Numerical Functional Analysis (5hp)

Schedule

Date	Time	Place	Hand in/Prepare	Key concepts
140929	13--15	1112	Read Chap. 1 (Metric spaces)	Continuity, Separability, Convergence
141010	13--15	2214	Inlupp #1, Read Chap. 2 (Banach spaces)	Completion, Compactness, Bounded linear operators
141023	13--15	2345	Inlupp #2, Read Chap. 3 (Hilbert spaces)	Orthogonal and Riesz's representations, Adjoints
141120	13--15	1245	Inlupp #3, Read Chap. 4+5+6 (Theorems and Applications), group presentations	"Big" Theorems and Applications
141128	13--15	2344	Group presentations	"Big" Theorems and Applications
141212			Draft mini-essay (2--6 pages) ready/sent to 2 referees (With CC to Stefan Engblom
141219			2 reviews (1/2--1 A4-page), sent to 2 authors (With CC to Stefan Engblom)
150116			Final version of mini-essay ready

Tips: room 2345 means "house 2, floor 3, room (23)45".

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

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 LaTeX-template below.

Suggested subjects for mini-essay, LaTeX-template

Author	Topic	Reviewers
Martin Almquist	Superconvergent functional output and dual-consistent approximations of PDEs	Samira Nikkar, Markus Wahlsten
Simon Sticko	The Babuska theory for non-coercive bilinear forms	Fatemeh Ghasemi, Hannes Frenander
Saleh Rezaeiravesh	A completeness theorem for non-selfadjoint eigenvalue problems	Markus Wahlsten, Siyang Wang
Samira Nikkar	Time dependent mapping from Cartesian coordinate systems into curvilinear coordinate systems and the geometric conservation law	Tomas Lundquist, Hanna Holmgren
Fatemeh Ghasemi	Fixed point proof of Lax-Milgram	Hannes Frenander, Siyang Wang
Gong Cheng	The Metric Lax and Applications	Andrea Ruggiu, Simon Sticko
Hanna Holmgren	Well-posedness for the Stokes equations	Viktor Linders, Saleh Rezaeiravesh
Siyang Wang	Lax equivalence theorem (for linear problems)	Cristina La Cognata, Martin Almquist
Markus Wahlsten	Ekeland's variational principle	Gong Cheng, Andrea Ruggiu
Tomas Lundquist	Arzelá-Ascoli theorem	Simon Sticko, Viktor Linders
Hannes Frenander	Schauder fixed point theorem	Saleh Rezaeiravesh, Cristina La Cognata
Andrea Ruggiu	Lax-Milgram, traditional proof	Martin Almquist, Samira Nikkar
Viktor Linders	Uniformly Best Wavenumber Approximations	Fatemeh Ghasemi, Gong Cheng
Cristina La Cognata	Construction of Sobolev spaces	Tomas Lundquist, Hanna Holmgren

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, ...

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

Material:
Quiz: Mega Quiz 1/2
Quiz: Mega Quiz 2/2

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	Martin Almquist, Simon Sticko, Saleh Rezaeiravesh	141128
Uniform boundedness (Banach-Steinhaus) theorem	Samira Nikkar, Fatemeh Ghasemi	141120
(Banach) Open mapping theorem	Gong Cheng, Hanna Holmgren, Siyang Wang	141128
(Banach) Closed graph theorem	Markus Wahlsten, Tomas Lundquist, Hannes Frenander	141128
Banach fixed point theorem	Andrea Ruggiu, Viktor Linders, Cristina La Cognata	141120

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 141127 @ 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 about 5 meetings in the form of short lectures and discussions
• About 4 shorter exercises - written and/or explained to the others
• 1 a bit more "in-depth" written assignment, (come up with your own suggestion if you like), written and corrected/criticized by others (you write one, and you give feedback to 2 other)

This is the first 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, (...)

Responsible: Stefan Engblom

Input obtained before the course:

• what background do you have in Functional Analysis?
• what would you hope with this course?
• what would you hope not with this course?
• something else?

