Exploring DFT+U parameter space with a Bayesian calibration assisted by Markov chain Monte Carlo sampling

AbstractThe density-functional theory is widely used to predict the physical properties of materials. However, it usually fails for strongly correlated materials. A popular solution is to use the Hubbard correction to treat strongly correlated electronic states. Unfortunately, the values of the Hubbard U and J parameters are initially unknown, and they can vary from one material to another. In this semi-empirical study, we explore the U and J parameter space of a group of iron-based compounds to simultaneously improve the prediction of physical properties (volume, magnetic moment, and bandgap). We used a Bayesian calibration assisted by Markov chain Monte Carlo sampling for three different exchange-correlation functionals (LDA, PBE, and PBEsol). We found that LDA requires the largest U correction. PBE has the smallest standard deviation and its U and J parameters are the most transferable to other iron-based compounds. Lastly, PBE predicts lattice parameters reasonably well without the Hubbard correction.

Download Full-text

OPTIMIZED NESTED MARKOV CHAIN MONTE CARLO SAMPLING: APPLICATION TO THE LIQUID NITROGEN HUGONIOT USING DENSITY FUNCTIONAL THEORY

10.1063/1.3295191 ◽

2009 ◽

Cited By ~ 1

Author(s):

M. Sam Shaw ◽

Joshua D. Coe ◽

Thomas D. Sewell ◽

Mark Elert ◽

Michael D. Furnish ◽

...

Keyword(s):

Density Functional Theory ◽

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Liquid Nitrogen ◽

Density Functional ◽

Monte Carlo Sampling ◽

Functional Theory

Download Full-text

Nested Markov chain Monte Carlo sampling of a density functional theory potential: Equilibrium thermodynamics of dense fluid nitrogen

The Journal of Chemical Physics ◽

10.1063/1.3200904 ◽

2009 ◽

Vol 131 (7) ◽

pp. 074105 ◽

Cited By ~ 8

Author(s):

Joshua D. Coe ◽

Thomas D. Sewell ◽

M. Sam Shaw

Keyword(s):

Density Functional Theory ◽

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Density Functional ◽

Monte Carlo Sampling ◽

Equilibrium Thermodynamics ◽

Functional Theory ◽

Dense Fluid ◽

Potential Equilibrium

Download Full-text

A Distributed Computer System for Parallel Markov Chain Monte Carlo (MCMC)

Inquiry@Queen's Undergraduate Research Conference Proceedings ◽

10.24908/iqurcp.9597 ◽

2018 ◽

Author(s):

Michael Hynes

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Parameter Space ◽

Heterogeneous Computing ◽

Peer To Peer ◽

Parallel Tempering ◽

Single Chain ◽

Distributed Computing Systems ◽

Mcmc Algorithms

A ubiquitous problem in physics is to determine expectation values of observables associated with a system. This problem is typically formulated as an integration of some likelihood over a multidimensional parameter space. In Bayesian analysis, numerical Markov Chain Monte Carlo (MCMC) algorithms are employed to solve such integrals using a fixed number of samples in the Markov Chain. In general, MCMC algorithms are computationally expensive for large datasets and have difficulties sampling from multimodal parameter spaces. An MCMC implementation that is robust and inexpensive for researchers is desired. Distributed computing systems have shown the potential to act as virtual supercomputers, such as in the SETI@home project in which millions of private computers participate. We propose that a clustered peer-to-peer (P2P) computer network serves as an ideal structure to run Markovian state exchange algorithms such as Parallel Tempering (PT). PT overcomes the difficulty in sampling from multimodal distributions by running multiple chains in parallel with different target distributions andexchanging their states in a Markovian manner. To demonstrate the feasibility of peer-to-peer Parallel Tempering (P2P PT), a simple two-dimensional dataset consisting of two Gaussian signals separated by a region of low probability was used in a Bayesian parameter fitting algorithm. A small connected peer-to-peer network was constructed using separate processes on a linux kernel, and P2P PT was applied to the dataset. These sampling results were compared with those obtained from sampling the parameter space with a single chain. It was found that the single chain was unable to sample both modes effectively, while the P2P PT method explored the target distribution well, visiting both modes approximately equally. Future work will involve scaling to many dimensions and large networks, and convergence conditions with highly heterogeneous computing capabilities of members within the network.

