### Sciences

## Subject: METODI STOCASTICI PER SIMULAZIONI (A.A. 2020/2021)

### master degree course in MATHEMATICS

Course year | 1 |
---|---|

CFU | 6 |

Teaching units |
Unit Metodi stocastici per simulazioni
Models and Application (lesson)
- TAF: Compulsory subjects, characteristic of the class SSD: MAT/07 CFU: 6
Cecilia VERNIA, Claudio GIBERTI |

Exam type | oral |

Evaluation | final vote |

Teaching language | Italiano |

### Teachers

*Cecilia VERNIA*

*Claudio GIBERTI*

### Overview

The aim of the course is to provide an introduction to stochastic methods for simulations. Starting from the basic elements of Markov chain theory, students are introduced to the description, analysis and control of mathematical models for the evolution of discrete-time stochastic dynamical systems. In particular, both through the theoretical description of Monte Carlo methods and the presentation of some simple examples, the course will enable the student to choose the proper stochastic algorithm in applications.

### Admission requirements

None formally required.

The student is assumed to have a basic knowledge of probability theory and basic scientific programming skills.

### Course contents

Markov chain;

Computer simulation of Markov chain;

Stationary distributions;

Reversible Markov chain;

Markov Chain Monte Carlo (MCMC) methods;

fast convergence of MCMC algorithms;

Metropolis algorithm;

Simulated annealing;

Boltzmann machine learning.

### Teaching methods

Class lectures with presentation of the content chapters by means of blackboard and slides.

### Assessment methods

Oral final exam on the topics of the section “Course Syllabus”. The exams might have to be done either in presence or in online form, depending on the evolution of COVID19 epidemic.

### Learning outcomes

Knowledge and understanding:

At the end of the course the student will have the basic knowledge and in-depth understanding of the stochastic methods for simulations, with particular regard to the Monte Carlo Markov Chains.

Applying knowledge and understanding:

At the end of the course, the student will have adequate knowledge to use stochastic methods for the simulation of problems arising from real applications..

Making judgements:

At the end of the course the student will be able to choose the appropriate methods for dealing with applicative problems.

Communicating skills:

At the end of the course the student must be able to explain the analyzed methods in a clear and rigorous way and to discuss their efficiency.

Learning skills:

At the end of the course the student will be able to go deep by himself into the main aspects of the subjects proposed in the course.

### Readings

- Appunti forniti dal docente;

- M.E. J Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press (1999).

- D. P. Landau, K. Binder, A guide ti Monte Carlo Simulations in Statistical Physics (Fourth Edition), Cambridge University Press (2015)