Mathematical foundations for computational science
Credits: 7.5hp
Time: The course will be given the first time Fall 2023 and then regularly every second year. The first lecture is scheduled at 13.15 September in room 106130.
Course structure: The course will be given as a series of lectures, which may be pre-recorded or live, in combination with seminars where the material is discussed.
Examination: The course will be examined through assignments and project work, including oral presentations to the group.
Level: The course is targeted to beginning graduate students with some background in mathematics and scientific computing. As an example, we would expect the students to have seen the finite difference method and the finite element method, and to be aware of machine learning on a conceptual level, but not to have advanced knowledge in these areas.
Content: The course will cover the most fundamental theory and tools that are needed to analyze and evaluate numerical methods for simulation and data analysis. The main focus will be on approximation theory and probabilistic methods, but there will also be some elements of functional analysis. Specifically the following topics will be included:
- Properties of univariate, multivariate and tensor product approximations.
- The relation of approximation theory to convergence estimates in optimal recovery problems.
- Polynomial approximations with error estimates and sampling inequalities, best N-term approximations.
- Machine learning from an approximation theory perspective. Deep learning, neural networks, density, convergence, and complexity.
- Probabilistic/stochastic convergence, weak convergence, law of large numbers. Monte Carlo method.
- Basics of Bayesian estimation and uncertainty quantification, aleatoric and epistemic uncertainty. Metropolis algorithm.
- Convergence of functions, function spaces.
Outline:
| Time | Block | Teacher(s) |
|---|---|---|
| 11/9-20/10 (6 weeks) | Approximation theory and functional analysis with applications | Gunilla Kreiss, Elisabeth Larsson |
| 23/10-24/11 (5 weeks) | Foundations of probabilistic modelling | Stefan Engblom |
| 27/11-15/12 (3 weeks) | Mathematical Foundations of Machine Learning | Mohammad Motamed |
Preliminary schedule for the first block (will be updated with more details)
| Date | Place | Content |
|---|---|---|
| 11/9 13-15 | room 106130 | Introduction to the course |
| 22/9 10-12 | room 106157 | |
| 27/9 10-12 | room 106157 | |
| 4/10 9-11 | room 106157 | |
| 12/10 13-15 | room 106190 | |
| 18/10 10-12 | room 106157 |
Contact persons: Elisabeth Larsson Gunilla Kreiss
Please let us know before September 5 if you are planning to follow the course, and if the preliminary schedule for the first block works for you.