Discrete Geodesic Calculus in the Space of Sobolev Curves

The Riemannian manifold of curves with a Sobolev metric is an important and frequently studied model in the theory of shape spaces. Various numerical approaches have been proposed to compute geodesics, but so far elude a rigorous convergence theory. By a slick modification of a temporal Galerkin discretization we manage to preserve coercivity and compactness properties of the continuous model and thereby are able to prove convergence for the geodesic boundary value problem. Likewise, for the numerical analysis of the geodesic initial value problem we are able to exploit the geodesic completeness of the underlying continuous model for the error control of a time-stepping approximation. In fact, we develop a convergent discretization of a comprehensive Riemannian calculus that in addition includes parallel transport, covariant differentiation, the Riemann curvature tensor, and sectional curvature, all important tools to explore the geometry of the space of curves. Selected numerical examples confirm the theoretical findings and show the qualitative behaviour. To this end, a low-dimensional submanifold of Sobolev curves with explicit formulas for ground truth covariant derivatives and curvatures are considered.