Collaborative Research: Stochastic Methods for Complex Systems

Drexel University, Philadelphia PA

Investigators

Abstract

This project addresses computational challenges in materials science, chemistry, uncertainty quantification, and related fields. Quantities of interest such as chemical reaction rates, strength of alloys, and more, can be estimated using a common mathematical modeling framework that computes mean values and provides quantification of the variance about the means. Estimating these quantities by computer simulation can be particularly challenging when the property that one wishes to study is rare and many repeated computer simulations would be required to estimate the mean value and the variance. While this challenge is somewhat alleviated by growth in computing power, some simulations, including chemical reaction rates, cannot be addressed via brute force computation. Rather than rely on raw computing power, the investigators intend to develop novel computer algorithms and approximations that will allow for more efficient and more accurate predictions. This includes the use of interacting copies of mathematical models, which communicate information between one another, resulting in higher quality estimates. These algorithms and approximations will allow more faithful prediction of quantities of interest and access to bigger models (such as larger, more complicated molecules). Mathematically the project will provide a rigorous understanding of the computer algorithms, providing confidence to scientists in a variety of fields. Multiscale distributions appear in a variety of applications, including materials science, chemistry, and uncertainty quantification. Given efficient sampling strategies, one can compute a variety of quantities of interest, including ensemble averages, mean first passage times, and probabilities of rare events. However, multiscale distributions in high number of dimensions are particularly challenging to sample. One example is the Boltzmann distribution induced by an energy landscape containing superbasins. Such a landscape features clusters of local minima that correspond to close groupings of modes in the distribution. This project will investigate four sampling algorithms: weighted ensemble sampling, parallel replica dynamics, local entropy smoothing, and piecewise deterministic Markov processes. Weighted ensemble sampling partitions state space into bins and then elects to sample within those bins in an optimal way. The project will investigate the choice of the sample allocation strategy and consider both finite and infinite system size limits for the method. Parallel replica dynamics also involves using an ensemble of samples, but, in contrast to weighted ensemble, it uses the replicas to efficiently find first exits out of one metastable region and into another. Local entropy smoothing removes the superbasin features of the energy landscape by performing local ensemble sampling and averaging. Finally, the investigators will use piecewise deterministic Markov processes to perform rejection free sampling without requiring estimates of gradients. These algorithms will be rigorously analyzed, and they will be tested on a variety of realistic high-dimensional problems including chemical reaction networks and stochastic molecular dynamics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.