  The approximation of probability measures on compact metric spaces and in
particular on Riemannian manifoldsby atomic or empirical ones is a classical
task in approximation and complexity theory with a wide range of applications.
Instead of point measures we are concerned with the approximation by measures
supported on Lipschitz curves. Special attention is paid to push-forward
measures of Lebesgue measures on the interval by such curves. Using the
discrepancy as distance between measures, we prove optimal approximation rates
in terms of Lipschitz constants of curves. Having established the theoretical
convergence rates, we are interested in the numerical minimization of the
discrepancy between a given probability measure and the set of push-forward
measures of Lebesgue measures on the interval by Lipschitz curves. We present
numerical examples for measures on the 2- and 3-dimensional torus, the
2-sphere, the rotation group on $\mathbb R^3$ and the Grassmannian of all
2-dimensional linear subspaces of $\mathbb{R}^4$. Our algorithm of choice is a
conjugate gradient method on these manifolds which incorporates second-oder
information. For efficiently computing the gradients and the Hessians within
the algorithm, we approximate the given measures by truncated Fourier series
and use fast Fourier transform techniques on these manifolds.
