Geir Bogfjellmo: Interpolation on non-flat spaces

Abstract:

Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic De Casteljau's algorithm, which leads to generalized Bézier curves. To construct C^2-splines from such curves is a complicated non-linear problem, until now lacking numerical methods. We have derived an iterative algorithm for C^2-splines on Riemannian symmetric spaces, and proven convergence. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. We demonstrate the algorithm for three geometries of interest: the n-sphere, complex projective space, and the real Grassmannian.