Ferrara

Variational methods for imaging

One of the most difficult challenges in scientific computing is the development of algorithms and software for large scale ill-posed inverse problems, such as imaging denoising and deblurring. Such problems are extremely sensitive to perturbations (e.g. noise) in the data. To compute a physically reliable approximation from given noisy data, it is necessary to incorporate appropriate regularization into the mathematical model. Numerical methods to solve the regularized problem require effective numerical optimization strategies and efficient large scale matrix computations. In these lectures we describe first and second-order methods, dual or primal-dual approaches, and Bregman-type schemes and how to efficiently implement the ideas with iterative methods on realistic large scale imaging problems.

Docenti: Valeria Ruggiero - Luca Zanni

Periodo: 18-20 Gennaio 2016 (15 ore)

Sede: Ferrara

 

Numerical methods for kinetic equations

In this course we consider the development and the mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. We review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic preserving methods and the construction of hybrid schemes.

Lecture I: Preliminaries on kinetic equations
Lecture II: Semi-Lagrangian schemes
Lecture III: Discrete velocity and spectral methods
Lecture IV: Breaking complexity: fast algorithms
Lecture V: Asymptotic-preserving schemes
Lecture VI: Fluid-kinetic coupling and hybrid methods
References
[1] G. Dimarco, L. Pareschi. Numerical methods for kinetic equations, Acta Numerica, 23 (2014), pp. 369–520.

 

Docente: Lorenzo Pareschi

Periodo: da stabilire con il docente 6/8 ore

 

Birational geometry

Abstract: An introduction to  MMP, rational connection, rationality and unirationality and their interplay in the realm of algebraic varieties.

Docente: Massimiliano Mella

Periodo: gennaio febbraio, 10 ore

 

http://docente.unife.it/claudia.menini/dispense/co-monads-and-descent/folder_contents​Monadi e loro applicazioni

Docente: C. Menini
Abstract: Beck's Theorem and its applications. Descent data, symmetries and connections associated to a monad.
Durata: 10-12 ore per un valore di 4 CFU.
Periodo: Seconda metà di Gennaio e Febbraio (indicativamente).
Lecture notes

 

 

 

Infinite Dimensional Analysis

docente A. Lunardi, M. Miranda

abstract This is an introductory course about analysis in abstract Wiener spaces, namely separable Banach or Hilbert spaces endowed with a nondegenerate Gaussian measure. Sobolev spaces and spaces of continuous functions will be considered. The basic differential operators (gradient and divergence) will be studied, as well as the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup, that are the Gaussian analogues of the Laplacian and the heat semigroup. The most important functional inequalities in this context, such as Poincare' and logarithmic Sobolev inequalities, will be proved. Hermite polynomials and the Wiener chaos will be described.

periodo:  prima fase ottobre 2015- febbraio 2016 seconda fase marzo 2016- maggio 2016 workshop finale giugno 2016.

 

Traveling waves in reaction-diffusion equations and in hyperbolic systems of conservation laws (a.a 14/15)

 

First part: Traveling waves in reaction-diffusion equations. Population dispersal models. Reaction-diffusion equations. Mono-stable and bi-stable reaction terms.  Degenerate diffusivities. Convective behaviors. Traveling wave solutions. Admissible speeds. Comparison-type techniques for degenerate boundary value problems. Asymptotic speed of propagation. (Luisa Malaguti, 15 hours)

 

Second part: Traveling waves in hyperbolic systems of conservation laws. Weak solutions of systems of conservation laws. Entropy solutions. The Riemann problem. An application to Euler equations. Traveling waves as viscous profiles of shock waves. Sketch of the proof of the construction of a solution by the vanishing viscosity method. (Andrea Corli, 15 hours)

Docenti: Andrea Corli (Ferrara), Luisa Malaguti (Modena)

Periodo: tra marzo e giugno, 30 ore (15 + 15)

 


Calculus of Variations and Geometric Measure Theory; application to the theory of BV functions. (a.a 14/15)

Docente
M. Miranda

Abstract
This course is an introduction to the theory of functions of bounded variations; the target is to describe fine properties of BV functions and set with finite perimeter using tools of geometric measure theory. These properties shall be used to prove properties of traces of BV functions on rectifiable codimension one surfaces. We shall also see applications to some PDE equations and minimization of some functional of the calculus of variations.

Periodo
Gennaio-Marzo 2015
Durata:
20 ore