Courses in Ferrara
Syllabus: The course consists of a weekly meeting, aimed at presenting and discussing some fundamental research papers on the topics of Mathematical Analysis. Some examples of topics that will be considered are:
- variational methods
- geometric and regularity properties of solutions to elliptic equations
- Gamma-convergence and applications
- functional inequalities
The course will consist of approximately 10 meetings of 2 hours each. During each meeting the participants will deliver a talk on a paper (or a series of papers) proposed by the teachers, related to the aforementioned topics.
Dates 2021/2022: November 2021/February 2022. In case of problems due to the pandemics, the course will be shifted to March 2022/May 2022
Title and Credits: Steady Problems in Fluid Dynamics and Elasticity, 3 CFU
Teacher: Vincenzo Coscia
Syllabus: Six lectures of approximately 2.5 hours each.
- Steady problems for the Navier-Stokes equations: basic questions and open problems.
- The functional spaces of hydrodynamics. Hydrodynamic potentials.
- The boundary value problem for the Stokes equations. Existence and uniqueness in bounded and exterior domains.
- The boundary value problem for the Navier-Stokes equations in bounded and exterior domains.
- The boundary value problem in elastostatics.
- Recent results on steady problems in fluid dynamics and elasticity in bounded and exterior domains.
The students will be required to participate in the course solving the assigned exercises. At the end of the course each participant will have to carry out a seminar on a prescribed topic.
Dates 2021/2022: March-April 2022
Title and Credits: BV functions and applications to variational problems; Mumford-Shah, 4 CFU
Teacher: Michele Miranda
Syllabus: This course is an introduction to the theory of functions of bounded variations; we describe fine properties of BV functions and sets with finite perimeter using tools of geometric measure theory. We shall prove the decomposition of the total variation measure defined by a BV function. The notion of special functions with bounded variation, SBV functions, will be introduced and its characterisation via the chain rule will be given; we shall prove closure and compactness results of SBV functions. These properties will be used in the study of the Mumford-Shah functional that has applications in variational problems with free discontinuities (for instance, image reconstruction). Then the notion of Gamma-convergence will be introduced and the Ambrosio-Tortorelli approximation of the Mumford-Shah functional will be described.
Dates 2021/2022: January-February 2022, around 20 hours
Syllabus: The course aims to provide an introduction to numerical methods for uncertainty quantification with specific reference to PDEs. After defining the main concepts in the field of uncertainty quantification, including some references to probability theory, the course focuses on two main approaches. The Monte Carlo method, in its variants characterized by multi-fidelity techniques, and the methods based on generalized polynomial chaos expansions, both in intrusive and non-intrusive form. Specific applications to the case of hyperbolic systems with relaxation terms and reaction-diffusion equations will be considered. In-depth study by students through specific reading of articles will also be suggested.
Dates 2021/2022: 12h lectures + reading course + home assignments
Title and credits: Computational intelligence and gradient-free optimization, 3 CFU
Syllabus: This course provides an introductory overview of key concepts in computational intelligence with a focus on metaheuristic methods for global optimization. These include Genetic Algorithms (bitstring and integer vector genotype representations) and Particle Swarm Optimization (constrained PSO, quantum-inspired PSO, and a multi-swarm version of quantum-inspired PSO), extended with adaptation mechanisms to provide support for dynamic optimization problems. The main algorithms will be illustrated with the help of simple implementations in Matlab and/or R language. In the last part of the course, using a mean-field approach, rigorous convergence results for some of the methods will be presented.
Dates: Around 12h lectures + 4h assignments, February-June 2022
Title and Credits: (Modal) Symbolic Learning, 2CFU+2CFU (optional, for some research work)
Teacher: Guido Sciavicco
Syllabus: Symbolic learning is the sub-discipline of machine learning that is focused on symbolic (that is, logic-based) methods. As such, it contributes to the foundations of modern Artificial Intelligence. Symbolic learning is usually based on propositional logic, and in part, on first-order logic. Modal symbolic learning is the extension of symbolic learning to modal (and therefore, temporal, spatial, spatio-temporal) logics, and it deals with dimensional data. In this course we shall lay down the logical foundations of symbolic learning, prove some basic properties, and present the modal extensions of classical learning algorithms, highlighting which ones of those properties are preserved, and which ones are not.
Dates 2021/2022: September 2022, 4 lectures, 8 hours