Analisi complessa e funzionale - Functional Analysis and Complex Analysis Nicola Arcozzi with Pavel Mozolyako ended
Integrazione gaussiana e meccanica statistica Pierluigi Contucci 15 h
Introduzione alla geometria di Cartan e sue applicazioni Emanuele Latini – Andrea Santi ended
Ottimizzazione convessa Alessandro Lanza 10 April 14-16 11 April 14-16 15 April 14-16 17 April 14-16 29 April 14-16 6 May 14-16 7 May 9-12 program: In this course, we give an overview of convex optimization methods for signal/image processing applications.
Topics which we would like to cover:
Variational methods for Imaging
Review on standard tools in convex optimization such as, e.g., (strong) convexity, sub-differentials, gradient descent.
Proximal methods for non-smooth optimization (acceleration à la Nesterov)
Alternating Direction Method of Multipliers (ADMM) and Majorize-Minimize approach.
Numerical implementation and simulations in MATLAB for exemplar imaging problems.
Short cycles of specialistic seminars
Topics in Math Organizers: Giovanni Cupini and Giovanni Mongardi 20 h
Metodi variazionali e PDE per l’elaborazione delle immagini Luca Calatroni 10 - 12 a.m. room Seminario I on May 9, 10, 13, 14, 15, 16 and 17; 2 - 3 p.m room Seminario I on May 15. In this course we will present some classical and recent approaches for some problems in image reconstruction (denoising, deblurring, inpainting, shadow-removal…) formulated in terms of appropriate minimisation problems in infinite-dimensional functional spaces. We will further draw connections between these minimisation problems and parabolic Partial Differential Equations (PDEs) based on non-linear diffusion and possibly combined with transport terms. For the practical implementation of the models above, we will review standard finite difference stencils discussing their extensions to anisotropic diffusion and diffusion-transport problems. The course will be complemented by some practical MATLAB classes where simple exemplar problems will be solved by means of some reference iterative algorithms.
Classical examples of imaging problems (denoising, deblurring, inpainting, segmentation..). Formulation as ill-posed inverse problems. Variational regularisation methods: regularisation term VS data fitting. Statistical interpretation: MAP estimation (2h)
Sobolev spaces, standard methods in calculus of variations: a review. Total variation, the space of functions of bounded variations (2h)
Second-order parabolic PDEs for image processing: heat equation, mean-curvature flow. Applications to image processing: linear VS non-linear PDEs. Regularisation of non-smoothness: lagged diffusivity. Anisotropic diffusion and diffusion-transport problems. (4h)
Finite differences stencils for PDE-based imaging models. (2h)
Numerical implementation and simulations in MATLAB for PDE-based models for image reconstruction (deblurring, inpainting, face fusion). (5h)
Microlocal methods for dynamical systems Maciej Zworski June 3, 4, 6 and 7 / 10:00 -13:00 Scattering resonances replace bound states/eigenvalues for spectral problems in which escape (scattering) to infinity is possible. These states have rates of oscillation and decay and that information is elegantly encoded in considering the corresponding ``eigenvalues" as poles of the meromorphic continuation of Green functions. The most famous ``pure maths" example is given by zeros of the Riemann zeta function which can be interpreted as resonances for scattering on the modular surface. In ``applied maths" they appear anywhere from gravitational waves to MEMS (Micro-Electro-Mechanical Systems). The mini course will provide a gentle introduction in the setting of potential scattering in dimension three. Only basic functional analysis will be a prerequisite. 1. One dimensional scattering: intuition behind outgoing and incoming waves and the definition of scattering resonances. 2. Analytic Fredholm theory and, as application, meromorphic continuation of Green's function for potentials scattering in dimension three. 3. Resonance free regions and expansion of waves in terms of resonances. 4. Counting resonances: upper bounds and existence (and some open problems). Complex valued potentials with no resonances. Section 2 of https://math.berkeley.edu/~zworski/revres.pdf (Bull Math Sci '17) will provide a reference with a more detailed presentation in the forthcoming book http://math.mit.edu/~dyatlov/res/ (AMS '19, to appear).
The de Branges theory of Hilbert spaces of entire functions and its applications to spectral theory of differential operators Anton Baranov info and schedule
Topics in global analysis Gerardo A. Mendoza 14 h - May 15 - 30, 2019.