MENU

You are here

Home

Mathematical Analysis, Modelling, and Applications

The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The present themes are multiscale analysis, mechanics of materials, micromagnetics, modelling of biological systems, and problems related to control theory.The applications of mathematics developed in this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming from the real world, from industrial applications, and from complex systems.

Topics in continuum mechanics

This is a 60-hours introductory course on continuum mechanics and its applications. The aim is to provide first year students with a solid understanding of the fundamental principles of the subject.

Advanced geometry 2

 

Smooth manifolds and differential topology.

 

Topics in computational fluid dynamics

Topics/Syllabus

  • Introduction to CFD, examples.
  • Constitutive laws
  • Incompressible flows.
  • Numerical methods for potential and thermal flows
  • Boundary layer theory
  • Thermodynamics effects, energy equation, enthalpy and entropy
  • Vorticity equations
  • Introduction to turbulence
  • Numerical methods for viscous flows: steady Stokes equations
  • Stabilisation (SUPG) and inf-sup condition

Introduction to geometric control

The course of 10 lectures will provide an introduction to geometric control theory. The first part of the course will be devoted to controllability, the second part will discuss stabilization, while the last part will focus on optimal control. No prior knowledge of control theory is required.

Course program:
1. Some basic questions in the control formalism, some examples of control systems.
2. Controllability of linear systems. Lie brackets and their relation with controlled motions.

Invariant manifolds for PDEs and some applications

Invariant manifolds are fundamental tools in the study of dynamical systems generated by differential equations. They provide coordinates in which the systems can be partially decoupled and can be used to track the asymptotic behaviors of the orbits. Therefore, starting with Poincare, Hadamard, Lyapunov, Perron and et al., people have studied extensively their existence, smoothness, and persistence under small perturbations (such as those due to the modelling procedure,  small noises, or computational round-off error, etc.).

Singular perturbations in fractional Sobolev spaces

Singular perturbations are a classical way to tackle problems where either a solution is not ensured by a lack of coerciveness, or there are too many solutions, typically due to a lack of strict convexity.

Advanced reduced order models in scientific machine learning

This course aims to provide a wide overview on novel strategies combining ideas from Reduced Order Modeling (ROM) and Scientific Machine Learning (SciML). The main goal is to investigate the great potential and the possible limitations of state-of-the-art methodologies to efficiently retrieve solutions of parametrized PDEs for computational mechanics problems.

Pages

Sign in