arXiv subjects

Jonathan Weare

Publications and source records attributed to Jonathan Weare.

At least 55 records · Page 3Linked to original sources

arXiv2020

Integrated VAC: 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. In VAC, the search for slowly decorrelating patterns is formalized as a variational problem solved by the eigenfunctions of the system's transition operator. VAC computes solutions to this variational problem by optimizing a linear or nonlinear model of the eigenfunctions using time series data. Here, we build on VAC's success by addressing two practical limitations. First, VAC can give poor eigenfunction estimates when the lag time parameter is chosen poorly. Second, VAC can overfit when using flexible parameterizations such as artificial neural networks with insufficient regularization. To address these issues, we propose an extension that we call integrated VAC (IVAC). IVAC integrates over multiple lag times before solving the variational problem, making its results more robust and reproducible than VAC's.

physics.data-an↗

arXiv2020

Long-timescale 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 et al. J. Phys. Chem. 150, 244111 (2019)], in which chemical kinetic statistics that satisfy equations of dynamical operators are represented by a basis expansion. Here, we reformulate this approach, clarifying (and reducing) the dependence on the choice of lag time. We present a new projection of the reactive current onto collective variables and provide improved estimators for rates and committors. We also present simple procedures for constructing suitable smoothly varying basis functions from arbitrary molecular features. To evaluate estimators and basis sets numerically, we generate and carefully validate a dataset of short trajectories for the unfolding and folding of the trp-cage miniprotein, a well-studied system. Our analysis demonstrates a comprehensive strategy for characterizing reaction pathways quantitatively.

physics.data-an↗

arXiv2020

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, H$_2$O, and N$_2$. 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 semi-stochastic extension, which involves exactly preserving a fixed subset of elements in each compression, improves statistical efficiency in some cases but reduces it in others. We also developed a new approach to sampling excitations that yields consistent improvements in statistical efficiency and reductions in computational cost. We discuss possible strategies based on our findings for improving the performance of stochastic quantum chemistry methods more generally.

physics.comp-ph↗

arXiv2020

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

The Eigenvector Method for Umbrella Sampling (EMUS) 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 general method for computing averages over arbitrary target distributions. In particular, we show that EMUS can be dramatically more efficient than direct MCMC when the target distribution is multimodal or when the goal is to compute tail probabilities. To illustrate these theoretical results, we present a tutorial application of the method to a problem from Bayesian statistics.

stat.ME↗

arXiv2020

Path properties of atmospheric transitions: illustration with a low-order sudden stratospheric warming model

Many rare weather events, including hurricanes, droughts, and floods, dramatically impact human life. To accurately forecast these events and characterize their climatology requires specialized mathematical techniques to fully leverage the limited data that are available. Here we describe \emph{transition path theory} (TPT), a framework originally developed for molecular simulation, and argue that it is a useful paradigm for developing mechanistic understanding of rare climate events. TPT provides a method to calculate statistical properties of the paths into the event. As an initial demonstration of the utility of TPT, we analyze a low-order model of sudden stratospheric warming (SSW), a dramatic disturbance to the polar vortex which can induce extreme cold spells at the surface in the midlatitudes. SSW events pose a major challenge for seasonal weather prediction because of their rapid, complex onset and development. Climate models struggle to capture the long-term statistics of SSW, owing to their diversity and intermittent nature. We use a stochastically forced Holton-Mass-type model with two stable states, corresponding to radiative equilibrium and a vacillating SSW-like regime. In this stochastic bistable setting, from certain probabilistic forecasts TPT facilitates estimation of dominant transition pathways and return times of transitions. These "dynamical statistics" are obtained by solving partial differential equations in the model's phase space. With future application to more complex models, TPT and its constituent quantities promise to improve the predictability of extreme weather events, through both generation and principled evaluation of forecasts.

physics.ao-ph↗

arXiv2019

A Kinetic Monte Carlo Approach for Simulating Cascading Transmission Line Failure

In this work, cascading transmission line failures are studied through a dynamical model of the power system operating under fixed conditions. The power grid is modeled as a stochastic dynamical system where first-principles electromechanical dynamics are excited by small Gaussian disturbances in demand and generation around a specified operating point. In this context, a single line failure is interpreted in a large deviation context as a first escape event across a surface in phase space defined by line security constraints. The resulting system of stochastic differential equations admits a transverse decomposition of the drift, which leads to considerable simplification in evaluating the quasipotential (rate function) and, consequently, computation of exit rates. Tractable expressions for the rate of transmission line failure in a restricted network are derived from large deviation theory arguments and validated against numerical simulations. Extensions to realistic settings are considered, and individual line failure models are aggregated into a Markov model of cascading failure inspired by chemical kinetics. Cascades are generated by traversing a graph composed of weighted edges representing transitions to degraded network topologies. Numerical results indicate that the Markov model can produce cascades with qualitative power-law properties similar to those observed in empirical cascades.

