Jérôme Michaud

Jérôme Michaud, Sergey Korotov

We study a longest-edge based refinement scheme for triangulations, termed the longest-edge altitude bisection (LEAB), in which each triangle is subdivided by dropping the altitude from the vertex opposite to its longest edge. Using the normalized shape space of triangles introduced by Perdomo and Plaza in: Properties of triangulations obtained by the longest-edge bisection. \emph{Cent. Eur. J. Math.}, 12(12) (2014), 1796-1810, we show that LEAB admits a simple geometric description: the normalized left and right children of a triangle in focus are obtained by intersecting the geodesic of right triangles with rays issued from the endpoints of the longest edge and explicit formulas for the mappings are derived. This characterization implies an interesting observation that the associated refinement dynamics collapse the entire shape space onto the right-triangle geodesic in a single step and that every point on this geodesic is fixed. Two-sided bounds for the contraction of the mesh size (discretization parameter) are derived. Also, applications and limitations of the method are briefly discussed.

Jérôme Michaud

In this paper, we are concerned with the study of the Isotropic Diffusion Source Approximation (IDSA) (Baxter et al., Phys. Rev. E 73, 046118, 2006) of radiative transfer. After having recalled well-known limits of the radiative transfer equation, we present the IDSA and adapt it to the case of the homogeneous sphere. We then show that for this example the IDSA suffers from severe numerical difficulties. We argue that these difficulties originate in the min-max switch coupling mechanism used in the IDSA. To overcome this problem we reformulate the IDSA to avoid the problematic coupling. This allows us to access the modeling error of the IDSA for the homogeneous sphere test case. The IDSA is shown to overestimate the streaming component, hence we propose a new version of the IDSA which is numerically shown to be more accurate than the old one. Analytical results and numerical tests are provided to support the accuracy of the new proposed approximation.

Jérôme Michaud, Anna Jon-and

Recent advances in large language models using deep learning techniques have renewed interest on how languages can be learned from data. However, it is unclear whether or how these models represent grammatical information from the learned languages. In addition, the models must be pre-trained on large corpora before they can be used. In this work, we propose an alternative, more transparent and cognitively plausible architecture for learning language. Instead of using deep learning, our approach uses a minimal cognitive architecture based on sequence memory and chunking. The learning mechanism is based on the principles of reinforcement learning. We test our architecture on a number of natural-like toy languages. Results show that the model can learn these artificial languages from scratch and extract grammatical information that supports learning. Our study demonstrates the power of this simple architecture and stresses the importance of sequence memory as a key component of the language learning process. Since other animals do not seem to have a faithful sequence memory, this may explain why only humans have developed complex languages.