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MATH 6650-100 (8706), Spring 2024

Numerical Analysis: Approximation Methods

Syllabus

Catalog Description:

In-depth treatment of numerical approximation techniques, including differentiation and integration.

Learning Outcomes: (Students will be able to...)

Apply approximation techniques for functions.
Apply approximation techniques for integrals and derivatives.
Analyze the errors in such approximations.
Prove the fundamental theorems upon which such analysis is based.

Prerequisites:

MATH 5600 - Introduction to Numerical Analysis

Class hours/ location:

Monday, Wednesday, and Friday 10:45 AM - 11:40 AM in Morton Hall 314.

Web page:

http://www.ohiouniversityfaculty.com/mohlenka/2245/6650/.

Instructor:

Martin J. Mohlenkamp, mohlenka@ohio.edu, Morton Hall 321C, 740-593-1259. The best ways to reach me are email and Teams.

Office hours:

Monday and Friday 9:40 AM - 10:35 AM. I am available to meet at other times, just contact me for an appointment.

Computing Environment:

We will use python in Jupyter notebooks within CoCalc. Although CoCalc accounts are free, I recommend you purchase a basic license or use pay-as-you-go upgrades. Unless otherwise specified, I will look for assignments in CoCalc when they are due, so you do not have to upload or otherwise submit them.

Attendance:

We will have many in-class learning activities, so attendance is required, and will count in your grade.

Materials:

Walter Gautschi Numerical Analysis, Birkhauser Boston; 2nd edition, 2011. ISBN-10: 0817682589. (Accessible through the university library.)
Armin Iske Approximation theory and algorithms for data analysis, Texts in Applied Mathematics 68, Springer, Cham, 2018. ISBN 978-3-030-05227-0 and 978-3-030-05228-7, doi:10.1007/978-3-030-05228-7. (Accessible through the university library.)
MathSciNet (Accessible through the university library.)

Content Plan:

I aspire to cover Gautschi

Chapter 3 Numerical Differentiation and Integration

and portions of Iske

Chapter 1 Introduction
Chapter 2 Basic Methods and Numerical Algorithms
Chapter 3 Best Approximations
Chapter 4 Euclidean Approximations
Chapter 5 Chebyshev Approximations
Chapter 6 Asymptotic Results
Chapter 7 Basic Concepts of Signal Approximation

and perhaps a little on

tensor approximations
approximating with neural newtorks.

Final project:

As a culminating experience, you will do a project to reproduce a recent (2023 or 2024) published paper on a topic related to this course (Mathematics Subject Classification 41 Approximations and expansions or 65D Numerical approximation and computational geometry (primarily algorithms)). Your goal is to demonstrate what you have learned in this course and in particular to demonstrate that you have met the Learning Outcomes listed at the start of this syllabus.

The deliverables for the project are:

A presentation, during the week of April 15 or 22.
A report, due April 29.

Explorations:

As preparation for the final project, each week you will do an assignment related to a paper that could be a project topic. Early in the semester, you will be reading different papers to look for topics. Later in the semester, you will be proposing a project and doing drafts of different aspects of your project.

Homework, presentations, quizzes, etc.

We will have some

Homework problems on theory
Homework problems doing algorithms by hand
Homework problems involving coding
Small in-class presentations of examples or proofs from the texts
Small quizzes on key theory or algorithms

Some homework problems will be graded both on content and on presentation, using the Good Problems system of handouts.

Late Work:

Late work is penalized 5% per day (or part thereof) late.

Grade:

Your grade is based on attendance 5%; homework, presentations, and quizzes combined 45%; explorations 15%; final project presentation 5%; and final project report 30%. An average of 90% guarantees you at least an A-, 80% a B-, 70% a C-, and 60% a D-. Grades are not the point.

Religious Accommodations

In accordance with the university's Interim policy on reasonable religious accommodations [40.003]:

You may be absent for up to three (3) days each academic semester, without penalty, to take time off for reasons of faith or religious or spiritual belief system or to participate in organized activities conducted under the auspices of a religious denomination, church, or other religious or spiritual organization. You are required to notify me in writing of specific dates requested for alternative accommodations no later than fourteen (14) days after the first day of instruction. These requests will remain confidential. For more information about this policy, students/you may contact the Director and Title IX Coordinator, Equity and Civil Rights Compliance, Lindley Hall 006, 740-593-9140, equity@ohio.edu.

If your absence causes a delay in finishing an assignment, you will receive a 1-day extension on it.

Academic (mis)conduct:

You are allowed to use most resources, but there are some limitations.

Unlimited use, without specific acknowledgment:

The textbook.
Discussions with me.
Any in-class, full-class discussions.
Python documentation.

