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arXiv · 2001.08599

A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds

Marie Billaud-Friess·Antonio Falcó·Anthony Nouy
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Abstract

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.

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Marie Billaud-Friess, Antonio Falcó, Anthony Nouy. 2021-01-09. A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds. https://doi.org/10.1007/s10543-021-00884-x

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