Download Full-text

Bayes or Bootstrap? A Simulation Study Comparing the Performance of Bayesian Markov Chain Monte Carlo Sampling and Bootstrapping in Assessing Phylogenetic Confidence

Molecular Biology and Evolution ◽

10.1093/molbev/msg028 ◽

2003 ◽

Vol 20 (2) ◽

pp. 255-266 ◽

Cited By ~ 612

Author(s):

M. E. Alfaro

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Simulation Study ◽

Monte Carlo Sampling

Download Full-text

Bayesian calibration of terrestrial ecosystem models: a study of advanced Markov chain Monte Carlo methods

Biogeosciences ◽

10.5194/bg-14-4295-2017 ◽

2017 ◽

Vol 14 (18) ◽

pp. 4295-4314 ◽

Cited By ~ 12

Author(s):

Dan Lu ◽

Daniel Ricciuto ◽

Anthony Walker ◽

Cosmin Safta ◽

William Munger

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Likelihood Function ◽

Terrestrial Ecosystem ◽

Error Model ◽

Model Parameters ◽

Posterior Distributions ◽

Ecosystem Models ◽

Bayesian Calibration

Abstract. Calibration of terrestrial ecosystem models is important but challenging. Bayesian inference implemented by Markov chain Monte Carlo (MCMC) sampling provides a comprehensive framework to estimate model parameters and associated uncertainties using their posterior distributions. The effectiveness and efficiency of the method strongly depend on the MCMC algorithm used. In this work, a differential evolution adaptive Metropolis (DREAM) algorithm is used to estimate posterior distributions of 21 parameters for the data assimilation linked ecosystem carbon (DALEC) model using 14 years of daily net ecosystem exchange data collected at the Harvard Forest Environmental Measurement Site eddy-flux tower. The calibration of DREAM results in a better model fit and predictive performance compared to the popular adaptive Metropolis (AM) scheme. Moreover, DREAM indicates that two parameters controlling autumn phenology have multiple modes in their posterior distributions while AM only identifies one mode. The application suggests that DREAM is very suitable to calibrate complex terrestrial ecosystem models, where the uncertain parameter size is usually large and existence of local optima is always a concern. In addition, this effort justifies the assumptions of the error model used in Bayesian calibration according to the residual analysis. The result indicates that a heteroscedastic, correlated, Gaussian error model is appropriate for the problem, and the consequent constructed likelihood function can alleviate the underestimation of parameter uncertainty that is usually caused by using uncorrelated error models.

Download Full-text

Estimation of trace gas fluxes with objectively determined basis functions using reversible-jump Markov chain Monte Carlo

Geoscientific Model Development ◽

10.5194/gmd-9-3213-2016 ◽

2016 ◽

Vol 9 (9) ◽

pp. 3213-3229 ◽

Cited By ~ 22

Author(s):

Mark F. Lunt ◽

Matt Rigby ◽

Anita L. Ganesan ◽

Alistair J. Manning

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Parameter Space ◽

Spatial Domain ◽

Monte Carlo Algorithm ◽

Trace Gas ◽

Reversible Jump ◽

Pseudo Data ◽

Atmospheric Trace Gas

Abstract. Atmospheric trace gas inversions often attempt to attribute fluxes to a high-dimensional grid using observations. To make this problem computationally feasible, and to reduce the degree of under-determination, some form of dimension reduction is usually performed. Here, we present an objective method for reducing the spatial dimension of the parameter space in atmospheric trace gas inversions. In addition to solving for a set of unknowns that govern emissions of a trace gas, we set out a framework that considers the number of unknowns to itself be an unknown. We rely on the well-established reversible-jump Markov chain Monte Carlo algorithm to use the data to determine the dimension of the parameter space. This framework provides a single-step process that solves for both the resolution of the inversion grid, as well as the magnitude of fluxes from this grid. Therefore, the uncertainty that surrounds the choice of aggregation is accounted for in the posterior parameter distribution. The posterior distribution of this transdimensional Markov chain provides a naturally smoothed solution, formed from an ensemble of coarser partitions of the spatial domain. We describe the form of the reversible-jump algorithm and how it may be applied to trace gas inversions. We build the system into a hierarchical Bayesian framework in which other unknown factors, such as the magnitude of the model uncertainty, can also be explored. A pseudo-data example is used to show the usefulness of this approach when compared to a subjectively chosen partitioning of a spatial domain. An inversion using real data is also shown to illustrate the scales at which the data allow for methane emissions over north-west Europe to be resolved.

Download Full-text