72687 - Applied Mathematics (2nd cycle)

Course Unit Page

SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Industry, innovation and infrastructure

Academic Year 2021/2022

Learning outcomes

The student will have knowledge of Physical and Engineering models formulated through partial derivatives and operations research techniques. In particular the student will be able to: i) identify and manage some well defined problems formulated through differential equations of the second order of parabolic type (diffusion), elliptic (Laplace/Poisson), hyperbolic (wave equations); ii) formulate a decision and optimization model and analyze its complexity. The student will be able to define advanced solution algorithms for these problems.

Course contents

Module 1 (Optimization):

  • Models of optimization and decision problems.
  • Linear and Nonlinear optimization Models.
  • Computational complexity.
  • Heuristic algorithms for linear and non linear problems.
  • Construnction Algorithms
  • Local Search Methods
  • Classical MetaHeuristics:Tabu Search, Simulated Annealing, Genetic Algorithms
  • Iterated Local/Tabu Search
  • Variable Neighborhood Search
  • Very Large Neighborhood Search/Ruin and Recreate
  • Local Search methods

Use of the proposed techniques to solve families of optimization problems in specific application areas (Network design, frequency assignment, packing, location...) 

Module 2

1. Diffusion. The heat/diffusion equation with one-dimensional space variable. Dirichlet, Neumann and Cauchy problems. Existence, uniqueness and continuos dependence of solutions.

2. Laplace/PoissonHarmonic functions of two real variables, average formula, maximum principle. Dirichlet and Neumann problem for the Laplace operator in a circle.

3. Waves and vibrationsThe D’Alembert equation with one-dimensional space variable. Dirichlet, Neumann and Cauchy problems. Existence, uniqueness and continuos dependence of solutions.

Readings/Bibliography

Slides of the lectures available online

Teaching methods

Frontal lectures and exercises

Assessment methods

Module 1

The exam aims to test the understanding of course content through the development of a project. The project can also be carried out in groups with a maximum of two students. Groups must register by the end of the module by sending an email to the lecturer. The teacher assigns the group a specific problem and the objectives to be achieved by a fixed date, usually the first exam date after the conclusion of the course. Students can interact with the teacher during this period by appointment. At the end of the period, students must submit to the teacher a short report that describes the project and summarizes the results achieved. The teacher evaluates the report and fixes a meeting with the students in which the content is examined and it is verified the individual contribution of each student to the results achieved.

For students who do not attend to the course and do not carry out the project during the course it will be defined by the teacher a specific timing according to the specifications above.

The evaluation of the module is expressed as a mark in thirtieths.

Module 2

The exam aims to test the understanding of course content through an oral test including theory questions and exercises.

The evaluation of the module is expressed as a mark in thirtieths.

Overall evaluation

The overall mark is the average of the marks obtained in the two modules.

Teaching tools

Teaching Material, slides and exercise available online

Office hours

See the website of Daniele Vigo

See the website of Eleonora Cinti