Seminar Details
2021-04-27 (14h) : Welcome Seminar for new PhD students (#2)
At MS Teams
Organized by Mathematical Engineering
Section 1: Matrix Completion Methods for Exoplanet Detection
Speaker :
Daglayan Sevim, Hazan
Abstract :
Exoplanet detection by direct imaging is a difficult process because the planet is faint in the light of the star. The presence of quasi-static speckles in the images, which obscure the astrophysical signatures, also makes detecting the planet difficult. Angular Differential Imaging, which takes advantage of the earth's rotation to take several images in a night to differentiate between rotating planets and quasi-static speckles, is one of the most commonly recommended methods for removing these speckles. A low-rank matrix approximation of the images can capture the speckle noise while ignoring moving planets. As a result, the difference between the original image and the low-rank matrix approximation can help to detect the planets. But in this method, the signal of the planet can also be absorbed into the model. We aim to propose a new technique that uses matrix completion. We start by assuming the presence of a planet at a specific location and calculating its path through the data cube. Then we take this path out of the cube and use matrix completion to fill in the blanks and obtain a low-rank approximation. We also propose a maximum-likelihood estimator to compute the intensity of the planet while comparing the model to the original data along the path.
Section 2: Discrete Exterior Calculus framework and proximal alternating minimization for solving discrete Mumford-Shah and Amborsio-Tortorelli models
Speaker :
Le Hoang Trieu Vy (UCLouvain and ENS Lyon)
Abstract :
This work is dedicated to joint image restoration and contour detection. We consider both the discrete Mumford-Shah and the Ambrosio-Tortorelli models, which can be formulated as a minimization involving the sum of a data fidelity term, a term enforcing smoothness everywhere in the image except at the location of the contours, and a third term that penalizes the length of the contours, being the discrete counterpart of the 1D Hausdorff measure. A particular attention is paid for the derivation of the coupling and the penalization terms relying on the derivative operator and expressed by means of the Discrete Exterior Calculus framework.
In this work, we derive proximal alternating minimization schemes with convergence guarantees (PALM-AT and SL-PAM) to estimate a critical point of our energy. We compare their behaviors when the discrete counterpart of the 1D Hausdorff measure is modeled either by an l1-norm or Ambrosio-Tortorelli penalization. The closed-form expressions of the involved proximity operators are provided. A thorough numerical study is conducted to evaluate the performance of both numerical schemes as well as comparisons between l1-norm and Ambrosio-Tortorelli penalization.
Section 3: Alternating Simulation on Hierarchical Abstractions
Speaker :
Calbert, Julien
Abstract :
Abstraction techniques provide formal guarantees for generic optimal control problems on nonlinear and hybrid systems.
Computing an abstraction solving the problem over the whole state-space is computationally demanding in high-dimensional spaces.
We circumvent this curse of dimensionality by introducing a hierarchical abstraction approach for solving an optimal control
problem for nonlinear systems with three nested partitions.
These nested partitions allow the construction of auxiliary systems that characterize simulation relations, which are suitably exploited
to provide upper and lower bounds for a branch and bound algorithm to yield an optimal solution for the control problem.
Section 4: Categorification of path complete lyapunov function
Speaker :
Sébastien Mattenet
Abstract :
In this talk I will consider the proof of stability of a variety of different system with the same general algorithm.
In mathematic, one of the several types of proof at your disposal, is the proof by monovariant, where you introduce some value on the state of a system (like energy in the case of physical system) and show that it only goes one way, concluding something for the evolution of the system, without having to solve it exactly.
One important property we may ask of a given system is the so called Global Asymptotic Stability, roughly saying that there is one point of equilibrium and every starting point eventually tends toward it. Luckily such a property is equivalent to the existence of a Lyapunov function, which is a particular example of a monovariant.
Given a particular system, several algorithm have been proposed to find some Lyapunov function, each with their limitation and narrow scope of application, moreover those algorithm are often only valid in one class of system. The goal of my research is to use the tool of category theory to extend the scope of application of existing tactics, in particular, to extend them to greater class of system.