# Stability in the weighted ensemble method

## Date

2022

## Authors

Lyons, Carter, author

Aristoff, David, advisor

Cheney, Margaret, committee member

Krapf, Diego, committee member

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## Abstract

In molecular dynamics, a quantity of interest is the mean first passage time, or average transition time, for a molecule to transition from a region A to a different region B. Often, significant potential barriers exist between A and B making the transition from A to B a rare event, which is an event that is highly improbable to occur. Correspondingly, the mean first passage time for a molecule to transition from A to B will be immense. So, using direct Markov chain Monte Carlo techniques to effectively estimate the mean first passage time is computationally infeasible due to the protracted simulations required. Instead, the Markov chain modeling the underlying molecular dynamics is simulated to steady-state and the steady-state flux from A into B is estimated. Then through the Hill relation, the mean first passage time is obtained as the reciprocal of the estimated steady-state flux. Estimating the steady-state flux into B is still a rare event but the difficulty has shifted from lengthy simulation times to a substantial variance on the desired estimate. Therefore, an importance sampling or importance splitting technique that emphasizes reaching B and reduces estimator variance must be used. Weighted ensemble is one importance sampling Markov chain Monte Carlo method often used to estimate mean first passage times in molecular dynamics. Broadly, weighted ensemble simulates a collection of Markov chain trajectories that are assigned a weight. Periodically, certain trajectories are copied while others are removed, to encourage a transition from A to B, and the trajectory weights are adjusted accordingly. By time-averaging the weighted average of these Markov chain trajectories, weighted ensemble estimates averages with respect to the Markov chain steady-state distribution. We focus on the use of weighted ensemble for estimating the mean first passage time from A to B, through estimating the steady-state flux from A into B, of a Markov chain where upon reaching B is restarted in A according to an initial, or recycle, distribution. First, we give a mathematical detailing of the weighted ensemble algorithm and provide an unbiased property, ergodic property, and variance formula. The unbiased property gives that the weighted ensemble average of many Markov chain trajectories produces an unbiased estimate for the underlying Markov chain law. Next, the ergodic property states that the weighted ensemble estimator converges almost surely to the desired steady-state average. Lastly, the variance formula provides exact variance of the weighted ensemble estimator. Next, we analyze the impact of the initial or recycle distribution, in A, on bias and variance of the weighted ensemble estimate and compare against adaptive multilevel splitting. Adaptive multilevel splitting is an importance splitting Markov chain Monte Carlo method also used in molecular dynamics for estimating mean first passage times. It has been studied that adaptive multilevel splitting requires a precise importance sampling of the initial, or recycle, distribution to maintain reasonable variance bounds on the adaptive multilevel splitting estimator. We show that the weighted ensemble estimator is less sensitive to the initial distribution since importance sampling the initial distribution frequently does not reduce the variance of the weighted ensemble estimator significantly. For a generic three state Markov chain and one dimensional overdamped Langevin dynamics, we develop specific conditions which must be satisfied for initial distribution importance sampling to provide a significant variance reduction on the weighted ensemble estimator. Finally, for bias, we develop conditions on A, such that the mean first passage time from A to B is stable with respect to changes in the initial distribution. That is, under a perturbation of the initial distribution the resulting change in the mean first passage time is insignificant. The conditions on A are verified with one dimensional overdamped Langevin dynamics and an example is provided. Furthermore, when the mean first passage time is unstable, we develop bounds, for one dimensional overdamped Langevin dynamics, on the change in the mean first passage time and show the tightness of the bounds with numerical examples.

## Description

## Rights Access

## Subject

importance sampling

molecular dynamics

variance reduction

Markov chain Monte Carlo

estimation

statistical mechanics