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In this paper, we propose a new splitting algorithm for dynamical low-rankapproximation motivated by the fibre bundle structure of the set of fixed rankmatrices. We first introduce a geometric description of the set of fixed rankmatrices which relies on a natural parametrization of matrices. More precisely,it is endowed with the structure of analytic principal bundle, with an explicitdescription of local charts. For matrix differential equations, we introduce afirst order numerical integrator working in local coordinates. The resultingalgorithm can be interpreted as a particular splitting of the projectionoperator onto the tangent space of the low-rank matrix manifold. It is provento be exact in some particular case. Numerical experiments confirm this resultand illustrate the robustness of the proposed algorithm.

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