Communications in Mathematical Sciences

Volume 18 (2020)

Number 8

Sampling from rough energy landscapes

Pages: 2271 – 2303



Petr Plecháč (Department of Mathematical Sciences, University of Delaware, Newark, Del., U.S.A.)

Gideon Simpson (Department of Mathematics, Drexel University, Philadelphia, Pennsylvania, U.S.A.)


We examine challenges to sampling from Boltzmann distributions associated with multiscale energy landscapes. The multiscale features—or “roughness”—correspond 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.


Markov chain Monte Carlo, random walk Metropolis, Metropolis adjusted Langevin, rough energy landscapes, multi-scale energy landscapes, mean squared displacement

2010 Mathematics Subject Classification

60J22, 65C05, 65C40

Received 14 June 2019

Accepted 7 July 2020

Published 22 December 2020