eess.SY↗

arXiv2019

Langevin Markov Chain Monte Carlo with stochastic gradients

Monte Carlo sampling techniques have broad applications in machine learning, Bayesian posterior inference, and parameter estimation. Often the target distribution takes the form of a product distribution over a dataset with a large number of entries. For sampling schemes utilizing gradient information it is cheaper for the derivative to be approximated using a random small subset of the data, introducing extra noise into the system. We present a new discretization scheme for underdamped Langevin dynamics when utilizing a stochastic (noisy) gradient. This scheme is shown to bias computed averages to second order in the stepsize while giving exact results in the special case of sampling a Gaussian distribution with a normally distributed stochastic gradient.

stat.ME↗

arXiv2019

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 [L.-H. Lim and J. Weare, SIAM Rev. 59, 547 (2017)]. These methods, which we term "FCI-FRI," stochastically impose sparsity during iterations of the power method and can be viewed as a generalization of full configuration interaction quantum Monte Carlo (FCIQMC) without walkers. In addition to the multinomial scheme commonly used to sample excitations in FCIQMC, we present a systematic scheme where excitations are not sampled independently. Performing ground-state calculations on five small molecules at fixed cost, we find that the systematic FCI-FRI scheme is 11 to 45 times more statistically efficient than the multinomial FCI-FRI scheme, which is in turn 1.4 to 178 times more statistically efficient than the original FCIQMC algorithm.

physics.chem-ph↗

arXiv2019

Maximizing simulated tropical cyclone intensity with action minimization

Direct computer simulation of intense tropical cyclones (TCs) in weather models is limited by computational expense. Intense TCs are rare and have small-scale structures, making it difficult to produce large ensembles of storms at high resolution. Further, models often fail to capture the process of rapid intensification, which is a distinguishing feature of many intense TCs. Understanding rapid intensification is especially important in the context of global warming, which may increase the frequency of intense TCs. To better leverage computational resources for the study of rapid intensification, we introduce an action minimization algorithm applied to the WRF and WRFPLUS models. Action minimization nudges the model into forming more intense TCs than it otherwise would; it does so via the maximum likelihood path in a stochastic formulation of the model, thereby allowing targeted study of intensification mechanisms. We apply action minimization to simulations of Hurricanes Danny (2015) and Fred (2009) at 6 km resolution to demonstrate that the algorithm consistently intensifies TCs via physically plausible pathways. We show an approximately ten-fold computational savings using action minimization to study the tail of the TC intensification distribution. Further, for Hurricanes Danny and Fred, action minimization produces perturbations that preferentially reduce low-level shear as compared to upper-level shear, at least above a threshold of approximately $4 \mathrm{\ m \ s^{-1}}$. We also demonstrate that asymmetric, time-dependent patterns of heating can cause significant TC intensification beyond symmetric, azimuthally-averaged heating and find a regime of non-linear response to asymmetric heating that has not been extensively studied in previous work.

physics.ao-ph↗

arXiv2019

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. We demonstrate the advantages of Quantile DMC compared to other sampling methods and discuss practical aspects of implementing Quantile DMC. To test the feasibility of Quantile DMC for extreme mesoscale weather, we sample extremely intense realizations of two historical tropical cyclones, 2010 Hurricane Earl and 2015 Hurricane Joaquin. Our results demonstrate Quantile DMC's potential to provide low-variance extreme weather statistics while highlighting the work that is necessary for Quantile DMC to attain greater efficiency in future applications.

physics.ao-ph↗

arXiv2019

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 estimation of dynamical quantities using a Markov state model. More generally, the boundary conditions impose restrictions on the choice of basis sets. We demonstrate how an alternative basis can be constructed using ideas from diffusion maps. In our numerical experiments, this basis gives results of comparable or better accuracy to Markov state models. Additionally, we show that delay embedding can reduce the information lost when projecting the system's dynamics for model construction; this improves estimates of dynamical statistics considerably over the standard practice of increasing the lag time.

