Mathematics faculty and graduate students are invited to attend the MS defense of Carter Lyons.

Date: Monday, April 4, 2022
Time: 9:00 a.m.
Location: Weber 201

You may also attend by Zoom:

Join Zoom Meeting: https://zoom.us/j/94978842101?pwd=Q21tZWVobTJzV1NSTVdONEthcnJEdz09
Meeting ID: 949 7884 2101
Passcode: 447888

Advisor: Dr. David Aristoff

Committee: Dr. Margaret Cheney, Dr. Diefo Krapf

Title: Stability in the Weighted Ensemble Method

Abstract: In molecular dynamics and computational chemistry a quantity of interest is the mean first passage time (MFPT) between two disjoint regions of state space A and B. A variety of biochemical studies such as protein folding and polymer translocation use the MFPT. Typically, a Markov chain is used to model the desired molecular dynamics where a naïve Monte Carlo estimate of the MFPT is given by averaging the time that multiple trajectories required to transition from A to B. Although, significant potential energy barriers often exist between A and B making the desired MFPT too large to estimate directly through naïve Monte Carlo. Fortunately, a technique of splitting and merging certain Markov chain trajectories can encourage the transition over potential energy barriers and provide computationally feasible MFPT estimates. Here we discuss the weighted ensemble (WE) algorithm, which provides an unbiased and ergodic implementation of the splitting and merging technique, applied to a Markov chain that upon reaching B is restarted in A according to an initial distribution. Furthermore, we discuss bias stability and variance stability in the WE algorithm. First, for bias stability, we analyze how the MFPT, starting from the initial distribution, changes under a perturbation of the initial distribution. Second, for variance stability, we develop conditions, for a three state and 1d Markov model, such that the variance of the WE estimator is significantly reduced by importance sampling the initial distribution when restarting Markov chain trajectories.