Simulation and Monte Carlo Methods

Year
1
Academic year
2023-2024
Code
02003122
Subject Area
Physics
Language of Instruction
Portuguese
Other Languages of Instruction
English
Mode of Delivery
Face-to-face
Duration
SEMESTRIAL
ECTS Credits
6.0
Type
Elective
Level
2nd Cycle Studies - Mestrado

Recommended Prerequisites

Programming capabilities at an intermediate level.

Teaching Methods

Lectures use the blackboard and occasionally slide projection.  They intend to be a discussion of the subjects and they include examples; students are encouraged to participate in these discussions. Examples discussed in lectures can and will, whenever possible, include case studies and typical applications, either in Physics or in other subjects.
We also aim to develop students creativity and curiosity by encouraging them to suggest ideas, themes, problems to be solved, etc.
Typical Monte Carlo dealt situations are also described and studied.

Learning Outcomes

1. The student should become aware of the limitations of pseudo-random numbers and of the different architectures of random number generators.
2. He must understand how Monte Carlo simulation works and how and when to apply it.
3. He should be able to simulate from a sample and to anticipate, through simulation, the response of a system.
4. He should also be able to model a physical process to predict and reproduce the outcome of a system.

Work Placement(s)

No

Syllabus

Monte Carlo method: Definition and implementation.
Random numbers – requirements. Types of random number generators. Tests.
Probabilities: discrete, continuous and cumulative. Uniform distribuition. Non-uniform distributions: exponential and Gaussian.
Change of probability density: inversion and Box-Muller methods. Change of variables.
Monte Carlo integration.
Importance sampling
Finding the root of equations: successive substitutions, Newton-Cotes, half intervals and regula-falsi methods.
Interpolation methods: polynomial interpolation, Lagrange formula and piecewise interpolation.
Finding the roots of functions: Newton-Raphson, secant, bisection and regula-falsi
Integration methods: Newton-Cotes quadrature (open and closed), middle point, trapezoid and Simpson rules. Gaussian quadrature.
Integration of differential equations: Euler, Neuer and Runge-Kutta methods.
Random walks, Markov chains, Metropolis algorithm. Examples and applications.

Head Lecturer(s)

Jaime Pedro Oliveira da Silva

Assessment Methods

Assessment
Presentation by the student of a topic included in the syllabus: 10.0%
Resolution Problems: 40.0%
Project: 50.0%

Bibliography

- Knuth, The Art of Computer Programming, 3rd vol, Addison-Wesley, 1999.
- Press et al., Numerical Recipies in c, Camb. Univ. Press, 1992.
- Wong, Computational Methods in Physics and Engineering, 2nd ed, Prentice-Hall, 1997.
- R. Gaylord, P. Wellin, Computer Simulations with Mathematica, Springer, 1995.