## MODELLING AND NUMERICAL SIMULATIONS

Italian

### 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.

### Prerequisites

Basic knowledge of algebra and calculus. Minimal knowledge of probability theory.

### Course contents summary

Short review of probability theory and statistics, with an emphasis on numerical techniques (probability functions generation, data analysis).

The problem of validating a hypothesis will be treated with some attention to mathematical 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.

Basics of percolation theory will be introduced as an example of how a simple model can model a variety of phenomena, in particular epidemiological models.

### Course contents

Basic review of probability theory (Bayes formula; binomial, Poisson, gaussian and exponential distributions). Exponential distribution for Poisson processes interarrival times.

Validation of hypotheses.

Review of law of large numbers and central limit theorem (in view of statistics good practice). Evaluation of mean and variance.

Markov processes.

The queue as a Markov process.

Introduction to percolation models.

Cluster finding algorithms.

Simple epidemiological models.

### 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

### Teaching methods

Lectures and exercises (with students involved). Students will be asked to have their laptop always with them (if they have one). Classes will be given at the University premises, although students who will be unable to attend physically will be able to attend connecting via Microsoft Teams (streaming). In case of worsening of Covid-19 pandemic, lectures will be delivered via streaming on the Microsoft Teams platform (we will in any case comply to the official schedule).

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.

### Assessment methods and criteria

Roughly in the middle of term a self-evaluation test on probability theory and statistics will be recommended, subject to compatibility with teaching stop dedicated to these activities (alternatively a homework will be evaluated). 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 analytical results completing results obtained during the course, comparison of expected results and results stemming from simulations, computations of errors. Students will hand over 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.

Exams will take place at the university premises. Once again, in case of worsening of the Covid-19 pandemic, we will move to Microsoft Teams meetings. Students will make sure they can share their screen.