Broad use, with acknowledgment:

Websites about numerical methods or programming.
Other textbooks or papers.
Explanations by other students.
Explanations by faculty, if ou make them aware that this is a class assignment.
Explanations by chatbots or other AI.

Acknowledge and describe this help in writing on the problem where it was used. For example, you might write

[Name] explained to me how to do [some part] of this problem.

I found an explanation of [concept] at the website [url].

I asked ChatGPT to [request].

Forbidden:

Any website or source that claims to have solutions to the exercises from our text.
Posting any homework problem to a website or forum.
Code produced by chatbots or other AI.
Direct copying.

A minor, first-time violation of this policy will receive a warning and discussion and clarification of the rules. Serious or second violations will result in a grade penalty on the assignment. Very serious or repeated violations will result in failure in the class and be reported to the Office of Community Standards and Student Responsibility, which may impose additional sanctions. You may appeal any sanctions through the grade appeal process.

Special Needs:

If you have specific physical, psychiatric, or learning disabilities and require accommodations, please let me know as soon as possible so that your learning needs may be appropriately met. You should also register with Student Accessibility Services to obtain written documentation and to learn about the resources they have available.

Responsible Employee Reporting Obligation:

If I learn of any instances of sexual harassment, sexual violence, and/or other forms of prohibited discrimination, I am required to report them. If you wish to share such information in confidence, then use the Office of Equity and Civil Rights Compliance.

Learning Resources:

Your classmates are your best resource. Use them!
Python
- Python home
- Python For Beginners
- Python documentation
- Python Tutorial
- NumPy home
- To get the documentation on a python function within a jupyter notebook, use "?". So, to get information about range, run range? in a python cell.
Markdown and $\LaTeX$
- Markdown language documentation and markdown syntax.
- $\LaTeX$ Wikibook
- When $\LaTeX$ is rendered on an html page using MathJax (like this page), you can see the underlying $\LaTeX$ code by right-clicking on the math, selecting Show Math As, and then selecting TeX Commands.
- In $\LaTeX$ either $...$ or $...$ can be used to indicate inline math. The form $...$ is better within html using MathJax, but markdown works better with $...$ . Similarly, markdown prefers $$...$$ over \[...\] for displayed math.

Schedule (Subject to change)

Many things will be filled in as we go along.

Week of January 15

Monday is the Martin Luther King, Jr. holiday, so no class.

Wednesday: Introductions
- Get set up in CoCalc.
  - Sign up for a free account, using your @ohio.edu email address.
  - Create a project. Within it hit Users. Within Add New Collaborators search for me mohlenka@ohio.edu and then add me as a collaborator. Your submissions of assignments go here.
  - Within the project, hit Explorer and within it hit Tour. Still within Explorer, hit the Library button, browse the available topics, select Markdown in CoCalc, hit Get a Copy, and go through markdown-intro.md and markdown-in-jupyter.ipynb. Try editing them to test markdown features.
  - Read about licenses and pay-as-you-go upgrades. Consider getting one.
- Due Friday at noon:
  - Upload this file jan19-intro.ipynb into your CoCalc project.
  - Edit the Mathematical Autobiography section to be about you.
  - Fill in the missing parts of the $\LaTeX$ Warm Up section.
Friday:
- Using MathSciNet (Accessible through the university library.)
- Due Monday at noon: Exploration: Choose a paper published in 2023 or 2024 with Mathematics Subject Classification 41 Approximations and expansions or 65D Numerical approximation and computational geometry (primarily algorithms). Upload it to CoCalc. Create a file jan22-exploration.ipynb and in it put your name, the full bibliographic information (title, authors, journal, year, etc.) for this paper, and the name of the pdf file you uploaded.
- Accessing the textbooks.
- For Monday, read Iske
  - 1.1 Preliminaries, Definitions and Notations
  - 1.2 Basic Problems and Outlook

Week of January 22

Monday
- Mini-presentations on Iske sections 1.1 and 1.2
Wednesday
- Continue mini-presentations on Iske sections 1.1 and 1.2
- For Friday, read Iske 2. Basic Methods and Numerical Algorithms introduction (page 9)
- Due Friday at noon: Exploration:
  - Repeat the exploration from last week for a different paper and then also
  - Write a one-paragraph (about 10 sentence) summary in your own words of the topic of the paper.
Friday
- Within your CoCalc project, hit Explorer, then hit the Library button, select A Whirlwind Tour of Python, and hit Get a Copy. In a group, go through the material, running cells and trying out things.
- For Monday, read Iske 2.1 Linear Least Squares Approximation.

