BAD-NEUS: Rapidly converging trajectory stratification.

An issue for molecular dynamics simulations is that events of interest often involve timescales that are much longer than the simulation time step, which is set by the fastest timescales of the model. Because of this timescale separation, direct simulation of many events is prohibitively computationally costly. This issue can be overcome by aggregating information from many relatively short simulations that sample segments of trajectories involving events of interest.

Accurate estimates of dynamical statistics using memory.

Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis.

Inexact iterative numerical linear algebra for neural network-based spectral estimation and rare-event prediction.

Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics, such as the likelihood and average time of events (predictions). Here, we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a dataset of short trajectories sampled at finite intervals.

Predicting rare events using neural networks and short-trajectory data.

Estimating the likelihood, timing, and nature of events is a major goal of modeling stochastic dynamical systems. When the event is rare in comparison with the timescales of simulation and/or measurement needed to resolve the elemental dynamics, accurate prediction from direct observations becomes challenging. In such cases a more effective approach is to cast statistics of interest as solutions to Feynman-Kac equations (partial differential equations).

Full Configuration Interaction Excited-State Energies in Large Active Spaces from Subspace Iteration with Repeated Random Sparsification.

We present a stable and systematically improvable quantum Monte Carlo (QMC) approach to calculating excited-state energies, which we implement using our fast randomized iteration method for the full configuration interaction problem (FCI-FRI). Unlike previous excited-state quantum Monte Carlo methods, our approach, which is based on an asymmetric variant of subspace iteration, avoids the use of dot products of random vectors and instead relies upon trial vectors to maintain orthogonality and estimate eigenvalues. By leveraging recent advances, we apply our method to calculate ground- and excited-state energies of challenging molecular systems in large active spaces, including the carbon dimer with 8 electrons in 108 orbitals (8e,108o), an oxo-Mn(salen) transition metal complex (28e,28o), ozone (18e,87o), and butadiene (22e,82o).

Augmented transition path theory for sequences of events.

Transition path theory provides a statistical description of the dynamics of a reaction in terms of local spatial quantities. In its original formulation, it is limited to reactions that consist of trajectories flowing from a reactant set A to a product set B. We extend the basic concepts and principles of transition path theory to reactions in which trajectories exhibit a specified sequence of events and illustrate the utility of this generalization on examples.

Computing transition path theory quantities with trajectory stratification.

Transition path theory computes statistics from ensembles of reactive trajectories. A common strategy for sampling reactive trajectories is to control the branching and pruning of trajectories so as to enhance the sampling of low probability segments. However, it can be challenging to apply transition path theory to data from such methods because determining whether configurations and trajectory segments are part of reactive trajectories requires looking backward and forward in time.

Error Bounds for Dynamical Spectral Estimation.

Dynamical spectral estimation is a well-established numerical approach for estimating eigenvalues and eigenfunctions of the Markov transition operator from trajectory data. Although the approach has been widely applied in biomolecular simulations, its error properties remain poorly understood. Here we analyze the error of a dynamical spectral estimation method called "the variational approach to conformational dynamics" (VAC).

Long-Time-Scale Predictions from Short-Trajectory Data: A Benchmark Analysis of the Trp-Cage Miniprotein.

Elucidating physical mechanisms with statistical confidence from molecular dynamics simulations can be challenging owing to the many degrees of freedom that contribute to collective motions. To address this issue, we recently introduced a dynamical Galerkin approximation (DGA) [Thiede, E. H.

Integrated Variational Approach to Conformational Dynamics: A Robust Strategy for Identifying Eigenfunctions of Dynamical Operators.

One approach to analyzing the dynamics of a physical system is to search for long-lived patterns in its motions. This approach has been particularly successful for molecular dynamics data, where slowly decorrelating patterns can indicate large-scale conformational changes. Detecting such patterns is the central objective of the variational approach to conformational dynamics (VAC), as well as the related methods of time-lagged independent component analysis and Markov state modeling.

Improved Fast Randomized Iteration Approach to Full Configuration Interaction.

We present three modifications to our recently introduced fast randomized iteration method for full configuration interaction (FCI-FRI) and investigate their effects on the method's performance for Ne, HO, and N. The initiator approximation, originally developed for full configuration interaction quantum Monte Carlo, significantly reduces statistical error in FCI-FRI when few samples are used in compression operations, enabling its application to larger chemical systems. The semistochastic extension, which involves exactly preserving a fixed subset of elements in each compression, improves statistical efficiency in some cases but reduces it in others.

Insulin Dissociates by Diverse Mechanisms of Coupled Unfolding and Unbinding.

The protein hormone insulin exists in various oligomeric forms, and a key step in binding its cellular receptor is dissociation of the dimer. This dissociation process and its corresponding association process have come to serve as paradigms of coupled (un)folding and (un)binding more generally. Despite its fundamental and practical importance, the mechanism of insulin dimer dissociation remains poorly understood.

Stratification as a general variance reduction method for Markov chain Monte Carlo.

The Eigenvector Method for Umbrella Sampling (EMUS) [46] belongs to a popular class of methods in statistical mechanics which adapt the principle of stratified survey sampling to the computation of free energies. We develop a detailed theoretical analysis of EMUS. Based on this analysis, we show that EMUS is an efficient method for computing averages over arbitrary target distributions.

