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Geometry-preserving lie group integrators for differential equations on the manifold of symmetric positive definite matrices

Lucas Drumetz 1, 2, 3 Alexandre Reiffers-Masson 4, 5 Naoufal El Bekri 6 Franck Vermet 6 
2 Lab-STICC_OSE - Equipe Observations Signal & Environnement
Lab-STICC - Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance : UMR6285
3 ODYSSEY - Océan Dynamique Observations Analyse
UBO UFR ST - Université de Bretagne Occidentale - UFR Sciences et Techniques, UR1 - Université de Rennes 1, IFREMER - Institut Français de Recherche pour l'Exploitation de la Mer, Inria Rennes – Bretagne Atlantique , IMT Atlantique - IMT Atlantique
5 Lab-STICC_MATHNET - Equipe Math & Net
Lab-STICC - Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance : UMR6285
Abstract : In many applications, one encounters signals that lie on manifolds rather than a Euclidean space. In particular, covariance matrices are examples of ubiquitous mathematical objects that have a non Euclidean structure. The application of Euclidean methods to integrate differential equations lying on such objects does not respect the geometry of the manifold, which can cause many numerical issues. In this paper, we propose to use Lie group methods to define geometry-preserving numerical integration schemes on the manifold of symmetric positive definite matrices. These can be applied to a number of differential equations on covariance matrices of practical interest. We show that they are more stable and robust than other classical or naive integration schemes on an example.
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Preprints, Working Papers, ...
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https://hal-imt-atlantique.archives-ouvertes.fr/hal-03815325
Contributor : Lucas Drumetz Connect in order to contact the contributor
Submitted on : Monday, October 24, 2022 - 1:44:26 PM
Last modification on : Friday, October 28, 2022 - 3:52:30 AM

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  • HAL Id : hal-03815325, version 2
  • ARXIV : 2210.08842

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Lucas Drumetz, Alexandre Reiffers-Masson, Naoufal El Bekri, Franck Vermet. Geometry-preserving lie group integrators for differential equations on the manifold of symmetric positive definite matrices. {date}. ⟨hal-03815325v2⟩

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