Week of January 29

Monday
- Mini-presentations on Iske sections 2.0 and 2.1
Wednesday
- Continue mini-presentations on Iske sections 2.0 and 2.1
- For Friday, read Iske 2.2 Regularization Methods
- Due Friday at noon: Exploration:
  - Repeat the exploration from last week (including the summary) for a different paper and then also
  - Identify one concept or method mentioned in the paper that you are not familiar with. Research it (usually Wikipedia is sufficient) and write a paragraph explaining it. Cite and link to your sources.
Friday: We will be working on python coding to
- Plot the unit 'circle' for various norms and candidate norms in two dimensions.
- Check through simulation if candidate norms satisfy the properties to be a norm.

Week of February 5

Monday
- Working session on norms using python.
- Due Wednesday at noon: python homework:
  - Clean up your code to test norm properties and plot the unit 'circle'. Be sure to include Docstrings and comments in the code.
  - Run your code on $\|\cdot\|_1$, $\|\cdot\|_{4/3}$, $\|\cdot\|_2$, $\|\cdot\|_4$, $\|\cdot\|_{\infty}$, $\|\cdot\|_1+\|\cdot\|_{\infty}$, $f(\mathbf{x})=2|x_1|+|x_2|$, $f(\mathbf{x})=2|x_1|-|x_2|$, and another function of your choice.
  - Interpret the results.
Wednesday
- Mini-presentations on Iske 2.2.
- Due Friday at noon: Exploration:
  - Repeat the exploration from last week for a different paper and then also
  - Identify the specific claims made in the paper (such as that their method is better than existing methods for certain problems).
Friday:
- Continue mini-presentations on Iske 2.2.
- For Monday, read Iske 2.3 Interpolation by Algebraic Polynomials

Week of February 12

Monday
- Finish mini-presentations on Iske 2.2.
- For Wednesday, read Iske 2.4 Divided Differences and the Newton Representation
Wednesday
- Start mini-presentations on Iske 2.3.
- Due Friday at noon: Exploration:
  - Repeat the exploration from last week for a different paper and then also
  - List the proofs in the paper and identify which you might be able to check by filling in all the details.
Friday
- Finish mini-presentations on Iske 2.3.
- Due next Friday at noon:
  - Make a python function that inputs:
    - xcoords : a list of $x$-coordinates $[x_0,\dots,x_{n-1}]$ at which to interpolate
    - k : an index $k$ with $0\le k \le n-1$
    - x : an $x$-coordinate at which to evaluate
    and outputs the value at $x$ of the Lagrange interpolating (basis) polynomial that is 1 at $x_k$ and 0 and the other $x_i\text{.}$
    Test your function by plotting it, using xcoords=[1,2,3,4] and k=2.
  - Make a python function that inputs:
    - xcoords : a list of $x$-coordinates $[x_0,\dots,x_{n-1}]$ at which to interpolate
    - ycoords : a list of $y$-coordinates $[y_0,\dots,y_{n-1}]$ corresponding to xcoords.
    - x : an $x$-coordinate at which to evaluate
    and outputs the the value at $x$ of the interpolating polynomial that passes through all $(x_i,y_i)\text{.}$ It should use the function you made above.
    Test your function by plotting it, using xcoords=[1,2,3,4] and ycoords=[-6,2,-10,12].
  - Use the functions you created to reproduce Iske figures 2.2(b) and 2.3.

Week of February 19

Monday
- Python group time.
- Start mini-presentations on Iske 2.4.
Wednesday
- Python group time.
- Continue mini-presentations on Iske 2.4.
- Read Iske 2.5 Error Estimates and Optimal Interpolation Points.
- Due next Monday at noon: Exploration:
  - Repeat the exploration from last week for a different paper and then also
  - List the numerical experiments in the paper and identify which you might be able to reproduce to (in)validate the claims of the paper.
Friday
- Continue mini-presentations on Iske 2.4.

Week of February 26

Monday
- Continue mini-presentations on Iske 2.4.
Wednesday
- Sick day, no class
- Due Friday at noon: Exploration:
  - Repeat the exploration from last week for a different paper and then also
  - Identify a python function that you have not used before and would likely be useful in trying to validate the results in this paper. Run a very simple calculation using this function.
Friday
- Continue mini-presentations on Iske 2.4.

Week of March 4

Monday
- Finish mini-presentations on Iske 2.4.
Wednesday
- Start mini-presentations on Iske 2.5.
- Read Iske 2.6 Interpolation by Trigonometric Polynomials
- Due Friday at noon: Exploration: Propose a paper to use for your final project. It could be one you used for an exploration or a new one. Explain why you think it is a good choice. Identify its specific claims, the proofs you plan to check, and the numerical experiments you plan to reproduce. Tips:
  - Avoid papers that give subtle improvements for complicated problems. They likely need too much coding.
  - Look for a paper whose numerical results are based on simple-to-reproduce test problems.
  - If the paper uses any data sets, make sure they are publically available.
Friday
- Finish mini-presentations on Iske 2.5.

