Showing papers on "Dynamic Monte Carlo method published in 2017"

Self-learning Monte Carlo method

Abstract: Monte Carlo simulation is an unbiased numerical tool for studying classical and quantum many-body systems. One of its bottlenecks is the lack of a general and efficient update algorithm for large size systems close to the phase transition, for which local updates perform badly. In this Rapid Communication, we propose a general-purpose Monte Carlo method, dubbed self-learning Monte Carlo (SLMC), in which an efficient update algorithm is first learned from the training data generated in trial simulations and then used to speed up the actual simulation. We demonstrate the efficiency of SLMC in a spin model at the phase transition point, achieving a 10--20 times speedup.

Variational Sequential Monte Carlo

Abstract: Many recent advances in large scale probabilistic inference rely on variational methods The success of variational approaches depends on (i) formulating a flexible parametric family of distributions, and (ii) optimizing the parameters to find the member of this family that most closely approximates the exact posterior In this paper we present a new approximating family of distributions, the variational sequential Monte Carlo (VSMC) family, and show how to optimize it in variational inference VSMC melds variational inference (VI) and sequential Monte Carlo (SMC), providing practitioners with flexible, accurate, and powerful Bayesian inference The VSMC family is a variational family that can approximate the posterior arbitrarily well, while still allowing for efficient optimization of its parameters We demonstrate its utility on state space models, stochastic volatility models for financial data, and deep Markov models of brain neural circuits

Learning Deep Latent Gaussian Models with Markov Chain Monte Carlo

Abstract: Deep latent Gaussian models are powerful and popular probabilistic models of highdimensional data. These models are almost always fit using variational expectationmaximization, an approximation to true maximum-marginal-likelihood estimation. In this paper, we propose a different approach: rather than use a variational approximation (which produces biased gradient signals), we use Markov chain Monte Carlo (MCMC, which allows us to trade bias for computation). We find that our MCMC-based approach has several advantages: it yields higher held-out likelihoods, produces sharper images, and does not suffer from the variational overpruning effect. MCMC’s additional computational overhead proves to be significant, but not prohibitive.

Self-learning Monte Carlo method: Continuous-time algorithm

Abstract: The self-learning Monte Carlo (SLMC) method speeds up the Monte Carlo simulation by designing and training an effective model to propose efficient global updates We implement the continuous-time quantum Monte Carlo algorithm for quantum impurity models in the framework of SLMC We introduce and train a diagram generating function (DGF) to model the probability distribution of field configurations in the continuous imaginary time at all orders of diagrammatic expansion By using the DGF to propose global updates, we show that the self-learning continuous-time Monte Carlo method can significantly reduce the computational complexity of the simulation

Self-learning quantum Monte Carlo method in interacting fermion systems

Abstract: The self-learning Monte Carlo method is a powerful general-purpose numerical method recently introduced to simulate many-body systems. In this work, we extend it to an interacting fermion quantum system in the framework of the widely used determinant quantum Monte Carlo. This method can generally reduce the computational complexity and moreover can greatly suppress the autocorrelation time near a critical point. This enables us to simulate an interacting fermion system on a $100\ifmmode\times\else\texttimes\fi{}100$ lattice even at the critical point and obtain critical exponents with high precision.

Inchworm Monte Carlo for exact non-adiabatic dynamics. I. Theory and algorithms

Abstract: In this paper, we provide a detailed description of the inchworm Monte Carlo formalism for the exact study of real-time non-adiabatic dynamics This method optimally recycles Monte Carlo information from earlier times to greatly suppress the dynamical sign problem Using the example of the spin-boson model, we formulate the inchworm expansion in two distinct ways: The first with respect to an expansion in the system-bath coupling and the second as an expansion in the diabatic coupling The latter approach motivates the development of a cumulant version of the inchworm Monte Carlo method, which has the benefit of improved scaling This paper deals completely with methodology, while Paper II provides a comprehensive comparison of the performance of the inchworm Monte Carlo algorithms to other exact methodologies as well as a discussion of the relative advantages and disadvantages of each

Circulant Embedding of Approximate Covariances for Inference From Gaussian Data on Large Lattices

Abstract: Recently proposed computationally efficient Markov chain Monte Carlo (MCMC) and Monte Carlo Expectation-Maximization (EM) methods for estimating covariance parameters from lattice data rely on successive imputations of values on an embedding lattice that is at least two times larger in each dimension. These methods can be considered exact in some sense, but we demonstrate that using such a large number of imputed values leads to slowly converging Markov chains and EM algorithms. We propose instead the use of a discrete spectral approximation to allow for the implementation of these methods on smaller embedding lattices. While our methods are approximate, our examples indicate that the error introduced by this approximation is small compared to the Monte Carlo errors present in long Markov chains or many iterations of Monte Carlo EM algorithms. Our results are demonstrated in simulation studies, as well as in numerical studies that explore both increasing domain and fixed domain asymptotics. We com...

