MA 402 (001) Spring 2024 Mathematics of Scientific Computing

Instructor: Chao Chen

Email: chao_chen@ncsu.edu (Please include course number in the subject.)

Times: Monday Wednesday 3:00PM - 4:15PM

Location: 330 Dabney Hall

Office hours: SAS 4236, MW 1:00PM - 2:00PM and by appointment

Introduction

This course covers five topics in scientific computing:

1. How numbers are represented by a computer?

2. One of the most useful matrix decompositions: SVD

3. Least squares method (invented by Gauss to discover the planetoid Ceres)

4. Monte Carlo methods (e.g., a simple algorithm for estimating pi)

5. Fourier analysis (without which viewing images or listening to music on your phone will be impossible)

This course will introduce some of the greatest algorithms in applied and computational mathematics, discuss the mathematics behind these algorithms, and demonstrate their applications in, e.g., image compression, singal processing.

Prerequisite

Math: linear algebra, probability, and calculus.

Programming proficiency: Matlab, or Python, or Julia, or C/C++, or Java, or R, …. (While this is NOT a programming course, the homework assignments will involve a substantial programming component.)

References and resources

1. Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares - Stephen Boyd and Lieven Vanderberghe.

2. Numerical Matrix Analysis: Linear Systems and Least Squares - Ilse Ipsen.

3. Monte Carlo theory, methods, and examples - Art Owen.

4. Scientific Computing with Case Studies, Dianne P. O'Leary.

5. The Elements of Statistical Learning - Trevor Hastie, Robert Tibshirani, Jerome Friedman.

6. Matrix Methods in Data Mining and Pattern Recognition - Lars Elden.

Learning Matlab

• MATLAB self-paced, online courses

• Matlab Guide - Des Higham and Nick Higham.

• Numerical Computing with MATLAB - Cleve Moler.

Using LaTeX
LaTeX can help produce beautifully typeset documents and is particularly useful for communicating mathematics. I highly encourage you to learn it.

• Overleaf online editor

• LaTeX NYU website

Grading

Numerical grade = 5 problem sessions (10%) + 5 homework (50%) + midterm (15%) + final (25%)

Letter grade

• 97 and above is A+, starting from 93 and below 97 is A, starting from 90 and below 93 is A-.

• Similar for the B, C, and D ranges.

• Anything below 60 is F.

Problem sessions

There will be an in-class problem session after each unit. During this session, students work through a problem sheet in groups of 2-3 (open book/notes). All members will receive the same grade. Every student needs to be in a different group each session. Submissions will be due after two days (before 11:59 PM).

Homework

There will be 5 assignments, one for each unit. Again, students work in groups of 2-3. All members will receive the same grade. Every student needs to be in a different group for every homework (it could be the same group in a problem session). Each submission should be a professional looking document (preferably using LaTeX). Thinking of how to best present your results (e.g., what and how to plot) is part of the assignment.

Midterm

The midterm will be held on Monday, February 26. It will be in-class and closed book. Please let me know ASAP if you can not make it and need to schedule a make-up exam.

Final

The final exam will be held at 3:30PM - 6:00PM on Monday, April 29. It will be closed book. The final exam can NOT be rescheduled for any reason.

Moodle

Lecture notes and homework assignments will be posted on Moodle.

Panopto

Lectures will be videotaped and made available via Panopto.

Learning outcomes

1. Know algorithms and the problems they address.

2. Understand mathematics of the algorithms.

3. Apply algorithms to solve related problems.

Homework submission instructions

• Please submit your projects by 11:59 PM on the due date. You should submit it on Moodle.

• If you notice an error in your project, you can use the “Edit submission” button prior to the due date/time.

• Please mention the names of your group mates.

• PDF is the only accepted format for homework files. You may find “save as PDF” or “print to PDF” useful if your file is not already in that format. You may submit only one file per homework. Applications such as acrobat or preview easily allow to concatenate existing PDF files if needed.

• Your file must be titled 402_lastname_HW#. If I were to turn in Homework 1, my file would be 402_chen_HW1.pdf.

• The use of LaTeX is strongly encouraged; for each homework, you will be given access to not only a PDF file but also to the LaTeX file that was used to generate it. You can use that LaTeX file as a starting point.

• Hand written assignments are acceptable but should be scanned. Cell phone quality photos may be rejected. Hand drawn graphs to display computational results are not acceptable.

• Your submission should be standalone: it should contain code, figures, and the output from the code. Please do not expect us to run your code; we will only run the code if something doesn't make sense.

• If your file is too large or unreadable, it may be rejected, in which case, you may resubmit.

• Late submissions are not accepted.

Tentative schedule

Week 1

• 1/8: Course overview, floating point representation

• 1/10: Round off error

Week 2

• 1/15: MLK (no class)

• 1/17: Catastrophic cancellation

Week 3

• 1/22: Vector norm and inner product

• 1/24: Orthogonal matrix

Week 4

• 1/29: Problem session 1

• 1/31: Geometric perspective of SVD

Week 5

• 2/5: Low-rank compression

• 2/7: Matrix norm

Week 6

• 2/12: Problem session 2

• 2/14: Image deblurring

Week 7

• 2/19: Least squares

• 2/21: QR decomposition

Week 8

• 2/26: Midterm

• 2/28: Gram-Schmidt orthogonalization

Week 9

• 3/4: Problem session 3

• 3/6: Written communication assessment (see below)

Week 10 (Spring break, no class)

Week 11

• 3/20: Householder QR

• 3/18: Estimation of Pi via Monte Carlo

Week 12

• 3/25: Law of large numbers and central limit theorem

• 3/27: Gaussian distribution

Week 13

• 4/1: Random number generation

• 4/3: Problem session 4

Week 14

• 4/8: Fourier series

• 4/10: Discrete Fourier series

Week 15

• 4/15: Fast Fourier transform

• 4/17: Problem session 5

Week 16

• 4/22: Review

Written communication assessment

Students in this course will complete a written communication assessment on March 6th, 2024. The assessment is completed online. You will need to bring your laptop to class, but you do not need to study or otherwise prepare. This is an important element of the assessment of the NC State University Pack Proficiencies. You are encouraged to do your best work! This information will help us determine the impact of our programs and guide future decisions regarding the education of students in the college of sciences. This assessment is administered by the DASA Assessment office. If you have an accessibility letter and require accommodations, please contact Madalene Compton at mtadams2@ncsu.edu. DASA Assessment will also provide laptops to students who need them to complete the assessment.

Academic integrity and honesty

Students are required to comply with the university policy on academic integrity found in the Code of Student Conduct 11.35.01 sections 8 and 9. Therefore, students are required to uphold the Pack Pledge: “I have neither given nor received unauthorized aid on this test or assignment.” Violations of academic integrity will be handled in accordance with the Student Discipline Procedures.

Please refer to the Academic Integrity web page for a detailed explanation of the University’s policies on academic integrity and some of the common understandings related to those policies.

Students with Disavilities

Reasonable accommondations will be made for students with verifiable disabilities. In order to take advantage of available accommodations, students must register with the Disability Servic3es Office. For more information on NC State's policy on working with students with disabilities, please see the Academic Accommodations for Students with Disabilities Regulation.