Week of March 11

Spring break, so no class.

Week of March 18

Monday
- Clean-up and planning.
- Start mini-presentations on Iske 2.6.
Wednesday
- Time to work on the final project.
- Due Friday at noon: Exploration: If your final project proposal was rejected, then propose a new paper. If your final project proposal was accepted, then
  - Identify a section in Iske that we have not covered, or a section in Gautschi, that has material that is useful background for your paper.
  - Make an outline of the theorems, examples, etc. in that section that would be most useful to discuss in class, and explain your selections.
  - (Prepare to present the things in your outline.)
Friday
- Time to work on the final project.

Week of March 25

Monday
- Continue mini-presentations on Iske 2.6.
- Read Iske 2.7 The Discrete Fourier Transform
Wednesday
- Finish mini-presentations on Iske 2.6.
- Start mini-presentations on Iske 2.7.
- Due Friday at noon: Exploration: Pick some self-contained topic that you will need for your final project, explain it, and do some related computation.
Friday
- Finish mini-presentations on Iske 2.7.

March 29 is the deadline to withdraw with WP or WF.

Week of April 1

Monday
- Mini-presentations on Iske Chapter 3 Best Approximations, introduction
- Start min-presentations on Iske 3.1 Existence
Wednesday
- Continue mini-presentations on Iske 3.1.
- Due Friday at noon: Exploration: Rough draft of final project report.
  See the report format and rubric in the final week. This draft requires a subset of the things in the final report. If you include more things, then I will also give you feedback on those. The report should be done as a single Jupyter notebook in CoCalc.
  - (2 points) Title, your name, paper bibliographic information, sections have headers, not many typos, complete sentences used, and other issues of presentation quality.
  - (2 points) Introduction: What the paper is about, what claims does it make, definitions needed to understand later parts of the report, ...
  - (3 points) Proof checking. For the theorem(s)/lemma(s) that you will check, include their original theorem/lemma statement(s) and proof(s) (you can insert as images).
  - (3 points) Numerical results checking. For the results that you will check, explain what they are testing and give their original results/ tables/ graphs (you can insert as images).
Friday
- Continue mini-presentations on Iske 3.1.

Week of April 8

Monday
- No class due to solar eclipse.
- Due Wednesday at noon: Exploration: Draft of presentation slides.
  The goal of this draft is to have all the slides you plan to use, even if many of them are placeholders. Placeholder slides should contain at least a sentence like "This slide will contain ....". If you include more things, then I will also give you feedback on those. Note that you will do two 10-minute presentations; it is your choice whether to do a single set of slides or two sets. The slides should be converted to pdf format and submitted through Blackboard.
  - (1 point) Title page, paper bibliographic information.
  - (2 points) Introduction part: Slides with titles and at least placeholder information.
  - (3 points) Proof checking part: Slides with titles and at least placeholder information.
  - (3 points) Numerical results checking part: Slides with titles and at least placeholder information.
  - (1 point) Conclusion part: Slides with titles and at least placeholder information.
Wednesday
- Finish mini-presentations on Iske 3.1.
Friday
- Clean-up and planning.
- Iske 3.2 Uniqueness

Week of April 15

Project presentations part I (10 minutes per person):
- Introduce the topic of the paper.
- Present a (significant) proof from the paper, filling in the details.

Week of April 22

Project presentations part II (10 minutes per person):
- Remind about the topic of the paper.
- Show some of the numerical results from the paper.
- Show your work trying to reproduce their numerical results.
Once you have presented, submit your slides in pdf format through Blackboard.

Week of April 29

Monday April 29: Project report due at 12:10pm, which is the end of our scheduled final examination time.
The report should be done as a single Jupyter notebook in CoCalc.
- (10 points) Title, your name, paper bibliographic information, sections have headers, not many typos, complete sentences used, and other issues of presentation quality.
- (10 points) Introduction: What the paper is about, what claims does it make, definitions needed to understand later parts of the report, ...
- (35 points) Proof checking. For the theorem(s)/lemma(s) that you will check, include their original theorem/lemma statement(s) and proof(s) (you can insert as images). Then give your expanded version, with full details and explanations.
- (35 points) Numerical results checking. For the results that you will check, explain what they are testing and give their original results/ tables/ graphs (you can insert as images). Then give your code and the corresponding results/ tables/ graphs you got. Discuss the extent to which you have verified their numerical results.
- (10 points) Conclusions: Summary of your conclusions from trying to verify their proof(s), summary of your conclusions from trying to verify their numerical result(s), disscussion of difficulties you encountered, advice for future students doing projects, self-assessment on how you demonstrated the course learning outcomes, ...

Martin J. Mohlenkamp

Last modified: Tue Apr 9 15:07:56 UTC 2024