# MODELLING AND NUMERICAL SIMULATIONS

## Learning outcomes of the course unit

At the end of the course the student will have better knowledge and understanding as she/he will: master the basics in probability theory; master the most basic foundations of statistical techniques like mean values and errors estimates, validation of hypotheses.

The student will be able to apply knowledge and understanding and in particular she/he will be able to: compute mean values and errors of a given data set; validate a simple hypothesis (which boils down to YES/NO alternatives) within a given confidence level; pin down the basic steps in setting up a simulation (singling out the relevant degrees of freedom, choosing a representation of the latter as data, choosing and implementing an algorithm for the simulation dynamic).

The student will be able to make judgements and in particular she/he will be able to: 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; understand whether the relevant degrees of freedom are to be looked for in the form of macrostates.

The student will also have acquired communication skills as she/he will be able to: present her/his results in a clean, precise and concise way; present her/his results both synthetically as for the overall picture and analytically as for the most delicate points; argue her/his thesis in public, in particular acting in a team.

Finally, the student will have acquired learning skills as she/he will be able to: understand whether numerical simulation solutions are due in the context of problems she/he will be facing in the context of future studies or work; progress in the study of solutions (e.g. algorithmic solutions) beyond what she/he has learnt in this course.

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. Exponential distribution for Poisson processes interarrival times.

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.

Informal introduction to percolation models and cluster finding algorithms.

## Recommended readings

Lecture notes provided by the lecturer, available on the ELLY platform.

A(ny) book on the theory of probability can be useful. One suggestion (due to its simplicity and clarity) is a book available in the library

- E.S. Ventsel, Teoria delle probabilità - Edizioni MIR

Notes provided by the lecturer.

## Teaching methods

Lectures and exercises (with students involved). Students will be asked to have their laptop always with them (if they have one).

Numerical work will be worked out in Matlab (with digressions in C/C++).

From time to time problems will be assigned to be worked out at home (proving a certain assertion, finishing a computation or pinning down a computer program). All the material will be made available on ELLY (codes, Matlab sessions diaries, lecture notes).

Style will be informal, always focusing problem solving. In this spirit, every subject will be supplemented with numerical experiments.

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

Roughly in the middle of term a self-evaluation test will be recommended, subject to compatibility with teaching stop dedicated to these activities (alternatively a homework will be evaluated). It will be about probability theory. That is mostly intended as self-evaluation, but if it is well done (and only in that case) it will be taken into account in the final grade (2 points added as a bonus).

Oral exam to which the candidate is admitted after having delivered a report on a project. In due advance of the exam session, a project will be assigned to the student. It will be a natural completion of subjects worked out during the lessons, with a clear assignement of what the student is supposed to do: numerical simulations, working out a few anaytical results completing results obtained during the course, comparison of expected results and results stemming from simulations, computations of errors. Students will present their written report at the latest 24 hours before the oral exam takes place.

Discussing the report will be the starting point of the oral exam. Besides a clean presentation of the technical solutions adopted and of the motivations for them, during the exam it could be required to reproduce a few results (for example: showing a program at work, verifying the correctness of a program). Both the report (and the results obtained) and the oral discussion will contribute to the final grade.

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.