This page gives an overview of the classes we teach regularly. For some earlier teaching by Michael Herbst, see https://michael-herbst.com/teaching/.
Errors are ubiquitous in computational science as neither models nor numerical techniques are perfect. With respect to eigenvalue problems motivated from materials science (transfer problems, atomistic modelling) we discuss, implement and apply numerical techniques for estimating simulation error.
Motivation for studying errors in eigenvalue problems
Types of simulation error
Residual-error relationships for eigenvalue problems
Perturbation theory and parametrised eigenvalue problems
Subtleties of infinite-dimensional eigenvalue problems
Discretisation and discretisation error
Plane-wave basis sets
Errors due to uncertain parameters (if time permits)
Non-linear eigenvalue problems (if time permits)
Exposure to numerical linear algebra
Exposure to numerical methods for solving partial differential equations (such as finite-element methods, plane-wave methods)
Exposure to implementing numerical algorithms (e.g. using Python or Julia)
This course delivers a mathematical viewpoint on materials modelling and it is explicitly intended for an interdisciplinary student audience. To keep it accessible, the key mathematical and physical concepts will both be revised as we go along. However, the learning curve will be steep and an interest to learn about the respective other discipline is required. The problem sheets and the projects require a substantial amount of work and feature both theoretical (proof-oriented) and applied (programming-based and simulation-based) components. While there is some freedom for students to select their respective focus, students are encouraged to team up across the disciplines for the course work.
There is no single textbook corresponding to the content of the course. Parts of the lectures have substantial overlap with the following resources, where further information can be found.
Youssef Saad. Numerical Methods for Large Eigenvalue Problems, SIAM (2011). A PDF is available free of charge on Youssef Saad's website.
Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms, SIAM (2002).
Peter Arbenz. Lecture notes on solving large scale eigenvalue problems, ETHZ. A PDF is available from Peter Arbenz' website.
Mathieu Lewin. Théorie spectrale et mécanique quantique, Springer (2022).
IS academia: MATH-251(b)
Target audience: Materials science Bachelor
Moodle link: https://go.epfl.ch/MATH-251_b
The students will learn key numerical techniques for solving standard mathematical problems in science and engineering. The underlying mathematical theory and properties are discussed.
The topics covered include:
Linear and non-linear systems of equations
Matrix factorisations and decompositions
Numerical differentiation and integration
Numerical solution of differential equations
Regression and least squares problems