Georgia Institute of Technology College of Sciences

The most popular model for Image Denoising is without any doubt the ROF (for Rudin-OsherFatemi) model. However, since the ROF approach has some drawbacks (the stair-case effect being one of them) practitioners have been looking for alternatives. One of them is the Elastica model, relying on the minimization in an appropriate functional space of the energy functional $J$ defined by

$$ J(v)=\varepsilon \int_{\Omega} \left[ a+b\left| \nabla\cdot \frac{\nabla v}{|\nabla v|}\right|^2 \right]|\nabla v| d\mathbf{x} + \frac{1}{2}\int_{\Omega} |f-v|^2d\mathbf{x} $$

where in $J(v)$: (i) $\Omega$ is typically a rectangular region of $R^2$ and $d\mathbf{x}=dx_1dx_2$. (ii) $\varepsilon, a$ and $b$ are positive parameters. (iii) function $f$ represents the image one intends to denoise.

Minimizing functional $J$ is a non-smooth, non-convex bi-harmonic problem from Calculus of  Variations. Its numerical solution is a relatively complicated issue. However, one can achieve this task rather easily by combining operator-splitting and finite element approximations. The main goal of this lecture is to describe such a methodology and to present the results of numerical experiments which validate it.

Kinetic equations play an important role in multiscale modeling hierarchy. It serves as a basic building block that connects the microscopic particle models and macroscopic continuum models. Numerically approximating kinetic equations presents several difficulties: 1) high dimensionality (the equation is in phase space); 2) nonlinearity and stiffness of the collision/interaction terms; 3) positivity of the solution (the unknown is a probability density function); 4) consistency to the limiting fluid models; etc. I will start with a brief overview of the kinetic equations including the Boltzmann equation and the Fokker-Planck equation, and then discuss in particular our recent effort of constructing efficient and robust numerical methods for these equations, overcoming some of the aforementioned difficulties. This is joint work with Ruiwen Shu (University of Maryland).

The talk is concerned with low multilinear rank approximations to antisymmetric tensors, that is, multivariate arrays for which the entries change sign when permuting pairs of indices. Such tensors play a major role in quantum chemistry. We show which ranks can be attained by an antisymmetric tensor and discuss the adaption of existing approximation algorithms to preserve antisymmetry, most notably a Jacobi-type algorithm. Particular attention is paid to the special case when choosing the rank equal to the order of the tensor. It is shown that this case can be addressed with an unstructured rank-1 approximation. This allows for the straightforward application of the higher-order power method, for which we discuss effective initialization strategies. This is a joint work with Daniel Kressner (EPFL).

The analysis and decomposition of nonstationary and nonlinear signals in the quest for the identification
of hidden quasiperiodicities and trends is of high theoretical and applied interest nowadays.

Linear techniques like Fourier and Wavelet Transform, historically used in signal processing, cannot capture
completely nonlinear and non stationary phenomena.

For this reason in the last few years new nonlinear methods have been developed like the groundbreaking
Empirical Mode Decomposition algorithm, aka Hilbert--Huang Transform, and the Iterative Filtering technique.

In this seminar I will give an overview of this kind of methods and I will introduce two new algorithms,
the Fast Iterative Filtering and the Adaptive Local Iterative Filtering. I will review the main theoretical results
and outline the most intriguing open problems that still need to be tackled in the field.
Some examples of applications of these techniques to both artificial and real life signals
will be shown to give a foretaste of their potential and robustness.