Fast geodesics computation with the phase flow method

This paper introduces a novel approach for rapidly computing a very large number of geodesics on a smooth surface. The idea is to apply the recently developed phase flow method [L. Ying, E.J. Candès, The phase flow method, J. Comput. Phys., to appear], an efficient and accurate technique for constructing phase maps for nonlinear ordinary differential equations on invariant manifolds, which are here the unit tangent bundles of the surfaces under study. We show how to rapidly construct the whole geodesic flow map which then allows computing any geodesic by straightforward local interpolation, an operation with constant complexity. A few numerical experiments complement our study and demonstrate the effectiveness of our approach.