, 3.7.9
  • 3.8.5, 3.8.7, 3.8.10, 3.8.11, 3.8.14, 3.8.15
  • 3.9.2, 3.9.3, 3.9.7, 3.10.10, 3.10.11, 3.10.12, 3.10.13, 3.10.14, 3.10.15

Written assignment: hand in solutions to at most one of the exercises in boldface in each group of the list. The exercises marked with an exclamation is strongly recommended to at least try and I will be particularly happy if you choose to hand in those! The minimum total is in any case 6 exercises (of course you are welcome to bring in even more than that if you like!). Hand in your solutions at the fourth meeting.

Second meeting

Material:
Fast Quiz: No 2
Slow Quiz: No 2

Before the meeting: (as before) Read Chap. 2 and try to solve the exercises (finishing about half of them before class should be reasonable). It is recommended to try as many exercises as you can. The following is a selection which connects in various ways to Numerical Analysis (all from Chap. 2):

  • 2.1.5, 2.1.6, 2.1.9
  • 2.2.11, 2.2.13, 2.2.14
  • 2.3.2, 2.3.3, 2.3.4, 2.3.7, 2.3.8, !2.3.10!, 2.3.11, 2.3.12
  • 2.4.3, !2.4.4!, 2.4.5, 2.4.6, 2.4.8, 2.4.9
  • 2.5.4, 2.5.7, 2.5.8, !2.5.10!
  • 2.6.3, 2.6.4, 2.6.5, 2.6.7, 2.6.10, 2.6.12, 2.6.13, 2.6.15
  • 2.7.1, 2.7.2, 2.7.7, 2.7.8, 2.7.9, 2.7.10, 2.7.11, 2.7.12
  • 2.8.2, 2.8.4, !2.8.6!, 2.8.8
  • 2.9.9, 2.9.12, 2.9.13
  • 2.10.4, 2.10.6, 2.10.8, 2.10.9

Written assignment: hand in solutions to at least one of the exercises in boldface in each group of the list. The exercises marked with an exclamation is strongly recommended to at least try and I will be particularly happy if you choose to hand in those! The minimum total is in any case 6 exercises. Hand in your solutions at the third meeting.

First meeting

Material:
Task solved in groups: The Metric Lax.
Fast Quiz: No 1
Slow Quiz: No 1

Before the meeting: Read Chap. 1 and try to solve the exercises (finishing about half of them before class should be reasonable). It is recommended to try as many exercises as you can. The following is a selection which connects in various ways to Numerical Analysis (all from Chap. 1):

  • 1.1.6, 1.1.7, 1.1.8, 1.1.9
  • 1.2.4, 1.2.5, 1.2.11, 1.2.13, 1.2.14, 1.2.15
  • 1.3.3, 1.3.8, 1.3.12
  • 1.4.2, 1.4.8
  • 1.5.6, 1.5.8, 1.5.9, 1.5.15
  • 1.6.6, 1.6.13, 1.6.14

Written assignment: hand in solutions to at least one of the exercises in boldface in each group of the list (= each section of Chap. 1). The minimum total is thus 6 exercises. Hand in your solutions at the second meeting.

Overview

This course is an introduction to Functional Analysis with a particular emphasis on constructs and results that connect in various ways to Numerical Analysis. Hence the name Numerical Functional Analysis!

  • Reading course with 5 seminars in the form of theory expositions and discussions
  • 3 written assignments and 1 group presentation
  • 1 a bit more "in-depth" written assignment, (come up with your own suggestion if you like), written and corrected/reviewed by others (you write one, and you give feedback to 2)

This is the second time this course is given. Your continuous feedback will be very much appreciated. Thank you!

Book: Kreyzig, Introductory functional analysis with applications.

Contents: metric, normed, and inner product spaces, completeness,
Banach/Hilbert spaces, bases, strong/weak convergence, open mapping
theorem, Banach fixed point theorem, formal error analysis, stability, (...)

Input obtained before the course

Some students have a substantial experience in Numerical Analysis, some in Functional Analysis. None have both. Please appreciate that the course is trying to strike a good balance!

Hopes

  • Better grasp of the theoretical foundations (such as spaces) of numerics we apply in solving PDE's.
  • Lära mig funktionalanalys!
  • I would like to see how the theoretical functional analysis stuff applies itself to numerical analysis
  • That the gaps in my knowledge resulting from my patch-work introduction to the subject be filled.
  • Få bättre förståelse och nya verktyg att analysera framför allt PDE:er. Bredda mina kunskaper inom matematik.
  • Jag hoppas få en bra översikt i funktionalanalys, och gärna fördjupa mig specifikt i sådant som rör integralekvationer av Fredholm-typ.
  • I would like to open a random FEM paper and understand the spaces.
  • Att inte behöva bli svettig när det pratas Sobolevrum (och annat funktionalanalysrelaterat inom numerik).
  • get my hands dirty with FA
  • Att få intuition kring funktionsrum och operatorer. Att förstå hur jag kan tolka klassiska resultat med utgångspunkt från numerisk analys. Trots att jag läst funktionalanalys minns jag lite av den eftersom jag aldrig använder den. Jag hoppas på att bli inspirerad...
  • ...there will be some useful topics. I mean useful in my applications.
  • My biggest hope with this course is that I develop a good understanding of functional analysis which will be a gateway to developing a good understanding of many other areas of pure mathematics, not to mention the area of PDEs which is so deeply connected to numerical analysis.

Fears

  • Att jag har för dåliga förkunskaper
  • Den största utmaningen är kanske att det kommer vara mycket jobb med kursen under de veckor den pågår...
  • My biggest fear is that I might discover the topic to be too difficult, which would mean it might take time from my research activities.
  • Hinna med att ta sig igenom stoffet/inlämningarna till varje seminarie.
  • That I finish it without the necessary skills I will need in my work as a Phd student.
  • Do not know or perhaps I try not to think about this.
  • Att vi fokuserar på patologiska exempel... Jag är även orolig att vi ska ha finita elementmetoden som utgångspunkt för numeriska metoder. Enligt min erfarenhet innebär det en djupdykning i Sobolevrum.
  • no fears.
  • Functional analysis can be quite raw. I am hoping for some intuitive explanations.
  • steep learning curve
  • Tycker att funktionalanalys känns mystiskt och svårgenomträngligt. Så är väl lite rädd för att det kommer vara mystiskt och svårt att förstå.

Responsible: Stefan Engblom

Updated  2019-09-06 15:42:32 by Stefan Engblom.