MODELLING AND NUMERICAL SIMULATIONS
Learning outcomes of the course unit
By an elementary introduction to modeling and simulating techniques which are of common usage in Compuational Physics, the course aims at showing how these techniques turn out ot be powerful in a variety of applications. As a matter of fact fields of application range from Economics and market analysis to computer networks or Computational Biology.
First of all the course aims at triggering intellectual curiosity. A sensible aim is then of course to provide basic tools in modeling techinques. These can be useful to approach more advanced studies, but also to address a variety of situations in which a computer scientist is required to elaborate a conceptual understanding of the data and information which are to be processed.
The students should have attended in past years courses in Physics and Probabilty and Statistics.
Course contents summary
The course aims at introducing the basics of modelling and simulating techniques which are of common usage in Computational Physics, but which also turn out to be very powerful in a variety of different applications.
The course is held in the computer lab. The main numerical tool will be the Matlab programming enviroment.
Main topics will be the following:
A short, informal summary of Probability Theory and Statistics. Stochastic variables probability distributions: flat distributions and pseudo-random number generators. The gaussian distribution. General techniques to generate numbers according to a given distribution.
Error analysis. Samples analysis. The bootstrap method. Data fitting. Basics of data mining (this is an extra, subject to time availability).
Basic introduction to Stochastic Differential Equations. The free Brownian Motion (and, subject to time availability, the general case of external forces plus gaussian noise): the Langevin Equation. Basics on different applications of the Langevin Equation.
Markov Chains and Dynamic Montecarlo methods. Introduction to the simulation of queues. Basics of Statistical Mechanics simulations. Molecolar Dynamics and its applications (this is an extra, subject to time availability).
Choice of a final simuation project (upon which the students and the lecturer will agree) which will be the subject of the final part of the course. Possible choices include:
simulating percolation (and its possible applications);
stochastic methods for the solution of the tree-cutting problem;
simulation of simple queue models.
The choice of the final project (which will be eventually further developed by the students for their final examination) crucially depends on how the program can be covered in the lectures, which in turns depends on time availability and acquired skills.
N.B.: The course qualifies as advanced because students can address advanced subjects: rigour will anyway always give way to intelligibility.
Students will mainly make use of notes provided by the lecturer (made available on the web).
For the introductory part of the course students can find it useful (but not essential) to refer to:
E. Ventsel, Teoria della Probabilità (ed MIR)
J. Taylor, Introduzione all'analisi degli errori (ed Zanichelli) (original english version available)
References for the second half of the course are available on the web (preprints eletronic archives) and will be indicated in case they are needed.
Main emphasys will be on aquiring problem solving skills. In view of this lectures will always be supported by numerical solutions to problems.
Final examination will consist in providing a simulation project. Students will be required to finish on their own the project which will be picked for the final part of the course.