Alexis Marie Frederic Goujon

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Delaunay-Triangulation-Based Learning With Hessian Total-Variation Regularization

Alexis Marie Frederic Goujon, Mehrsa Pourya, Michaël Unser

2023

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Macroscale analogues(1-3) of microscopic spin systems offer direct insights into fundamental physical principles, thereby advancing our understanding of synchronization phenomena(4) and informing the design of novel classes of chiral metamaterials(5-7). Here we introduce hydrodynamic spin lattices (HSLs) of 'walking' droplets as a class of active spin systems with particle-wave coupling. HSLs reveal various non-equilibrium symmetry-breaking phenomena, including transitions from antiferromagnetic to ferromagnetic order that can be controlled by varying the lattice geometry and system rotation(8). Theoretical predictions based on a generalized Kuramoto model(4) derived from first principles rationalize our experimental observations, establishing HSLs as a versatile platform for exploring active phase oscillator dynamics. The tunability of HSLs suggests exciting directions for future research, from active spin-wave dynamics to hydrodynamic analogue computation and droplet-based topological insulators. A macroscopic analogue of a spin system is shown to emerge in an ensemble of droplets bouncing on the surface of a vibrating bath, revealing symmetry-breaking phenomena such as 'magnetic' ordering.

NATURE PORTFOLIO2021

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Shortest-support multi-spline bases for generalized sampling

Shayan Aziznejad, Alexis Marie Frederic Goujon, Alireza Naderi, Michaël Unser

Generalized sampling consists in the recovery of a function f, from the samples of the responses of a collection of linear shift-invariant systems to the input f . The reconstructed function is typically a member of a finitely generated integer-shift invariant space that can reproduce polynomials up to a given degree M. While this property allows for an approximation power of order (M + 1), it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least (M+1). Following this result, we introduce the notion of shortest basis of degree M, which is motivated by our desire to minimize computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications such as fast derivative sampling with arbitrarily high approximation power. (C) 2021 Elsevier B.V. All rights reserved.

2021

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