72687 - Applied Mathematics (2nd cycle)

Learning outcomes

At the end of the course, the student has in-depth knowledge of models of Physics and Engineering formulated both through partial differential equations and through Operations Research techniques. In particular, the student will be able to: i) recognize and treat some problems well posed for differential equations of the second order of parabolic (diffusion), elliptic (Laplace / Poisson), hyperbolic (wave equation); ii) formulate an optimisation and decision-making problem in an appropriate way and analyse its complexity. Finally, the student will be able to define models of optimization and decision problems and to define advanced algorithms for their resolution.

Course contents

Module 1 (Optimization):

• Models of optimization and decision problems.
• Linear and nonlinear optimization models.
• Computational complexity of problems and algorithms.
• Heuristic algorithms for linear and nonlinear problems
• Constructive algorithms
• Local search methods
• Classical metaheuristic algorithms: Tabu Search, Simulated Annealing, Genetic Algorithms
• Iterated Local/Tabu Search
• Variable Neighborhood Search
• Very Large Neighborhood Search/Ruin and Recreate

Use of the techniques described for the resolution of families of optimization problems in various application sectors (Network design, frequency assignment, packing, location ..)

Module 2

1. Diffusion. Model heat/diffusion equation in a space variable. Dirichlet and Neumann problems in the case of finite length and Cauchy problem in the case of infinite length. Existence, uniqueness and continuous dependence on data.

2. Laplace/Poisson equation. Harmonic functions in two variables, mean formulas and maximum principle. Dirichlet and Neumann problems for the Laplacian in the circle.

3. Waves and vibrations. D'Alembert model in a space variable Dirichlet and Neumann problems in the case of finite length and Cauchy problem in the case of infinite length. Existence, uniqueness and continuous dependence on data.

Readings/Bibliography

Teaching Material, slides and exercise 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.

Office hours

See the website of Daniele Vigo

See the website of Eleonora Cinti