# MODELING AND NUMERICAL SIMULATION

## Learning outcomes of the course unit

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.

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

Basics of calculus and linear algebra.

Basics of calculus and linear algebra.

## Course contents summary

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 could 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.

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 could 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.

## Recommended readings

Notes provided by the lecturer.

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.

The approach 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

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, whose discussion will be the starting point for the oral examination.

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, the discussion of which will be starting point for the oral examination.