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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.
