38 Topics for Presentations/Posters/Papers
Warning
To be updated….
Presentations/posters will take place on the week beginning April, 27. Papers will be due the same week.
- You can choose to do a presentation/poster/paper for 10% of your grade (see syllabus for details),
- If you have your own ideas on topics please let me know and I can advise if they are suitable,
- The main aim is for you to go away and learn something new and tell us about it in an interesting way.
Presentations.
- Presentations will be 12 minutes plus 3 minutes for questions (I will have to be strict on timings to fit everyone in),
- You will present to the whole group in one of our normal class times,
- You may present however you think best illustrates the key ideas (you may use the whiteboards, slides, jupyter notebook, ….),
- Note that 15 minutes is not very long! Pick something interesting but be wary of the time constraints,
- It would be nice if your presentation included some numerical implementation. You may submit accompanying numerical experiments,
Posters.
- Walk around the department to find some examples - the following rubric also mostly applies to you if you do a poster
- There will be time in lectures to answer questions on your poster,
Papers.
- Produce written notes on a topic of your choice,
- It would be great if we could upload these to the webpage for everyone to benefit from,
- There will be time in lectures for questions,
38.1 Rubric
Here are the things that I’ll be looking for in your presentations/posters/papers:
Content (4 points)
- Accuracy: the method/results/proof/examples should be correct,
- Originality: tell us something that we haven’t seen in lectures but links to existing topics or extends the content completed in class or on the homework assignments,
- Give examples to motivate/explain key ideas,
Numerical Implementation (2 points)
- Show us some numerical results/experimentation
Presentation (2 points)
- You will be graded on how clearly you present the key ideas,
- Think about how you want to present (whiteboard/slides/notebook) - what is the best way to convey the main point?
Questions (2 points)
- How well are you able to answer the questions at the end?
- Bonus points for asking good questions
(The following is in no particular order. To be updated….)
38.2 Convergence of Jacobi and Gauss-Seidel
In the midterm we prove that Jacobi and GS converge for SDD matrices. You can actually prove convergence under a weaker assumption. Show that Jacobi and GS converge under the assumption that A is irreducibly diagonally dominant.
38.3 “From potential theory to matrix iterations in six steps”
Pick out something that you find interesting from this paper Driscoll, Toh, and Trefethen (1998). [more than one person can do this: just coordinate so that you are not giving the same presentation].
38.4 Convergence of Conjugate Gradient
Give some ideas in the proof of Theorem 11.2 from the lecture notes.
38.5 Preconditioned CG
Use e.g. Incomplete LU to improve the conditioning of a matrix. Give numerical examples that show this improves the convergence behaviour.
38.6 Arnoldi Iteration
38.7 GMRES
38.8 MINRES
When A is symmetric (or Hermitian) the Arnoldi iteration simplifies to the Lanczos iteration (and GMRES becomes MINRES)
38.9 Solving Poisson’s equation in 2d
38.10 (other) Applications of the Discrete Fourier Transform
Find something that you find interesting in e.g. “Brigham, E. O. (1988). The Fast Fourier Transform and Its Applications”. For example, how can you use the FFT to compute the product of two polynomials? Why is this useful?
38.11 Richardson-Lucy iteration for image de-blurring
Suppose you have some grayscale image represented by some matrix in [0,1]^{n\times m} that has been flattened into a vector b \in [0,1]^{nm} (e.g. b represents a NASA image from the Hubble space telescope) and a blurring matrix A. The observed image is then b = Ax where x is the original (sharp) image. The Richardson-Lucy method is an iterative method that tries to recover x.
[ Info: Saved animation to c:\Users\math5\Math 5485\Math5486\pics\math5486.gif
38.12 Fornberg algorithm
Finite differences on arbitrary nodes
38.13 Google PageRank and the Power Method
Explain how Google uses the power method (see midterm) in their page ranking algorithm.
Driscoll, Tobin A, and Richard J Braun. 2018. Fundamentals of Numerical Computation. https://fncbook.com/.
Driscoll, Tobin A, Kim-Chuan Toh, and Lloyd N Trefethen. 1998. “From Potential Theory to Matrix Iterations in Six Steps.” SIAM Rev. Soc. Ind. Appl. Math. 40 (3): 547–78.