Jonathan Weare

On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method.

An issue of general interest in computer simulations is to incorporate information from experiments into a structural model. An important caveat in pursuing this goal is to avoid corrupting the resulting model with spurious and arbitrary biases. While the problem of biasing thermodynamic ensembles can be formulated rigorously using the maximum entropy method introduced by Jaynes, the approach can be cumbersome in practical applications with the need to determine multiple unknown coefficients iteratively. A popular alternative strategy to incorporate the information from experiments is to rely on restrained-ensemble molecular dynamics simulations. However, the fundamental validity of this computational strategy remains in question. Here, it is demonstrated that the statistical distribution produced by restrained-ensemble simulations is formally consistent with the maximum entropy method of Jaynes. This clarifies the underlying conditions under which restrained-ensemble simulations will yield results that are consistent with the maximum entropy method.

Particle filtering with path sampling and an application to a bimodal ocean current model

This paper introduces a recursive particle filtering algorithm designed to filter high dimensional systems with complicated non-linear and non-Gaussian effects. The method incorporates a parallel marginalization (PMMC) step in conjunction with the hybrid Monte Carlo (HMC) scheme to improve samples generated by standard particle filters. Parallel marginalization is an efficient Markov chain Monte Carlo (MCMC) strategy that uses lower dimensional approximate marginal distributions of the target distribution to accelerate equilibration. As a validation the algorithm is tested on a 2516 dimensional, bimodal, stochastic model motivated by the Kuroshio current that runs along the Japanese coast. The results of this test indicate that the method is an attractive alternative for problems that require the generality of a particle filter but have been inaccessible due to the limitations of standard particle filtering strategies.

AN AFFINE-INVARIANT SAMPLER FOR EXOPLANET FITTING AND DISCOVERY IN RADIAL VELOCITY DATA

Markov chain Monte Carlo (MCMC) proves to be powerful for Bayesian inference and in particular for exoplanet radial velocity fitting because MCMC provides more statistical information and makes better use of data than common approaches like chi-square fitting. However, the nonlinear density functions encountered in these problems can make MCMC time-consuming. In this paper, we apply an ensemble sampler respecting affine invariance to orbital parameter extraction from radial velocity data. This new sampler has only one free parameter, and does not require much tuning for good performance, which is important for automatization. The autocorrelation time of this sampler is approximately the same for all parameters and far smaller than Metropolis-Hastings, which means it requires many fewer function calls to produce the same number of independent samples. The affine-invariant sampler speeds up MCMC by hundreds of times compared with Metropolis-Hastings in the same computing situation. This novel sampler would be ideal for projects involving large data sets such as statistical investigations of planet distribution. The biggest obstacle to ensemble samplers is the existence of multiple local optima; we present a clustering technique to deal with local optima by clustering based on the likelihood of the walkers in the ensemble. We demonstrate the effectiveness of the sampler on real radial velocity data.

Data Assimilation in the Low Noise Regime with Application to the Kuroshio

AbstractOnline data assimilation techniques such as ensemble Kalman filters and particle filters lose accuracy dramatically when presented with an unlikely observation. Such an observation may be caused by an unusually large measurement error or reflect a rare fluctuation in the dynamics of the system. Over a long enough span of time it becomes likely that one or several of these events will occur. Often they are signatures of the most interesting features of the underlying system and their prediction becomes the primary focus of the data assimilation procedure. The Kuroshio or Black Current that runs along the eastern coast of Japan is an example of such a system. It undergoes infrequent but dramatic changes of state between a small meander during which the current remains close to the coast of Japan, and a large meander during which it bulges away from the coast. Because of the important role that the Kuroshio plays in distributing heat and salinity in the surrounding region, prediction of these transit...

Efficient Monte Carlo sampling by parallel marginalization

Markov chain Monte Carlo sampling methods often suffer from long correlation times. Consequently, these methods must be run for many steps to generate an independent sample. In this paper, a method is proposed to overcome this difficulty. The method utilizes information from rapidly equilibrating coarse Markov chains that sample marginal distributions of the full system. This is accomplished through exchanges between the full chain and the auxiliary coarse chains. Results of numerical tests on the bridge sampling and filtering/smoothing problems for a stochastic differential equation are presented.

