Sampling from Rough Energy Landscapes

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

Numerical Analysis of Parallel Replica Dynamics

Parallel replica dynamics is a method for accelerating the computation of processes characterized by a sequence of infrequent events. In this work, the processes are governed by the overdamped Langevin equation. Such processes spend much of their time about the minima of the underlying potential, occasionally transitioning into different basins of attraction. The essential idea of parallel replica dynamics is that the exit time distribution from a given well for a single process can be approximated by the minimum of the exit time distributions of $N$ independent identical processes, each run for only 1/N-th the amount of time. While promising, this leads to a series of numerical analysis questions about the accuracy of the exit distributions. Building upon the recent work in Le Bris et al., we prove a unified error estimate on the exit distributions of the algorithm against an unaccelerated process. Furthermore, we study a dephasing mechanism, and prove that it will successfully complete.Comment: 37 pages, 4 figures, revised and new estimates from the previous versio

Stability of Solitary Waves for a Generalized Derivative Nonlinear Schr\"odinger Equation

We consider a derivative nonlinear Schr\"odinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, their velocity. We illustrate these results with numerical simulations.Comment: 29 pages, 4 Figure

Focusing Singularity in a Derivative Nonlinear Schr\"odinger Equation

We present a numerical study of a derivative nonlinear Schr\"odinger equation with a general power nonlinearity, $|\psi|^{2\sigma}\psi_x$. In the $L^2$-supercritical regime, $\sigma>1$, our simulations indicate that there is a finite time singularity. We obtain a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schr\"odinger equation with supercritical power law nonlinearity.Comment: 24 pages, 17 figure