Answers:

• I have learned a little functional analysis by myself before. But I don't know what it would be in the 'numerical' area. I would like to learn more on mathematics and numerical analysis. And I would like to have some connections to FEM.

• I almost have no background in functional analysis. The only chance I met this topic was in finite element method I course. My intention of taking this course is that it helps me to understand the theoretical part of a conference talk about finite elements, and to be able to discuss with finite element people on this subject. I probably won´t use functional analysis in my research in a direct way. It will be good if you give some lectures after we read the book so that we understand more.

• Har läst lite i Kreyszig. (hope:) Att kursen fokuserar på funktionalanalys som är relevant för numerik. Exempel: Jag är alltid i ett hilbert rum. Jag är inte så intreserad av vad L^p 1<=p<=inf har för egenskaper, det kanske räcker att veta vad som gäller i L^2. (hope not:) Att kursen håller en super-stringent nivå, diskuterar detaljer om sigmaalgebras och topologi etc.

• I have not taken any course in Functional analysis before. I hope to get a good feeling for what functional analysis is and when it can be used. I hope that the course is not too "flummig" :) (I want it clear and structured...)

• I have basically zero background in functional analysis. I hope to understand some concepts that often appear in a numerical analysis context, and hence understand more at seminars, conferences etc.

• My only previous experience with "Functional Analysis" comes back to a graduate-level course which was mainly devoted to the Navier-Stokes equations. In that course, the definitions and properties of the Hilbert and Sobolev Spaces were reviewed in order to define turbulence concepts. I have no idea what "Numerical Functional Analysis" could be and that is the most powerful motivation for me to take this course. Furthermore, it is a brilliant opportunity to review the "Functional Analysis" theories and basics!

• I have not taken any course in Functional analysis but I studied by myself. When I was a master student, I participated in a course called " Applied Functional analysis" without registering . I hope to learn it deeply. Its applications in numerical analysis is also important for me.

• I've studied Functional Analysis at Politecnico di Milano, on fall 2011. I have also followed a course, held by Sandro Salsa, of PDE's involving Functional Analysis on spring 2012. In my master degree's courses I was used to program in finite elements and I was used to use some elements of Functional Analysis applied to PDEs: anyway I want to have a general recap of the theory and I would also focus my previous knowledge to the numerical applications.

• I have one unfinished Ph.D course in functional analysis; I only read half of it. I'm from the division of computational mathematics, and hope that this course is relevant for my education.

• Jag har tidigare läst en kurs i funktionalanalys på masternivå, men har i övrigt inte arbetat så mycket inom ämnet. Jag hoppas framför allt att kursen ska ge mig en välbehövlig repetition av grunderna samt fördjupning inom valda delar. Kopplingar till numeriska tillämpningar är förstås extra intressant, så jag hoppas att kursen inte fastnar allt för mycket i komplicerad matematisk bevisföring kring allmänna resultat.

• During my master i attended a basic course in functional analysis with the following topics: Metric spaces, Banach spaces, Ascoli-Arzelá theorem, Hilbert spaces, L^p spaces. and another one more advanced with following topics: Distributions theory, weak formulations and related theorems. (what would you hope with this course?) review my knowledge about functional analysis and extend it to numerical analysis

• I had one course in Functional Analysis at master level. I hope to learn Functional analysis useful to my research. I hope the course is not too abstract.

• Jag har ingen nämnvärd erfarenhet av funktionalanalys annat än vad jag sporadiskt läst själv. Jag hoppas på en klar och koncis kurs som poängterar de aspekter som är relevanta för numerisk analys, snarare än en överskådande kurs utan tydligt fokus.

• I have studied mechanical engineering as my background, so I have used it just as much as I needed in the applications that I have worked with. I never had a course in functional analysis though after the high school. I would hope to be able to catch the basics, and know how to solve more advanced problems that I encounter with later in my research. (what would you hope not with this course?) Not so much less useful theoretical stuff. Not so unnecessary complicated notations.