Computing regularized splines in the Riemannian manifold of probability measures

In this paper, we give new numerical methods to solve C² splines for interpolating a finite set of probability measures. Due to the difficulties of generalizing interpolations and solving them with explicit expressions on manifolds, we first introduce a theoretical analysis with different structures on the Riemannian manifold of probability measures P ₊(I). Then, by combining standard numerical methods and a new mathematical modeling we generalize interpolating splines as a minimizer of an energy functional on P ₊(I). Finally, we show the efficiency of the proposed methods and their computational relevance with several applications.