In my computational problem, I have a Riemann submanifold $S^{1000}$ embedded in $\mathbb{R}^{300000}$. I can numerically compute the induced metric tensor and the Jacobian. I have no analytical expression of these quantities; they all come from large-scale computations.
I would like to compute the exponential map and logarithmic map on my $S^{1000}$ manifold. I expect to do so by numerically solving a large number of equations, which I can afford with high-peformance computing. I want to have the exponential and logarithmic maps to compute geodesics on $S^{1000}$, eventually.
My question is: How can the exponential maps and logarithmic maps be computed when there is no analytical expression? What systems of equations do I need to solve?
Thank you, Luke