Steered transition path sampling

We introduce a path sampling method for obtaining statistical properties of an arbitrary stochastic dynamics. The method works by decomposing a trajectory in time, estimating the probability of satisfying a progress constraint, modifying the dynamics based on that probability, and then reweighting to calculate averages. Because the progress constraint can be formulated in terms of occurrences of events within time intervals, the method is particularly well suited for controlling the sampling of currents of dynamic events. We demonstrate the method for calculating transition probabilities in barrier crossing problems and survival probabilities in strongly diffusive systems with absorbing states, which are difficult to treat by shooting. We discuss the relation of the algorithm to other methods.

Variance Reduction for Particle Filters of Systems With Time Scale Separation

We present a particle filter construction for a system that exhibits time-scale separation. The separation of time scales allows two simplifications that we exploit: 1) the use of the averaging principle for the dimensional reduction of the dynamics for each particle during the prediction step and 2) the factorization of the transition probability for the Rao-Blackwellization of the update step. The resulting particle filter is faster and has smaller variance than the particle filter based on the original system. The method is tested on a multiscale stochastic differential equation and on a multiscale pure jump diffusion motivated by chemical reactions.

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., Verlet algorithm), is available to propagate the system from time ti (trajectory positions and velocities xi = (ri, vi)) to time ti + 1 (xi + 1) by xi + 1 = fi(xi), the dynamics problem spanning an interval from t0[ellipsis (horizontal)]tM can be transformed into a root finding problem, F(X) = [xi - f(x(i - 1)]i = 1, M = 0, for the trajectory variables. The root finding problem is solved using a variety of root finding techniques, including quasi-Newton and preconditioned quasi-Newton schemes that are all unconditionally convergent. The algorithms are parallelized by assigning a processor to each time-step entry in the columns of F(X). The relation of this approach to other recently proposed parallel in time methods is discussed, and the effectiveness of various approaches to solving the root finding problem is tested. We demonstrate that more efficient dynamical models based on simplified interactions or coarsening time-steps provide preconditioners for the root finding problem. However, for MD and AIMD simulations, such preconditioners are not required to obtain reasonable convergence and their cost must be considered in the performance of the algorithm. The parallel in time algorithms developed are tested by applying them to MD and AIMD simulations of size and complexity similar to those encountered in present day applications. These include a 1000 Si atom MD simulation using Stillinger-Weber potentials, and a HCl + 4H2O AIMD simulation at the MP2 level. The maximum speedup (serial execution/timeparallel execution time) obtained by parallelizing the Stillinger-Weber MD simulation was nearly 3.0. For the AIMD MP2 simulations, the algorithms achieved speedups of up to 14.3. The parallel in time algorithms can be implemented in a distributed computing environment using very slow transmission control protocol/Internet protocol networks. Scripts written in Python that make calls to a precompiled quantum chemistry package (NWChem) are demonstrated to provide an actual speedup of 8.2 for a 2.5 ps AIMD simulation of HCl + 4H2O at the MP2/6-31G* level. Implemented in this way these algorithms can be used for long time high-level AIMD simulations at a modest cost using machines connected by very slow networks such as WiFi, or in different time zones connected by the Internet. The algorithms can also be used with programs that are already parallel. Using these algorithms, we are able to reduce the cost of a MP2/6-311++G(2d,2p) simulation that had reached its maximum possible speedup in the parallelization of the electronic structure calculation from 32 s/time step to 6.9 s/time step.

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. There is increasing evidence that the nucleotide regulation (and other characteristics) of actin cannot be fully described from the minimum energy structure, but rather depends on a dynamic equilibrium between conformations. In this work we explore the conformational mobility of the actin monomer (G-actin) in a coarse-grained subspace using umbrella sampling to bias all-atom molecular-dynamics simulations along the variables of interest. The results reveal that ADP-bound actin subunits are more conformationally mobile than ATP-bound subunits. We used a multiscale analysis method involving coarse-grained and atomistic representations of these simulations to characterize how the nucleotide affects the low-energy states of these systems. The interface between subdomains SD2-SD4, which is important for polymerization, is stabilized in an actin filament-like (F-actin) conformation in ATP-bound G-actin. Additionally, the nucleotide modulates the conformation of the SD1-SD3 interface, a region involved in the binding of several actin-binding proteins.

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.