Monte Carlo-simuloinnit, 2+2 ov (4+4 op) / Monte Carlo simulations, 2+2 sw (4+4 op)

Course materials by Kai Nordlund, spring 2006 and earlier

• Basics of Monte Carlo simulations
• Monte Carlo simulations in physics

• Grunder för Monte Carlo-simuleringar
• Monte Carlo-simuleringar i fysik

Timetable

Exam for part I was Thursday Mar 16 10.00-14.00 in the Accelerator lab.

Lectures

I will try to put out the lecture notes here at least one day before the lecture. After the lecture I may still correct errors and put a new version here 'for the record'.

Part I

Part II

Exercises

Part I

• Data file bimodal.dat

• Data file dsa_nonegative.dat

Part II

• Coordinates for 20 largest Finnish cities (by year 2000 population)

• Sudoku example 1
• Sudoku example 2
• Sudoku example 3
• Sudoku example 4

• Sudoku example : pretty hard (not required)
• Sudoku example : really hard (not required)
• Sudoku example : really hard 2 (not required)

Exercise solutions

List of contents: "Basics of MC"

1. Introduction (1/2 hours)
   • Times, lectures, exercises

2. Monte Carlo - introduction (1 hour)
   • Definition
   • History: first MC simulation
   • Confusing terminology, and trying to avoid it
   • Even most MD is sort of MC!
   • List of some varieties of MC

3. Generating random numbers (4 hours)
   • Importance of doing this well
   • Testing generators: 2D, MC simulation, etc...
   • Generating a uniform distribution
   • Generating non-uniform distributions
     • Analytical approach
     • von Neumann rejection method
     • Special case: Gaussian distribution
   • Generating points on a surface of a sphere: the infamous pitfall
   • Non-random random numbers: these may sometimes be better

4. Monte Carlo integration (2-3 hours)
   • MC integration of many-dimensional functions
   • When is this advantageous compared to a grid?
   • Weight functions

5. MC simulation of experimental data (2 hours)
   • Quick-and-dirty: the bootstrap method
   • Test: how good is this actually?
   • Simulation of physical process

6. Cellular automata (2-4 hours)
   • Basic concepts
   • Numerical rule schemes
   • Applications: sand piles, forest fires etc.

List of contents: MC simulations in physics

1. Random walks: introduction (2 hours)
   • Drunken sailor in 1 and 2D
   • Relation to diffusion
   • Lattice gas
   • Length of polymers

2. Kinetic Monte Carlo (2 hours)
   • Motion of activated objects
   • Trivial and residence-time algorithm
   • Lattice and non-lattice
   • Example: defect migration in Si

3. MC simulation of ensembles (4 hours)
   • MC and statistical physics
   • Simulating atomic systems
   • Metropolis algorithm: NVT
   • Demon algorithm: NVE
   • NPT and muPT algorithms
   • Calculating thermodynamic quantities

4. Simulated annealing (1 hour)
   • Basic idea: Metropolis + T lowering (as in metals annealing)
   • example: Annealing a metal
   • example: Traveling salesman problem
   • Exercise: solve the salesman problem
5. Quantum Monte Carlo
   • Variational principle
   • Model case: particle in a box
   • Random Walk QMC

   Time permitting and according to attendees interests possibly also some of the following:

6. Ising model (1-2 hours) ?
   • Definition of problem, algorithm
   • Computer code

7. Solving equations (2-6 hours) ?
   • Linear sets of equations
   • Integral equations

8. Transport simulations (4-8 hours) ?
   • Relation to radiation effects
   • Simulating ionic collisions

Sisältökuvaus: osa 1 Opetusajankohta: kl. 2006 ti, to 10-12 (luennot pääasiassa tiistaisin, laskuh. torstaisin) 17.1-2.3. Ensimmäinen luento Ti 17.1. Paikka: Kiihdytinlaboratorion seminaarihuone (Pietari Kalmin katu 2) Luennoija: Prof. Kai Nordlund Assistentti: Eero Kesälä Laajuus: 2 ov, 4 op Kesto: 7 viikkoa Ajankohta opintojen ajoitusmallissa: cum laude tai erikoistumisvaihe Esitiedot: Matematiikka. Fortran-, C- tai java-kielen perusteiden tuntemus.

Simple forest fire MC simulation

Kurssilla käsitellään useilla luonnontieteen aloilla hyvin tärkeiden Monte Carlo-simulointimenetelmien perusteita. Kurssilla käsitellään satunnaislukujen generointia eri jakaumissa, Monte Carlo-integrointia ja sen virhelaskentaa, synteettisen datan generointia sekä soluautomaatteja

Lasku/ohjelmointiharjoitukset ja loppukoe.

Luentomonisteet.

Tukena voi käyttää useitakin kirjoja, kts. englanninkielinen osuus alla.

Kurssi pidetään englanniksi pyydettäessä. Laskuharjoituksiin ja kokeisiin voi vastata suomeksi, ruotsiksi tai englanniksi.

Description: part 1 "Basics of Monte Carlo simulations"

Description: part 2 "Monte Carlo simulations" in physics

During the course we will give an introduction to the so called Monte Carlo simulation methods which are very important in many branches of modern science.

Programming and mathematical exercises are given during the course, but not every week. They are graded by the lecturer For the more demanding exercises more than 1 week of solution time is given.

The programming exercises should be preferably solved in an Unix environment, but also solutions written under other environments in strict adherence to the Fortran90, ANSI C or java standards (so that they can be compiled anywhere) are acceptable.

Exercises (50 %)
Final exam (50 %)

Lecture notes.

Some parts of the material are well described in

• W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C; The Art of Scientific Computing, Cambridge University Press, New York, second edition, 1995
  
• M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford, England, 1989
  
• Gould, Tobochnik: An Introduction to Computer Simulation Methods : Applications to Physical Systems, Harvey Gould, et al

Taustatietoa/Background information

History of MC

• HISTORY OF MONTE CARLO METHOD by Sabri Pllana . Also has nice Buffon's needle java applet.
• History of simulations in Manhattan project (MC was according to this not used there).
• History of Metropolis MC and MD: An Interview with Bernie Alder

• A bit of history of random number generators

• The Mersenne twister source code in C
• The Mersenne twister source code in C, 2002 version
• The Mersenne twister source code in Fortran

• Program CA1.c from Gould-Tobochnik, graphics-free version
  
• Program CA1.f from Gould-Tobochnik, graphics-free version
  

• An article about Stephen Wolfram

• Jan Tobochnik
• Harvey Gould