NUMERICAL ALGORITHMS

This course will enable each student to understand the meaning and the potentialities of simple numerical algorithms; the students will learn to implement those Algorithms, and to perform simple correctness tests.

Each student can handle basic Numerical Methods. He is able to implement in Matlab simple Numerical Algorithms.

Each student must know the fundamental concepts of Infinitesimal Calculus in one and more variables and Linear Algebra.

Floating-point numbers. Notions of programming in MATLAB
- Numerical solution of nonlinear equations: Picard and Newton methods
- Data approximation and numerical interpolation
- Numerical integration
- Numerical methods for the solution of ordinary differential equations: Euler methods, Cranck-Nicolson, Runge-Kutta methods
- Finite difference and finite element methods for the solution of elliptic partial differential equations
- Conjugate gradient method for the solution of linear systems

A. Quarteroni, F. Saleri, e P. Gervasio. Scientific Computing with MATLAB and Octave. Springer Verlag, 2010.

During the course it will be required to carry out four assignments in which the students will implement and test the theoretical methods seen in class.

1) The first assignment will deal with a marketing problem that can be solved through a non-linear equation in one variable. Students will have to compare different numerical algorithms to solve this problem. In the second part of the assignment students will have to extend the algorithms implemented to the case in more variables.

2) The second assignment will concern the methods for data approximation and interpolation, with a particular application to the epidemiological data of COVID-19 collected in Italy in the first months of the epidemic.

3) The third assignment will concern the comparison of the methods analyzed in class for the solution of a high-dimensional, sparse linear system.

4) The fourth assignment will focus on the mathematical modeling of an epidemic through a system of non-linear differential equations. Students will first have to compare the different numerical integrators seen in class on a linear test equation, and then use these solvers for modeling the first months of the COVID-19 epidemic in Italy.

For each assignment, students must submit a report presenting the results obtained.

During the oral exam students will be asked to discuss the results presented in the reports. The student's ability to link these results to the numerical properties of the algorithms used will be verified. In particular, students will be required to write the mathematical equations of these algorithms and to recall the main steps of the proofs on their properties.

Classroom lessons.
PC-based activities.
The moodle platform is exploited in order to propose assignments, and supplementary material.

English

Classes will be in English

written and oral