We examine challenges to sampling from Boltzmann distributions associated
with multiscale energy landscapes. The multiscale features, or "roughness,"
corresponds to highly oscillatory, but bounded, perturbations of a smooth
landscape. Through a combination of numerical experiments and analysis we
demonstrate that the performance of Metropolis Adjusted Langevin Algorithm can
be severely attenuated as the roughness increases. In contrast, we prove that
Random Walk Metropolis is insensitive to such roughness. We also formulate two
alternative sampling strategies that incorporate large scale features of the
energy landscape, while resisting the impact of fine scale roughness; these
also outperform Random Walk Metropolis. Numerical experiments on these
landscapes are presented that confirm our predictions. Open questions and
numerical challenges are also highlighted.Comment: 34 pages, first revisio