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A good overview of the main content of this course (density-functional theory) provdies
More details on the mathematical structure of quantum mechanics can be found in
Additionally the following resources provide further reading on selected aspects of the course:
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.
Content
Overview of atomistic modelling of materials: first-principle models, atomistic machine learning, prediction of material properties
Revision of key concepts from quantum mechanics
Revision of key concepts from functional analysis
Periodic boundary conditions, Bloch theory and plane-wave basis sets
Density-functional theory (DFT): From the mathematical formulation to practical calculations
Robust numerical methods for DFT simulations
Error control in density-functional theory simulations: Perturbation-based techniques to estimate discretisation errors; uncertainty propagation; inverse problems
Error estimation and propagation in atomistic machine learning (as 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
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.
MATH-500: Mathematical aspects of materials modelling
°persist_js_state¤mimeŠtext/html˛last_run_timestampËAÚş˛'çŽ ˇhas_pluto_hook_featuresÂŹrootassigneeŔ§cell_idŮ$9261420d-3754-4820-a4f0-c5bda988157ašdepends_on_disabled_cells§runtimeÎ [ďľpublished_object_keys¸depends_on_skipped_cells§erroredÂŮ$0ccf77bd-0c5d-4c3c-82b5-b25c4adbc3baŚqueued¤logs§runningÂŚoutput¤bodyÚnPrerequisites
Analysis
Linear algebra
Exposure to numerical methods for solving differential equations (such as finite-element methods, finite-difference approaches, plane-wave methods)
Exposure to implementing numerical algorithms (e.g. using Python or Julia)
This course delivers a mathematical viewpoint on materials modelling and it is explicitly intended for an interdisciplinary student audience. To keep it accessible, the key mathematical and physical concepts will both be revised as we go along. However, the learning curve will be steep and an interest to learn about the respective other discipline is required. The problem sheets and the project require a substantial amount of work and feature both theoretical (proof-oriented) and applied (programming-based and simulation-based) components. While there is some freedom for students to select their respective focus, students are encouraged to team up across the discplines for the course work.
Past participants from materials science found it useful to take this course after they followed the lectures on Fundamentals of solid state materials.
Course outline
°persist_js_state¤mimeŠtext/html˛last_run_timestampËAÚş˛'ę>ˇhas_pluto_hook_featuresÂŹrootassigneeŔ§cell_idŮ$c0da4104-9a5a-4553-8df0-fdadd0194c14šdepends_on_disabled_cells§runtimeÎ üľpublished_object_keys¸depends_on_skipped_cells§erroredÂŮ$854d367b-1609-4c00-9c62-9439874a03d6Śqueued¤logs§runningÂŚoutput¤bodyÚ$EPFL resources
If you are an EPFL student, you can find the following course resources: