Seminar Details
2024-02-20 (14h) : Newcomers seminars
At Euler building (room a.002)
Organized by Mathematical Engineering
Section 1: Analysis and development of almost-linear time Laplacian solvers for finite element problems.
Speaker :
Christophe Heneffe (PhD UCLouvain/INMA)
Abstract :
Many applications require solving Laplacian systems as fast as possible. It is the case, for example, with finite element simulations. There exist many methods to solve such systems that work well in practice, but they don't enjoy strong theoretical guarantees except in very specific cases. In the last decades, many theoretical algorithms have been created which allows to solve any Laplacian systems in almost linear time. However, most have never been implemented in practice. These algorithms are very general since they work for any Laplacian system. Adapting them by taking advantage of the properties of the finite element meshes could lead to very efficient solvers for FEM problems. The purpose of this project is the development, analysis and implementation of almost-linear time algorithms specifically adapted to solve FEM problems.
Section 2: Efficient, Complete G-Invariance for G-Equivariant Networks via Algorithmic reduction.
Speaker :
Mataigne, Simon
Abstract :
Group-Equivariant Convolutional Neural Networks (G-CNNs) generalize the translation-equivariance of traditional CNNs to group-equivariance, using more general symmetry transformations such as rotation for weight tying. For tasks such as classification, such transformations are removed at the end of the network, to achieve group-invariance, typically by taking a maximum over the group. While this is indeed invariant, it is excessively so; two inputs that are non-identical up to group action can yield the same output, resulting in a general lack of robustness to adversarial attacks. Sanborn and Miolane (2023) proposed an alternative method for achieving invariance without loss of signal structure, called the $G$-triple correlation ($G$-TC). While this method yields demonstrable gains in accuracy and robustness, it comes with a significant increase in computational cost. In this paper, we introduce a new invariant layer based on the Fourier transform of the $G$-TC: the $G$-bispectrum. Operating in Fourier space allows us to significantly reduce the computational cost. Our main theoretical result provides a reduction of the $G$-bispectrum that conserves the selective invariance of the $G$-TC, while only requiring $\mathcal{O}(|G|)$ coefficients. In a suite of experiments, we demonstrate that our approach retains all of the advantages of the $G$-TC, while significantly reducing the computational cost
Section 3: An event-triggered data-driven predictive control
Speaker :
Amir Mehrnoosh (PhD UCLouvain/INMA)
Abstract :
We develop a data-driven model predictive control (MPC) design procedure to control unknown linear time-invariant systems. This algorithm only requires measured input-output data to drive the system to the reference signal. We add filters on desired inputs and outputs in the cost function to improve the transient response. Moreover, the Hankel matrices are updated online based on a multi-step event-triggered MPC scheme to deal with the uncertainties. This also reduces the computational cost and balances it with the closed-loop performance. Simulation results illustrate the effectiveness of the proposed approach.