Selected Topics in Mathematical Modelling and Computational Sciences 1
Winter Semester 2024/2025
Master in Mathematics
Practicalities
Instructor: Jack S. Hale (jack.hale@uni.lu).
Dates
Please see the ACME system.
Teaching units
Following the rules set out by the Vice Rectorate for Academic Affairs the course will be taught as a continuous block of 2 TUs of 45 minutes beginning at the scheduled start in the ACME system.
Description
In this seminar series we will discover some classic papers that have had a huge impact on the field of Computational Sciences and Mathematical Modelling. As well as understanding the technical content of the papers, you will be encouraged to form a broader view of the history of the field, to understand the context of these important developments, and to see how those advances have influenced modern industrial practice. The course and its content will be student-led following the framework outlined below.
Assessment
Each student will select a paper. They will be the project leader for their chosen paper. Every student is expected to contribute fully to delivering on the coursework for each paper, following the overall directions of the project leader.
My expectation is that all students will collaborate fully on the course and produce high quality work. If you are not meeting expectations, I will contact you via email during the course. If the issue does not improve, I will deduct marks on an individual basis.
Students will be jointly assessed on the following pieces of coursework. There is no examination.
The reports should consist of:
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A written report of approximately ten A4 pages that touches equally on the five roles detailed below. The report should be prepared using LaTeX.
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Supporting code examples should be uploaded to a GitHub repository and results placed directly in the report.
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Joint presentation of 25 minutes plus a question and answer session as shown in the schedule.
The reports should be submitted by the project lead as a PDF and, if applicable, a GitHub link via email by 1700 on the date shown in the schedule below. Please cc: the whole class.
Attendance policy
This is a practical course and attendance is mandatory. A maximum of 2 TU can be missed across the semester. Further TUs missed will require a meeting with the instructor and study programme director.
Retake policy
There is no retake possible - register for the course again at the next available semester.
Schedule
- 17/09: Introduction, practicalities, planning.
- 24/09: Group discussion.
- 1/10: Group discussion.
- 8/10: Group discussion.
- 15/10: Group discussion.
- 22/10: Report and presentation 1 due.
- 29/11: Group discussion.
- 5/11: Group discussion.
- 12/11: Group discussion.
- 19/11: Group discussion.
- 26/11: Report and presentation 2 due.
- 03/12: Group discussion.
- 10/12: Group discussion.
- 17/12: Presentation 3 due.
- 20/12: Report 3 due.
Outline
The following roles provide suggestions for the content and structure of the report and presentation:
- The Historians. Understand the historical context of the paper; prior works, the authors biography, or the general technological/political/cultural context of the time.
- The Technicians. Will be responsible for reading the paper with care and producing a summary presentation and report of the main results.
- The Experimentalists. If appropriate, will re-produce some of the results from the paper in a simple dynamic programming language, e.g. Python or MATLAB.
- The Futurists. Understand the impact of the paper in future academic research and in an industrial context. Discover how the presentation of the key ideas of the paper has evolved in e.g. modern textbooks.
- Critics. Will ‘critique’ the article in a constructive way, e.g. What was clear? What was not clear? Is the information laid out in a good order? Are there any omissions? Are the arguments strong? What could be improved?
Papers
Suggestions from the students are welcome!
| [1] | Achi Brandt. Multi-Level Adaptive Solutions to Boundary-Value Problems. Mathematics of Computation, 31(138):333--390, 1977. Publisher: American Mathematical Society. |
| [2] | F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numer, 8(2):129--151, 1974. |
| [3] | C. G. Broyden. A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation, 19(92):577--593, 1965. |
| [4] | J. C. Butcher. A history of Runge-Kutta methods. Applied Numerical Mathematics, 20(3):247--260, March 1996. |
| [5] | James W. Cooley and John W. Tukey. An Algorithm for the Machine Calculation of Complex Fourier Series. Mathematics of Computation, 19(90):297--301, 1965. Publisher: American Mathematical Society. |
| [6] | Carl de Boor. On calculating with B-splines. Journal of Approximation Theory, 6(1):50--62, July 1972. |
| [7] | E. W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1(1):269--271, December 1959. |
| [8] | G. Golub and W. Kahan. Calculating the Singular Values and Pseudo-Inverse of a Matrix. Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis, 2(2):205--224, 1965. Publisher: Society for Industrial and Applied Mathematics. |
| [9] | Gene H. Golub and John H. Welsch. Calculation of Gauss Quadrature Rules. Mathematics of Computation, 23(106):221--s10, 1969. Publisher: American Mathematical Society. |
| [10] | John L. Gustafson. Reevaluating Amdahl's law. Communications of the ACM, 31(5):532--533, May 1988. |
| [11] | John H. Halton. A Retrospective and Prospective Survey of the Monte Carlo Method. SIAM Review, 12(1):1--63, January 1970. Publisher: Society for Industrial and Applied Mathematics. |
| [12] | Michael Held and Richard M. Karp. A Dynamic Programming Approach to Sequencing Problems. Journal of the Society for Industrial and Applied Mathematics, 10(1):196--210, 1962. Publisher: Society for Industrial and Applied Mathematics. |
| [13] | Magnus Rudolph Hestenes and Eduard Stiefel. Methods of conjugate gradients for solving linear systems, volume 49. NBS Washington, DC, 1952. |
| [14] | Nicholas J. Higham. The Accuracy of Floating Point Summation. SIAM Journal on Scientific Computing, 14(4):783--799, July 1993. |
| [15] | T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39):4135--4195, October 2005. |
| [16] | Leslie Lamport. Time, clocks, and the ordering of events in a distributed system. Communications of the ACM, 21(7):558--565, July 1978. |
| [17] | Stanley Osher and James A Sethian. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 79(1):12--49, November 1988. |
| [18] | Youcef Saad and Martin H. Schultz. GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM Journal on Scientific and Statistical Computing, 7(3):856--869, July 1986. Publisher: Society for Industrial and Applied Mathematics. |
This file was generated by bibtex2html 1.99.
Acknowledgements
Ideas of Nick Trefethen on running an effective mathematics seminar series (https://web.stanford.edu/class/cme324/classics/). Contributions to the paper list made by Francesco Viti, Stéphane P. A. Bordas, Ivan Nourdin and Xavier Besseron.