Georgia Institute of Technology College of Sciences

Inpainting, deblurring and denoising images are common tasks required for a number of applications in science and engineering. Since the seminal work of Rudin, Osher and Fatemi, image regularization by total variation (TV) became a standard heuristic for achieving these tasks. In this talk, I will introduce the TV regularization model and some connections with sparse optimization and compressed sensing. Later, I will summarize some of the fastest existing methods for solving TV regularization. Motivated by improving the super-linear (on the dimension) running time of these algorithms, we propose two heuristics for image regularization models: the first one is to replace the TV by the \ell^1 norm of the Laplacian, and the second is a new, to the best of our knowledge, approximation of the TV seminorm, based on a redundant parameterization of the gradient field. We prove that the latter regularizer is an O(log n) approximation of the TV seminorm. This proof is based on basic techniques from Discrete Fourier Analysis and an estimate of the fundamental solutions of the Laplace equation on a grid, due to Mangad. Finally, we present preliminary computational results for the three models, on mid-scale images. This talk will be self-contained. Joint work with Arkadi Nemirovski.

We prove an a-posteriori KAM theorem which applies to some ill-posed Hamiltonian equations. We show that given an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, there is a true solution nearby. Furthermore, the solution is "whiskered" in the sense that it has stable and unstable directions. We do not assume that the equation defines an evolution equation. Some examples are the Boussinesq equation (and system) and the elliptic equations in cylindrical domains. This is joint work with Y. Sire. Related work with E. Fontich and Y. Sire.

I will discuss regularity of fully nonlinear elliptic equations when the usual uniform upper bound on the ellipticity is replaced by bound on its $L^d$ norm, where $d$ is the dimension of the ambient space. Our estimates refine the classical theory and require several new ideas that we believe are of independent interest. As an application, we prove homogenization for a class of stationary ergodic strictly elliptic equations.

We use a problem in arithmetic dynamics as motivation to introduce new computational methods in algebraic number theory, as well as new techniques for studying quadratic points on algebraic curves.