How to compute the length of a geodesic on a Riemannian manifold with small error in arbitrary Sobolev norms

TL;DR

This paper introduces an algorithm that accurately computes geodesic lengths on Riemannian manifolds using polynomial interpolation of the eikonal equation, achieving arbitrary Sobolev norm accuracy and enabling applications in PDE solutions for finance and physics.

Contribution

It presents a novel polynomial interpolation method for the eikonal equation that approximates geodesic lengths with high Sobolev norm accuracy, incorporating geometric information for error estimation.

Findings

Achieves arbitrarily strong Sobolev norm error bounds

Provides a practical algorithm for geodesic length computation

Enables accurate PDE solutions relevant to finance and physics

Abstract

We compute the length of geodesics on a Riemannian manifold by regular polynomial interpolation of the global solution of the eikonal equation related to the line element $ds^2=g_{ij}dx^idx^j$ of the manifold. Our algorithm approximates the length functional in arbitrarily strong Sobolev norms. Error estimates are obtained where the geometric information is used. It is pointed out how the algorithm can be used to get accurate approximation of solutions of parabolic partial differential equations leading obvious applications to finance and physics.