Beyond Walkers in Stochastic Quantum Chemistry: Reducing Error Using Fast Randomized Iteration.

We introduce a family of methods for the full configuration interaction problem in quantum chemistry, based on the fast randomized iteration (FRI) framework [Lim, L.-H.; Weare, J.

Galerkin approximation of dynamical quantities using trajectory data.

Understanding chemical mechanisms requires estimating dynamical statistics such as expected hitting times, reaction rates, and committors. Here, we present a general framework for calculating these dynamical quantities by approximating boundary value problems using dynamical operators with a Galerkin expansion. A specific choice of basis set in the expansion corresponds to the estimation of dynamical quantities using a Markov state model.

Practical rare event sampling for extreme mesoscale weather.

Extreme mesoscale weather, including tropical cyclones, squall lines, and floods, can be enormously damaging and yet challenging to simulate; hence, there is a pressing need for more efficient simulation strategies. Here, we present a new rare event sampling algorithm called quantile diffusion Monte Carlo (quantile DMC). Quantile DMC is a simple-to-use algorithm that can sample extreme tail behavior for a wide class of processes.

Trajectory stratification of stochastic dynamics.

We present a general mathematical framework for trajectory stratification for simulating rare events. Trajectory stratification involves decomposing trajectories of the underlying process into fragments limited to restricted regions of state space (strata), computing averages over the distributions of the trajectory fragments within the strata with minimal communication between them, and combining those averages with appropriate weights to yield averages with respect to the original underlying process. Our framework reveals the full generality and flexibility of trajectory stratification, and it illuminates a common mathematical structure shared by existing algorithms for sampling rare events.

Eigenvector method for umbrella sampling enables error analysis.

Umbrella sampling efficiently yields equilibrium averages that depend on exploring rare states of a model by biasing simulations to windows of coordinate values and then combining the resulting data with physical weighting. Here, we introduce a mathematical framework that casts the step of combining the data as an eigenproblem. The advantage to this approach is that it facilitates error analysis.

Multiple Time-Step Dual-Hamiltonian Hybrid Molecular Dynamics - Monte Carlo Canonical Propagation Algorithm.

A multiple time-step integrator based on a dual Hamiltonian and a hybrid method combining molecular dynamics (MD) and Monte Carlo (MC) is proposed to sample systems in the canonical ensemble. The Dual Hamiltonian Multiple Time-Step (DHMTS) algorithm is based on two similar Hamiltonians: a computationally expensive one that serves as a reference and a computationally inexpensive one to which the workload is shifted. The central assumption is that the difference between the two Hamiltonians is slowly varying.

SHARP ENTRYWISE PERTURBATION BOUNDS FOR MARKOV CHAINS.

For many Markov chains of practical interest, the invariant distribution is extremely sensitive to perturbations of some entries of the transition matrix, but insensitive to others; we give an example of such a chain, motivated by a problem in computational statistical physics. We have derived perturbation bounds on the relative error of the invariant distribution that reveal these variations in sensitivity. Our bounds are sharp, we do not impose any structural assumptions on the transition matrix or on the perturbation, and computing the bounds has the same complexity as computing the invariant distribution or computing other bounds in the literature.

Finding Chemical Reaction Paths with a Multilevel Preconditioning Protocol.

Finding transition paths for chemical reactions can be computationally costly owing to the level of quantum-chemical theory needed for accuracy. Here, we show that a multilevel preconditioning scheme that was recently introduced (Tempkin et al. , , 184114) can be used to accelerate quantum-chemical string calculations.

Using multiscale preconditioning to accelerate the convergence of iterative molecular calculations.

Iterative procedures for optimizing properties of molecular models often converge slowly owing to the computational cost of accurately representing features of interest. Here, we introduce a preconditioning scheme that allows one to use a less expensive model to guide exploration of the energy landscape of a more expensive model and thus speed the discovery of locally stable states of the latter. We illustrate our approach in the contexts of energy minimization and the string method for finding transition pathways.

Nucleotide regulation of the structure and dynamics of G-actin.

Actin, a highly conserved cytoskeletal protein found in all eukaryotic cells, facilitates cell motility and membrane remodeling via a directional polymerization cycle referred to as treadmilling. The nucleotide bound at the core of each actin subunit regulates this process. Although the biochemical kinetics of treadmilling has been well characterized, the atomistic details of how the nucleotide affects polymerization remain to be definitively determined.

Relaxation of a family of broken-bond crystal-surface models.

We study the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. With computational experiments and theoretical arguments we are able to derive several partial differential equation limits identified (or nearly identified) in previous studies and to clarify the correct choice of surface tension appearing in the PDE and the correct scaling regime giving rise to each PDE. We also provide preliminary computational investigations of a number of interesting qualitative features of the large-scale behavior of the models.

Extending molecular simulation time scales: Parallel in time integrations for high-level quantum chemistry and complex force representations.

Parallel in time simulation algorithms are presented and applied to conventional molecular dynamics (MD) and ab initio molecular dynamics (AIMD) models of realistic complexity. Assuming that a forward time integrator, f (e.g.