physics.data-an↗

arXiv2018

Simulating the stochastic dynamics and cascade failure of power networks

For large-scale power networks, the failure of particular transmission lines can offload power to other lines and cause self-protection trips to activate, instigating a cascade of line failures. In extreme cases, this can bring down the entire network. Learning where the vulnerabilities are and the expected timescales for which failures are likely is an active area of research. In this article we present a novel stochastic dynamics model for a large-scale power network along with a framework for efficient computer simulation of the model including long timescale events such as cascade failure. We build on an existing Hamiltonian formulation and introduce stochastic forcing and damping components to simulate small perturbations to the network. Our model and simulation framework allow assessment of the particular weaknesses in a power network that make it susceptible to cascade failure, along with the timescales and mechanism for expected failures.

physics.soc-ph↗

arXiv2017

Umbrella sampling: a powerful method to sample tails of distributions

We present the umbrella sampling (US) technique and show that it can be used to sample extremely low probability areas of the posterior distribution that may be required in statistical analyses of data. In this approach sampling of the target likelihood is split into sampling of multiple biased likelihoods confined within individual umbrella windows. We show that the US algorithm is efficient and highly parallel and that it can be easily used with other existing MCMC samplers. The method allows the user to capitalize on their intuition and define umbrella windows and increase sampling accuracy along specific directions in the parameter space. Alternatively, one can define umbrella windows using an approach similar to parallel tempering. We provide a public code that implements umbrella sampling as a standalone python package. We present a number of tests illustrating the power of the US method in sampling low probability areas of the posterior and show that this ability allows a considerably more robust sampling of multi-modal distributions compared to the standard sampling methods. We also present an application of the method in a real world example of deriving cosmological constraints using the supernova type Ia data. We show that umbrella sampling can sample the posterior accurately down to the $\approx 15σ$ credible region in the $Ω_{\rm m}-Ω_Λ$ plane, while for the same computational work the affine-invariant MCMC sampling implemented in the {\tt emcee} code samples the posterior reliably only to $\approx 3σ$.

astro-ph.IM↗

arXiv2017

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. We demonstrate the power of the framework by defining strata in terms of both points in time and path-dependent variables for efficiently estimating averages that were not previously tractable.

cond-mat.stat-mech↗

arXiv2017

Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra

We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of randomized iterative algorithms based on similar principles to address a variety of common tasks in numerical linear algebra. From the point of view of numerical linear algebra, the main novelty of the Fast Randomized Iteration schemes described in this article is that they work in either linear or constant cost per iteration (and in total, under appropriate conditions) and are rather versatile: we will show how they apply to solution of linear systems, eigenvalue problems, and matrix exponentiation, in dimensions far beyond the present limits of numerical linear algebra. While traditional iterative methods in numerical linear algebra were created in part to deal with instances where a matrix (of size $\mathcal{O}(n^2)$) is too big to store, the algorithms that we propose are effective even in instances where the solution vector itself (of size $\mathcal{O}(n)$) may be too big to store or manipulate. In fact, our work is motivated by recent DMC based quantum Monte Carlo schemes that have been applied to matrices as large as $10^{108} \times 10^{108}$. We provide basic convergence results, discuss the dependence of these results on the dimension of the system, and demonstrate dramatic cost savings on a range of test problems.

math.NA↗

arXiv2016

Ensemble preconditioning for Markov chain Monte Carlo simulation

We describe parallel Markov chain Monte Carlo methods that propagate a collective ensemble of paths, with local covariance information calculated from neighboring replicas. The use of collective dynamics eliminates multiplicative noise and stabilizes the dynamics thus providing a practical approach to difficult anisotropic sampling problems in high dimensions. Numerical experiments with model problems demonstrate that dramatic potential speedups, compared to various alternative schemes, are attainable.

stat.ME↗

arXiv2016

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. We discuss how the error scales with the number of windows. Then, we derive a central limit theorem for averages that are obtained from umbrella sampling. The central limit theorem suggests an estimator of the error contributions from individual windows, and we develop a simple and computationally inexpensive procedure for implementing it. We demonstrate this estimator for simulations of the alanine dipeptide and show that it emphasizes low free energy pathways between stable states in comparison to existing approaches for assessing error contributions. We discuss the possibility of using the estimator and, more generally, the eigenvector method for umbrella sampling to guide adaptation of the simulation parameters to accelerate convergence.

cond-mat.stat-mech↗

arXiv2015

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. Moreover, our bounds have a simple interpretation in terms of hitting times, which can be used to draw intuitive but rigorous conclusions about the sensitivity of a chain to various types of perturbations.

math.NA↗