This course starts from explaining internal circuits and digital processing inside a computer in order to understand possible sources of numerical calculation errors. Then fundamental mathematics and representative algorithms of numerical analyses will be reviewed, which includes differentiation, integration, equation solvers, optimization, Fourier transform, linear algebra etc. It covers those applications such as molecular dynamics, first-principles quantum calculations, finite element method, and phase-field method.
At the end of this course, students will be able to:
1) Understand what is going in a computer and know how to estimate possible errors in numerical calculations.
2) Understand fundamental ways of thinking how to perform differentiation, integration, optimization etc with assistance of a computer, and learn representative algorithms.
3) Learn fundamental physics and mathematics of microscopic simulations such as molecular dynamics and first-principles calculations, and understand to what problems they can be appropriately applied.
4) Learn fundamental physics and mathematics of macroscopic simulations such as finite element method and phase-field method, and understand to what problems they can be appropriately applied.
Numerical analysis, Molecular Dynamics, First-principles calculation, Density functional theory, Electronic structure, Finite element method, Phase field method, micro-structure
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Questions are accepted anytime during each class.
Course schedule | Required learning | |
---|---|---|
Class 1 | Fundamental of digital processing and numerical errors in computer | Understand the circuits and operation fundamental of computer, and possible error sources of numerical calculations |
Class 2 | Numerical differentiation, integration, solution of differential equation | Understand difference method and its applications to differentiation, integration, and differential equation |
Class 3 | Applications using solvers of differential equation, interpolation, smoothing | Learn examples of applications of differential equation solvers. Understand fundamentals of data processing such as interpolation and smoothing. |
Class 4 | Linear least-squares method, optimization, solution of complex equation | Understand ways of thinking for optimization and learn representative algorithms. |
Class 5 | Non-linear least-squares method, Fourier transform, linear algebra | Learn fundamentals and algorithms of non-linear optimization and Fourier transform. Representative linear algebra used in programming will also be introduced. |
Class 6 | Monte Carlo Method I | Understand numerical methods (Monte Carlo method) that use random numbers, and its application to higher-dimensional integrals. |
Class 7 | Monte Carlo Method II | Understand Markov chain, Metropolis algorithm for simulating statistical properties. |
Class 8 | Finite Element Method I | Understand the formulation of finite element method for solving Poisson's equations. |
Class 9 | Finite Element Method II | Understand the application of finite element method for mechanics, heat, electronics, and chemistry. |
Class 10 | Phase-Field Method | Understand the phase-field method that is used to describe microstructure evolution such as crystal growth and grain growth. |
Class 11 | Molecular Dynamics: Fundamentals | Learn molecular dynamics as a simulation technique for the properties of an ensemble of classical particles. |
Class 12 | Molecular Dynamics: Applications | Overview several examples of molecular dynamics simulations of gases, liquids, and solids. |
Class 13 | First Principles Calculations: Fundamentals | Learn first principles electronic state calculations based on the density functional theory. |
Class 14 | First Principles Calculations: Applications | Overview several basic examples of first principles electronic state calculations (charge distribution, band dispersions, density of states, Fermi surface, etc.). |
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook will be specified at the class. Related text and materials will be distributed.
None required
Students will be evaluated by a term-end examination
Fundamental mathematics studies such as differential calculus, integral calculus, and linear algebra
Fundamental physics and chemistry studies such as classical dynamics and quantum mechanics