Section: New Results

Miscellaneous

Material Coherence from Trajectories via Burau Eigenanalysis of Braids

Participant : David Cohen-Steiner.

In collaboration with Melissa Yeung and Mathieu Desbrun (Caltech).

In this paper [58], we provide a numerical tool to study material coherence from a set of 2D Lagrangian trajectories sampling a dynamical system, i.e., from the motion of passive tracers. We show that eigenvectors of the Burau representation of a topological braid derived from the trajectories have levelsets corresponding to components of the Nielsen-Thurston decomposition of the dynamical system. One can thus detect and identify clusters of space-time trajectories corresponding to coherent regions of the dynamical system by solving an eigenvalue problem. Unlike previous methods, the scalable computational complexity of our braid-based approach allows the analysis of large amounts of trajectories. Studying two-dimensional flows and their induced transport and mixing properties is key to geophysical studies of atmospheric and oceanic processes. However, one often has only sparse tracer trajectories (e.g., positions of buoys in time) to infer the overall flow geometry. Fortunately, topological methods based on the theory of braid groups have recently been proposed to extract structures from such a sparse set of trajectories by measuring their entan-glement. This braid viewpoint offers sound foundations for the definition of coherent structures. Yet, there has been only limited efforts in developing practical tools that can leverage topological properties for the efficient analysis of flow structures: handling a larger number of tra-jectories remains computationally challenging. We contribute a new and simple computational tool to extract Lagrangian structures from sparse trajectories by noting that the eigenstructure of the Burau matrix representation of a braid of particle trajectories can be used to reveal coherent regions of the flows. Detection of clusters of space-time trajectories corresponding to coherent regions of the dynamical system can thus be achieved by solving a simple eigenvalue problem. This paper establishes the theoretical foundations behind this braid eigenanalysis approach, along with numerical validations on various flows.

Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space

Participants : Alex Delalande, Frédéric Chazal.

In collaboration with Quentin Mérigot (Institut de Mathématiques d'Orsay).

In [55], we study an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)Holder continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent Hölder-stability results for optimal transport maps.