# MODELLING AND NUMERICAL SIMULATIONS

## Learning outcomes of the course unit

Students are supposed to understand the power of modeling and simulation techniques. They should distinguish cases in which a problem can be directly simulated and cases in which a modeling phase is compelling, capturing the relevant degrees of freedom (typically in the form of macrostates) and finally accomplishing a problem they can tackle by computational tools. They should be able to quote an error for sample averages, recognizing statistically independent and dependent data. They should be able to validate simple hypotheses, which boil down to alternatives (YES/NO). They should fairly master the Matlab basic tools. They should be fairly able to present results of their work to colleagues.

The course aims at providing an elementary introduction to modeling and numerical simulations techniques, which are of common usage in Computational Physics. Though born in the framework of scientific applications, these techniques are as a matter of fact of general relevance. As a results, they proved to be effective in a variety of fields (economics and finance, computer networks, computational biophysics). In view of this, the course will be to a large extent a collection of topics presented in a seminar style. On top of providing conceptual and technical tools, the course will aim in the end at tackling a project, in which the students will finalize one of the simulations introduced during the classes. This activity will be the subject of the final examination.

## Prerequisites

Basic knowledge of algebra and calculus.

No prerequisites.

## Course contents summary

First content will the basics of probability theory and statistics, with an emphasis on numerical techniques (probability functions generation, data analysis). The problem of validating an hypothesis will be treated with a minimal rigour. A large fraction of the course will be devoted to applications of Markov processes theory. Modeling of queues will be the main application of the formalism. Simple examples of dynamic MonteCarlo will be proposed to Physics students (if interested in). An elementary introduction to stochastic differential equations will be proposed to Maths students (if interested in), in particular facing Langevin equation (brownian motion and tree-cutting problem). Basics of percolation theory will be introduced as an example of how a simple model can model a variety of phenomena.

First of all, we will aim at introducing the basics of probability theory and statistics, with an emphasis on numerical techniques (probability functions generation, data analysis). Data analysis will give the chance to introduce modeling in the simple form of data fitting. A large fraction of the course will be devoted to applications of Markov processes theory. Modeling of queues will be the main application of the formalism. Simple examples of dynamic MonteCarlo will be proposed to Physics students (if any). An elementary introduction to stochastic differential equations will be proposed to Maths students (if any), in particular facing Langevin equation (brownian motion and tree-cutting problem). Basics of percolation theory will be introduced as an example of how a simple model can model a variety of phenomena.

## Course contents

Basics of combinatorics and probability theory.

Bayes formula.

Binomial, Poisson, gaussian, exponential distributions.

Validation of hypotheses.

Cebysev inequality, law of large numbers and central limit theorem.

Evaluation of mean and variance

Markov processes.

The queue as a Markov process.

Percolation models (optional)

## Recommended readings

Notes provided by the lecturer, available on the ELLY platform.

Notes provided by the lecturer.

## Teaching methods

Style will be mostly informal, giving emphasis to problem solving. In view of this, every subject will be tackled also via numerical experiments. Students will asked to have always with them their laptop (if the have one). All the materials will be made available on the ELLY platform (codes, matlab session files, notes by the lecturer).

Style will be mostly informal, giving emphasis to problem solving. In view of this, every subject will be tackled also via numerical experiments.

## Assessment methods and criteria

A self-evaluation test on probability theory will be offered to students. Before the exams session students will be assigned a project. This will be the follow-up of work done during the term, with a clear indication of assignments, i.e. what the student is supposed to work out: numerical tests and simulations, final completion of a few analytical results worked out by the lecturer, comparison of simulation results and expected results, computation of errors. Students will return a written report (roughly 24 hours before the exam), whose discussion will be the core of the oral exam. The exam is oral as it is mainly the discussion of the students reports on their projects; some basic work on numerical data presented by the student could be involved (e.g. showing a code at work).

Evaluation will be in part in itinere, via the assignement of numerical exercises. At the end every student will be assigned a problem to be solved via numerical simulations. Students will present their solution toghether with a report.