Computation of ground-state properties in molecular systems: back-propagation with auxiliary-field quantum Monte Carlo

Abstract: We address the computation of ground-state properties of chemical systems and realistic materials within the auxiliary-field quantum Monte Carlo method. The phase constraint to control the Fermion phase problem requires the random walks in Slater determinant space to be open-ended with branching. This in turn makes it necessary to use back-propagation (BP) to compute averages and correlation functions of operators that do not commute with the Hamiltonian. Several BP schemes are investigated, and their optimization with respect to the phaseless constraint is considered. We propose a modified BP method for the computation of observables in electronic systems, discuss its numerical stability and computational complexity, and assess its performance by computing ground-state properties in several molecular systems, including small organic molecules.

On perturbed proximal gradient algorithms

Abstract: We study a version of the proximal gradient algorithm for which the gradient is intractable and is approximated by Monte Carlo methods (and in particular Markov Chain Monte Carlo). We derive conditions on the step size and the Monte Carlo batch size under which convergence is guaranteed: both increasing batch size and constant batch size are considered. We also derive non-asymptotic bounds for an averaged version. Our results cover both the cases of biased and unbiased Monte Carlo approximation. To support our findings, we discuss the inference of a sparse generalized linear model with random effect and the problem of learning the edge structure and parameters of sparse undirected graphical models.

Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems

Abstract: We are interested in computing the expectation of a functional of a PDE solution under a Bayesian posterior distribution. Using Bayes's rule, we reduce the problem to estimating the ratio of two related prior expectations. For a model elliptic problem, we provide a full convergence and complexity analysis of the ratio estimator in the case where Monte Carlo, quasi-Monte Carlo, or multilevel Monte Carlo methods are used as estimators for the two prior expectations. We show that the computational complexity of the ratio estimator to achieve a given accuracy is the same as the corresponding complexity of the individual estimators for the numerator and the denominator. We also include numerical simulations, in the context of the model elliptic problem, which demonstrate the effectiveness of the approach.

DSMC modeling of flows with recombination reactions

Abstract: An empirical microscopic recombination model is developed for the direct simulation Monte Carlo method that complements the extended weak vibrational bias model of dissociation. The model maintains the correct equilibrium reaction constant in a wide range of temperatures by using the collision theory to enforce the number of recombination events. It also strictly follows the detailed balance requirement for equilibrium gas. The model and its implementation are verified with oxygen and nitrogen heat bath relaxation and compared with available experimental data on atomic oxygen recombination in argon and molecular nitrogen.

Currents and Green's functions of impurities out of equilibrium: Results from inchworm quantum Monte Carlo

Abstract: We generalize the recently developed inchworm quantum Monte Carlo method to the full Keldysh contour with forward, backward, and equilibrium branches to describe the dynamics of strongly correlated impurity problems with time-dependent parameters. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We then illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state.

Inchworm Monte Carlo for exact non-adiabatic dynamics. II. Benchmarks and comparison with established methods.

Abstract: In this second paper of a two part series, we present extensive benchmark results for two different inchworm Monte Carlo expansions for the spin-boson model Our results are compared to previously developed numerically exact approaches for this problem A detailed discussion of convergence and error propagation is presented Our results and analysis allow for an understanding of the benefits and drawbacks of inchworm Monte Carlo compared to other approaches for exact real-time non-adiabatic quantum dynamics

A Model to Represent Correlated Time Series in Reliability Evaluation by Non-Sequential Monte Carlo Simulation

Abstract: This paper proposes a model that represents statistically dependent time-varying quantities, such as loads, wind power generation, and water inflows, and can be applied to evaluate power systems composite reliability by Non-Sequential Monte Carlo Simulation (MCS). This proposal is based on nonparametric stochastic models, which do not require a priori characterizations of the probability density functions of the random variables. Additionally, the maximal information coefficient, which allows mapping nonlinear relationships between the variables, and Bayesian network structures, which allow handling high dimensionality problems when multiple time series are represented, are applied. The proposed model allows reliability indices to be obtained with the same accuracy as the Sequential MCS but with computational costs on the order of the Non-Sequential MCS. The model is flexible enough to represent the relationships between variables with different levels of discretization, as in the case of the wind power generation and water inflows of a hydroelectric system.

