Numerical Functional Analysis (7.5hp)
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This course is given during the spring 2022.
Schedule
| Occasion | Date | Time | Place | Hand in/Prepare | Key concepts |
|---|---|---|---|---|---|
| L0 | 0203 | 13--14 (c:a) | Online | Come as you are! | Welcome lecture; overview of key concepts, Why, Who, and What |
| S1 | 0217 | 13--16 | 101130 | Read Chap. 1 (Metric spaces), Inlupp #1 | Continuity, Separability, Convergence |
| S2 | 0310 | 13--16 | 101158 | Deadline Inlupp #1, Read Chap. 2 (Banach spaces), Inlupp #2 | Completion, Compactness, Bounded linear operators |
| S3 | 0331 | 13--16 | 4101 | Deadline Inlupp #2, Read Chap. 3 (Hilbert spaces), Inlupp #3 | Orthogonal and Riesz's representations, Adjoints |
| S4 | 0407 | 13--16 | 80115 | Deadline Inlupp #3, Read Chap. 4+5+6 (Theorems and Applications), group presentations | "Big" Theorems and Applications |
| S5 | 0421 | 13--16 | 101142 | Group presentations | "Big" Theorems and Applications |
| Deadline | 0522 | Draft mini-essay (2--6 pages) ready/sent to 2 referees (With CC to Stefan Engblom) | |||
| Deadline | 0605 | 2 reviews (1/2--1 A4-page), sent to 2 authors (With CC to Stefan Engblom) | |||
| Deadline | 0619 | Final version of mini-essay ready |
Notes: Date "0604" means "June 4th". Place: TBD. All seminars are scheduled for 2+"a bit more" hours, meaning 2 full 45-minutes seminars with 15 min breaks, and a somewhat shorter wrap-up by the end (about 20--30 minutes).
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. Ideas and suggestions will be distributed but you are also more than welcome to come up with something more personal! 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 |
|---|---|---|
| Hakan Runvik | Stability analysis with the Koopman and Perron-Frobenius operators | Erik Blom, Tuan Anh Dao |
| Erik Blom | Fixed point proof of existence and uniqueness for Stochastic Differential Equations | Gustav Eriksson, Tuan Anh Dao |
| Gustav Eriksson | An adjoint method for optimal control problems | Anh Tung Nguyen, Gesina Menz |
| Tuan Anh Dao | Thermodynamic consistency and entropy principles of the magnetohydrodynamics equations in presence of non-zero magnetic divergence | Gesina Menz , Jonas Evaeus |
| Gesina Menz | The Banach-Saks-Mazur Theorem | Anh Tung Nguyen, David Krantz |
| Anh Tung Nguyen | Nonlinear control meets functional analysis | Jonas Evaeus, Ivy Weber |
| Jonas Evaeus | Abstract analysis of Bayesian methods | David Krantz, Hakan Runvik |
| David Krantz |
Solvability of integral equations on Lipschitz domains | Erik Blom, Ivy Weber |
| Ivy Weber | Sobolev inner-product spaces and applications to finite element methods | Gustav Eriksson, Hakan Runvik |
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 one of the 5 "Big Theorems".
| Result | Group | Presents |
|---|---|---|
| (Banach) Closed graph theorem | Håkan Runvik, David Krantz
|
220421 |
| Hahn-Banach theorem | Jonas Evaeus, Anh Tung Nguyen | 220421 |
| Uniform boundedness (Banach-Steinhaus) theorem | Erik Blom, Gesina Menz | 220421 |
| (Banach) Open mapping theorem | Ivy Weber, Sribalaji Coimbatore Anand | 220407 |
| Banach fixed point theorem | Gustav Eriksson, Tuan Anh Dao | 220407 |
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).
Third meeting
Material:
Fast Quiz: No 3
Slow Quiz: No 3
Before the meeting: Read Chap. 3 and try to solve the exercises. It is recommended to try as many exercises as you can. The following is a selection which connects in various ways to Numerical Analysis or similar (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: Read Chap. 2 and try to solve the exercises. It is recommended to try as many exercises as you can. The following is a selection which connects in various ways to Numerical Analysis or similar (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. It is recommended to try as many exercises as you can. The following is a selection which connects in various ways to analysis of computations (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 solving numerical and computational problems. 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 third time this course is given. Your continuous feedback will be very much appreciated. Thank you!
Book: Kreyszig, 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 Probability Theory, in Integration Theory, some in Functional Analysis. Very few have all! Please appreciate that the course is trying to strike a balance!
Hopes
- Fun mathematics, maybe helping to view problems I work on with a different (functional analysis/operator-ish) lens
- During my MSc I had a few courses where my understanding of parts of the course was limited because I didn't know a lot of the background when it came to different types of spaces so I would like to learn more about this to avoid the same issue in the future.
- Filling my knowledge gap and becoming more confident when working with tools within the field.
- I hope that it will be useful when I do continuous analysis for PDEs.
- En snabbtitt i boken säger mig att jag borde känna igen en del från typ matematik-tunga kvantmekanikkurser (t.ex. kap. 3), men aldrig läst ngt särskilt under rubriken "functional analysis".
- I hope that we will have a good chance to discuss Banach space and Hilbert space from basic to advanced level, from fundamental theory to actual applications
- In my research, I use vector norms, normed spaces, etc. Probability spaces, sigma algebra. I would like to understand these terms more clearly and in depth.
- I am currently working with finite element methods which rely quite heavily on piecewise polynomial functions, so any information and results about either polynomials or piecewise smooth functions would be extremely helpful. Material about derivatives in general would also be good.
- To better understand the fundamentals of my research (PDE analysis).
- My hope is to unravel the mystery of the tools used, and how these are applied to investigate the numerical realm. I also hope to find the time to go through the course material properly and to do the final project.
Fears
- Not to be lost in abstract math.
- Not too much implementation.
- It only being a theoretical course with no applications.
- I don't think I have a clear answer for this.
- Actually, I am not really interested in numerical methods and also do not have much knowledge of them. I hope that there will not be so much stuff related to numerical methods.
- Nothing specific comes to mind.
- Too much emphasis on specific applications
- I don't think this is likely given the focus on numerical functional analysis, but I'm hoping for a course with a greater focus on practical results than the abstract mathematical background (a 60-40 balance between the two would be ideal)
- Ser fram emot första doktorandkursen med mycket tillfällen att diskutera och samarbeta på olika sätt!
Responsible: Stefan Engblom