• Important eigenvalue problems in materials science
• 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)
• Estimating errors in non-linear eigenvalue problems (if time permits)

• Analysis
• Linear algebra
• Exposure to numerical linear algebra
• 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.

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.

Project during the semester and oral exam

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).
• Nicholas J. Higham. *Accuracy and Stability of Numerical Algorithms*, SIAM (2002).
• Peter Arbenz. *Lecture notes on solving large scale eigenvalue problems*, ETHZ.
• Mathieu Lewin. *Théorie spectrale et mécanique quantique*, Springer (2022).

• [External resource] Lecture notes on solving large scale eigenvalue problems / Arbenzhttps://people.inf.ethz.ch/arbenz/ewp/Lnotes/lsevp.pdf
• Find the references at the Libraryhttps://slsp-epfl.primo.exlibrisgroup.com/discovery/search?query=course_code,contains,%22MATH-500%202025-2026%22,AND&tab=41SLSP_EPF_MyInst_and_CI&search_scope=MyInst_and_CI&sortby=rank&vid=41SLSP_EPF:prod&lang=fr&mode=advanced&offset=0

• https://matmat.org/teaching/error-controlhttps://matmat.org/teaching/error-control

• https://go.epfl.ch/MATH-500https://go.epfl.ch/MATH-500