Selected configuration interaction method using sampled first-order corrections to wave functions

Abstract: A new selected configuration interaction (CI) method was proposed for the potential energy surfaces of quasi-degenerate and excited states. Slater determinants are generated by sampling the first-order corrections to the target-state wave functions using the quantum Monte Carlo method in determinant space. As in the Monte Carlo (MC) CI method, the wave function is improved at each iteration by generating new determinants and applying a pruning step. Compared to the random generation in the MCCI calculations, the number of iterations before convergence is significantly reduced. Regarding the potential energy curves of the ground and excited states of C2, the non-parallelity errors were sufficiently small, thus indicating the method’s applicability to the calculations of potential energy surfaces.

Numerical Methods in Coupled Monte Carlo and Thermal-Hydraulic Calculations

Abstract: Large-scale reactor calculations with Monte Carlo (MC), including nonlinear feedback effects, have become a reality in the course of the last decade. In particular, implementations of coupled MC an...

Preliminary Coupling of the Monte Carlo Code OpenMC and the Multiphysics Object-Oriented Simulation Environment for Analyzing Doppler Feedback in Monte Carlo Simulations

Abstract: In recent years, the use of Monte Carlo methods for modeling reactors has become feasible due to the increasing availability of massively parallel computer systems. One of the primary challenges ye...

Coupled Electron-Ion Monte Carlo simulation of hydrogen molecular crystals

Abstract: We performed simulations for solid molecular hydrogen at high pressures (250GPa$\leq$P$\leq$500GPa) along two isotherms at T=200 K (phases III and VI) and at T=414 K (phase IV). At T=200K we considered likely candidates for phase III, the C2c and Cmca12 structures, while at T=414K in phase IV we studied the Pc48 structure. We employed both Coupled Electron-Ion Monte Carlo (CEIMC) and Path Integral Molecular Dynamics (PIMD) based on Density Functional Theory (DFT) using the vdW-DF approximation. The comparison between the two methods allows us to address the question of the accuracy of the xc approximation of DFT for thermal and quantum protons without recurring to perturbation theories. In general, we find that atomic and molecular fluctuations in PIMD are larger than in CEIMC which suggests that the potential energy surface from vdW-DF is less structured than the one from Quantum Monte Carlo. We find qualitatively different behaviors for systems prepared in the C2c structure for increasing pressure. Within PIMD the C2c structure is dynamically partially stable for P$\leq$250GPa only: it retains the symmetry of the molecular centers but not the molecular orientation; at intermediate pressures it develops layered structures like Pbcn or Ibam and transforms to the metallic Cmca-4 structure at P$\geq$450GPa. Instead, within CEIMC, the C2c structure is found to be dynamically stable at least up to 450GPa; at increasing pressure the molecular bond length increases and the nuclear correlation decreases. For the other two structures the two methods are in qualitative agreement although quantitative differences remain. We discuss various structural properties and the electrical conductivity. We find these structures become conducting around 350GPa but the metallic Drude-like behavior is reached only at around 500GPa, consistent with recent experimental claims.

Evaluation of Different Approximations for Correlation Coefficients in Stochastic FDTD to Estimate SAR Variance in a Human Head Model

Abstract: In this paper, the stochastic finite-difference time domain (S-FDTD) method is employed to calculate the standard deviation of specific absorption rate (SAR) in a two-dimensional (2-D) slice of human head. S-FDTD calculates both the mean and standard deviation of SAR caused by variability or uncertainty in the electrical properties of the human head tissues. The accuracy of the S-FDTD result is controlled by the approximations for correlation coefficients between the electrical properties of the tissues and the fields propagating in them. Hence, different approximations for correlation coefficients are tested in order to evaluate their effect on the standard deviation of SAR. The 1-D Monte Carlo correlation coefficient (MC-CC) approximation reported in our previous work is successfully extended to 2-D and also tested for the head model. Then, all the results are compared with that of full-fledged Monte Carlo method (considered as gold standard in statistical simulations). In order to accelerate the simulations, the proposed algorithm is run on graphics processing unit by exploiting OpenACC application program interface. Using different correlation coefficients shows that the extended 2-D MC-CC S-FDTD results are very close to that of Monte Carlo and yield more accurate results than other approximations in SAR calculations.

