"""metadatashow_logsèdisabled®skip_as_script«code_foldedënotebook_id$6d0915ea-c61c-11f0-2150-19b71f42064cbondscell_results$b1c510c9-4130-4f5e-8d15-3bfaf0a61033queued¤logsrunning¦outputbodyILiterature
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). A PDF for EPFL students is available from Springer Link.
Content
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)
Non-linear eigenvalue problems (if time permits)
Algorithm demonstrations and implementations will be based on the Julia programming language and interactive Pluto notebooks.
Resources on the Julia programming language
In this class we employ the Julia programming language for code examples and programming exercises. Helpful information and pointers to further resources on Julia can be found in the online material of the MATH-251(b) Numerical Analysis lecture:
Summary
Errors are ubiquitous in computational science as neither models nor numerical techniques are perfect. With respect to eigenvalue problems motivated from materials science and atomistic modelling we discuss, implement and apply numerical techniques for estimating simulation error.
If you spot an error feel free to make a pull request to the github repository generating this website.
MATH-500: Error control in scientific modelling
persist_js_state¤mimetext/htmllast_run_timestampAGhas_pluto_hook_features¬rootassigneecell_id$9261420d-3754-4820-a4f0-c5bda988157adepends_on_disabled_cells§runtime published_object_keysdepends_on_skipped_cells§errored$0ccf77bd-0c5d-4c3c-82b5-b25c4adbc3baqueued¤logsrunning¦outputbodyPrerequisites
Analysis
Linear algebra
Exposure to numerical linear algebra
Some experience with numerical methods for solving differential equations (such as finite-element methods, finite-difference approaches, plane-wave methods)
Exposure to implementing numerical algorithms (e.g. using Python or Julia)
This course delivers a mathematical viewpoint on materials modelling and it is explicitly intended for an interdisciplinary student audience. To keep it accessible, the key mathematical and physical concepts will both be revised as we go along. However, the learning curve will be steep and an interest to learn about the respective other discipline is required.
The problem sheets and the projects require a substantial amount of work and feature both theoretical (proof-oriented) and applied (programming-based and simulation-based) components. While there is some freedom for students to select their respective focus, students are encouraged to team up across the disciplines for the course work.
In the past participants from materials science found it useful to take this course after they followed the lectures on Fundamentals of solid state materials.
Course outline
persist_js_state¤mimetext/htmllast_run_timestampAGhas_pluto_hook_features¬rootassigneecell_id$c0da4104-9a5a-4553-8df0-fdadd0194c14depends_on_disabled_cells§runtime Hpublished_object_keysdepends_on_skipped_cells§errored$854d367b-1609-4c00-9c62-9439874a03d6queued¤logsrunning¦outputbody"EPFL resources
If you are an EPFL student, you can find the following course resources: