| Date | Speaker | Title/Abstract |
|---|---|---|
| 5 Oct | Carola-Bibiane Schönlieb University of Cambridge |
Higher-Order PDEs and Variational Techniques for Image Restoration Restoring the original image contents from distorted measurements is one of the most important tasks in image processing. It comprises the enhancement and reconstruction of images distorted by noise or blur (image denoising/deblurring) and the filling-in of gaps in images (image inpainting). Within various standard methodologies for the solution of these tasks, partial differential equations (PDEs) and variational methods constitute a rich toolbox of restoration approaches. These techniques are interesting from both an applicational viewpoint – because they are able to produce qualitatively good visual results and can be captured within automatable processing algorithms – but also from a mathematical analysis point of view – because they show some beautiful mathematical concepts and pose interesting analytical problems. In this presentation we shall concentrate on a specific class of PDE & variational techniques, namely higher-order approaches, e.g., third- and fourth-order PDEs. After spending some time on introducing the concept of such methods and giving a historical overview of some important contributions in this area, we will get to know some recently proposed higher-order methods, their mathematical properties and applications. The presentation will be furnished by various numerical examples and applications for image restoration. |
| 19 Oct | Clément Mouhot University of Cambridge |
The Boltzmann equation with polynomial tails and the exponential H-theorem We shall present some recent joint results Gualdani and Mischler about the Cauchy theory of the Boltzmann equation for hard spheres in a periodic box. The main theorem is the existence and uniqueness of close-to-equilibrium solutions which are $L^1$ in velocity with polynomial moments and $L^\infty$ in space. As an application we answer a conjecture about the exponential H-theorem with sharp rate. It relies on new tools in spectral theory about high-order factorization of non-symmetric operators and quantitative estimates on their semigroups, as well as new technical estimates on the Boltzmann equation. |
| 30 Nov | Sergio Segura de León University of Valencia |
Bounded solutions to the 1-Laplacian equation with a critical gradient term The Dirichlet problem for an elliptic equation in a bounded domain is studied; the 1-Laplacian operator and lower order terms appear in the equation. We introduce a suitable definition of solution and prove the existence of a bounded solution, which is a function of bounded variation having a negligible jump part. Moreover, a uniqueness result for small positive data is obtained, and explicit examples of solutions are shown. This is a joint work with F. Andreu and A. Dall'Aglio. |
| 14 Dec | Yves Capdeboscq University of Oxford |
On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients In this talk, we show that a topological argument allows to extend uniqueness results known for sufficiently smooth coefficients to locally piecewise smooth coefficients within a bounded $C^0$ domain. This is a joint work with John M. Ball and Basang Tsering-Xiao. We will also comment on the (non)-existence of quantitative unique continuation results independent of the frequency for piecewise smooth coefficients. |