Path integral Monte Carlo simulations of dense carbon-hydrogen plasmas

Abstract: Carbon-hydrogen plasmas and hydrocarbon materials are of broad interest to laser shock experimentalists, high energy density physicists, and astrophysicists. Accurate equations of state (EOS) of hydrocarbons are valuable for various studies from inertial confinement fusion (ICF) to planetary science. By combining path integral Monte Carlo (PIMC) results at high temperatures and density functional theory molecular dynamics (DFT-MD) results at lower temperatures, we compute the EOS for hydrocarbons at 1473 separate ($\rho,T$)-points distributed over a range of compositions. These methods accurately treat electronic excitation and many-body interaction effects and thus provide a benchmark-quality EOS that surpasses that of semi-empirical and Thomas-Fermi-based methods in the warm dense matter regime. By comparing our first-principles EOS to the LEOS 5112 model for CH, we validate the specific heat assumptions in this model but suggest that the Grueneisen parameter is too large at low temperature. Based on our first-principles EOS, we predict the Hugoniot curve of polystyrene to be 2-5% softer at maximum compression than that predicted by orbital-free DFT and SESAME 7593. By investigating the atomic structure and chemical bonding, we show a drastic decrease in the lifetime of chemical bonds in the pressure interval of 0.4-4 megabar. We find the assumption of linear mixing to be valid for describing the EOS and the shock Hugoniot curve of the dense, partially ionized hydrocarbons under consideration. We make predictions of the shock compression of glow-discharge polymers and investigate the effects of oxygen content and C:H ratio on their Hugoniot curve. Our full suite of first-principles simulation results may be used to benchmark future theoretical investigations pertaining to hydrocarbon EOS, and should be helpful in guiding the design of future gigabar experiments.

Excitation variance matching with limited configuration interaction expansions in variational Monte Carlo

Abstract: In the regime where traditional approaches to electronic structure cannot afford to achieve accurate energy differences via exhaustive wave function flexibility, rigorous approaches to balancing different states’ accuracies become desirable As a direct measure of a wave function’s accuracy, the energy variance offers one route to achieving such a balance Here, we develop and test a variance matching approach for predicting excitation energies within the context of variational Monte Carlo and selective configuration interaction In a series of tests on small but difficult molecules, we demonstrate that the approach is effective at delivering accurate excitation energies when the wave function is far from the exhaustive flexibility limit Results in C3, where we combine this approach with variational Monte Carlo orbital optimization, are especially encouraging

Modeling the Migration of Platinum Nanoparticles on Surfaces Using a Kinetic Monte Carlo Approach

Abstract: We propose a kinetic Monte Carlo (kMC) model for simulating the movement of platinum particles on supports, based on atom-by-atom diffusion on the surface of the particle. The proposed model was able to reproduce equilibrium cluster shapes predicted using Wulff-construction. The diffusivity of platinum particles was simulated both purely based on random motion and assisted using an external field that causes a drift velocity. The overall particle diffusivity increases with temperature; however, the extracted activation barrier appears to be temperature independent. In addition, this barrier was found to increase with particle size, as well as, with the adhesion between the particle and the support.

Transforming high-dimensional potential energy surfaces into sum-of-products form using Monte Carlo methods.

Abstract: We propose a Monte Carlo method, "Monte Carlo Potfit," for transforming high-dimensional potential energy surfaces evaluated on discrete grid points into a sum-of-products form, more precisely into a Tucker form. To this end we use a variational ansatz in which we replace numerically exact integrals with Monte Carlo integrals. This largely reduces the numerical cost by avoiding the evaluation of the potential on all grid points and allows a treatment of surfaces up to 15-18 degrees of freedom. We furthermore show that the error made with this ansatz can be controlled and vanishes in certain limits. We present calculations on the potential of HFCO to demonstrate the features of the algorithm. To demonstrate the power of the method, we transformed a 15D potential of the protonated water dimer (Zundel cation) in a sum-of-products form and calculated the ground and lowest 26 vibrationally excited states of the Zundel cation with the multi-configuration time-dependent Hartree method.