A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds
| Autor: | Antonio Falcó, Anthony Nouy, Marie Billaud-Friess |
|---|---|
| Přispěvatelé: | Producción Científica UCH 2022, UCH. Departamento de Matemáticas, Física y Ciencias Tecnológicas |
| Rok vydání: | 2020 |
| Předmět: |
Grassmann
Variedades de Differential topology Rank (linear algebra) Computer Networks and Communications Low-rank approximation Grassmann manifolds Manifolds (Mathematics) Matrix (mathematics) Variedades (Matemáticas) Local coordinates Tangent space FOS: Mathematics Geometría diferencial Fiber bundle Mathematics - Numerical Analysis Mathematics Applied Mathematics Geometry Differential Numerical Analysis (math.NA) Principal bundle Manifold Computational Mathematics 15A23 65F30 65L05 65L20 15A23 65F30 65L05 65L20 15A23 65F30 65L05 65L20 15A23 65F30 65L05 65L20 15A23 65F30 65L05 65L20 Algorithm Topología diferencial Software |
| Zdroj: | CEU Repositorio Institucional Fundación Universitaria San Pablo CEU (FUSPCEU) |
| DOI: | 10.48550/arxiv.2001.08599 |
| Popis: | Este artículo se encuentra disponible en la siguiente URL: https://link.springer.com/article/10.1007/s10543-021-00884-x Este es el postprint del siguiente artículo: Billaud-Friess, M., Falcó, A. & Nouy, A. (2022). A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds. Bit Numerical Mathematics, vol. 62, i. 2 (jun.), pp. 387?408, que se ha publicado de forma definitiva en https://doi.org/10.1007/s10543-021-00884-x This is the peer reviewed version of the following article: Billaud-Friess, M., Falcó, A. & Nouy, A. (2022). A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds. Bit Numerical Mathematics, vol. 62, i. 2 (jun.), pp. 387?408, which has been published in final form at https://doi.org/10.1007/s10543-021-00884-x In this paper, we propose a new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of the set of fixed rank matrices. We first introduce a geometric description of the set of fixed rank matrices which relies on a natural parametrization of matrices. More precisely, it is endowed with the structure of analytic principal bundle, with an explicit description of local charts. For matrix differential equations, we introduce a first order numerical integrator working in local coordinates. The resulting algorithm can be interpreted as a particular splitting of the projection operator onto the tangent space of the low-rank matrix manifold. It is proven to be exact in some particular case. Numerical experiments confirm this result and illustrate the robustness of the proposed algorithm. Postprint |
| Databáze: | OpenAIRE |
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