Mathematical aspects of materials modelling

Summary

The simulation of materials properties from first principles is a key component of modern materials science research. We discuss common simulation techniques (such as plane-wave density-functional theory) from a mathematical perspective, outline robust algorithms and error estimation techniques.

Content

Learning Prerequisites

Required courses

• Analysis
• Linear algebra
• Numerical analysis
• Exposure to numerical methods for solving differential equations (such as finite-element methods, finite-difference approaches, plane-wave methods)
• Exposure to implementing numerical algorithms (e.g. using Python or Julia)

Past participants from materials science found it further useful to take this course after they followed the lectures on Statistical Mechanics or Fundamentals of solid state materials.

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 project 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 discplines for the course work.

Expected student activities

Students are expected to attend lectures and participate actively in class and exercises. Exercises will include theoretical, programming and simulation-based assignments. Students also complete substantial group project that contain (to varying extend) theoretical and applied components.

Assessment methods

Project during the semester and oral exam

Resources

Bibliography

A good overview of the main content of this course (density-functional theory) provdies

• Eric Cancès and Gero Friesecke (Eds). "Density-functional theory", Springer (2023). DOI 10.1007/978-3-031-22340-2

More details on the mathematical structure of quantum mechanics can be found in

• Mathieu Lewin. *Théorie spectrale et mécanique quantique*, Springer (2022).

Additionally the following resources provide further reading on selected aspects of the course:

• Youssef Saad. *Numerical Methods for Large Eigenvalue Problems*, SIAM (2011).
• Nicholas J. Higham. *Accuracy and Stability of Numerical Algorithms*, SIAM (2002).
• Peter Arbenz. *Lecture notes on solving large scale eigenvalue problems*, ETHZ.

Ressources en bibliothèque

• Find the references at the Library
• [External resource] Lecture notes on solving large scale eigenvalue problems / Arbenz

Websites

• https://teaching.matmat.org/math-materials/

Moodle Link

• https://go.epfl.ch/MATH-500