Monte Carlo Formulation Research Articles

Adaptive formulation for probabilistic storm surge predictions through sharing of numerical simulation results across storm advisories

As a tropical storm/cyclone approaches landfall, real-time probabilistic predictions for the anticipated surge provide valuable information for emergency preparedness/response decisions. These probabilistic predictions are made through an uncertainty quantification process that involves: (i) generating a sufficiently large ensemble of storm scenarios based on the nominal storm advisory and the anticipated forecast errors; (ii) performing high-fidelity numerical simulations to obtain surge predictions for each storm scenario; and (iii) estimating surge statistics of interest by assembling the simulation results. This process is repeated whenever the nominal storm advisory is updated. The number of storm scenarios utilized in the analysis directly impacts the statistical accuracy of the probabilistic predictions; a larger ensemble improves accuracy but requires greater computational resources to provide predictions with the desired expediency to guide real-time decisions. This paper revisits two recently proposed Monte-Carlo (MC) frameworks that aim to improve accuracy without increasing the computational burden: adaptive importance sampling (AIS) and adaptive multi-fidelity Monte Carlo (AMFMC). The foundational concept behind them is similar: share numerical simulation results across the probabilistic predictions performed for different storm advisories to accelerate the MC estimation. This is achieved differently for each approach, through adaptive development of an importance sampling proposal density (for AIS) or a surrogate model (for AMFMC). Here, a direct comparison between these frameworks is established, focusing on the mechanisms for the information sharing and the challenges encountered in tuning the algorithm adaptive characteristics to provide probabilistic estimates across a large number of quantities of interest (QoIs), corresponding to the surge predictions for different locations within the coastal region of interest. As this large number results in conflicting choices for the adaptive characteristics, a compromise solution needs to be promoted. The efficacy of the two frameworks is examined in detail in this setting, comparing the accuracy of idealized implementations (adaptive decisions independently made for each QoI) to the accuracy of practical implementations (single, compromise decision within the MC implementation). The study also showcases the importance of information sharing across storm advisories in real-time probabilistic storm surge predictions and provides guidelines for an efficient adaptive MC formulation in such settings.

Optimal Allocation of Future Oil Production under Uncertainty: Lessons from Nigeria for Emerging Oil Producing Countries

Under the uncertainties presented by future upstream oil production, increasing domestic demand for petroleum products, volatile energy prices, a build-out of domestic refining capacity, and a global energy transition underway, this paper seeks an optimal allocation of Nigeria's projected oil production under scenarios representing possible versions of the future. Using the Reference Energy System developed by Gbakon et al. (2021) for crude oil utilization through a network of possible end-uses, the framework is tested under different scenarios. Furthermore, the framework is coupled to a Monte Carlo formulation of the problem, which allows greater flexibility in addressing questions of the likelihood of attaining desirable policy outcomes such as petroleum product self-sufficiency. Within this solution structure, a family of curves is generated, which represents the spread of outcomes. Scenario analysis, for example, shows that under 'Energy Transition' scenario, oil export ratio declines from 43% in 2025 to 6% in 2040. Whereas, under the 'Business-as-Usual' scenario, the oil export ratio declines from 37% in 2025 to 0% in 2034, with clear consequences for foreign exchange earnings. While under the ‘Stated Policy' scenario oil export ratio declines from 60% in 2025 to 54% in 2040. Implications for net system benefits and the respective drivers are further interrogated.

Quantum Monte Carlo formulation of the second order algebraic diagrammatic construction: Toward a massively parallel correlated excited state method.

The quantum Monte Carlo (QMC) algebraic diagrammatic construction (ADC) method is introduced, which solves the eigenvalue problem of the second-order ADC scheme for the polarization propagator stochastically within the framework of QMC methodology allowing for massively parallel computations. As common virtue of the Monte Carlo integration techniques, quantum Monte Carlo algebraic diagrammatic construction (QMCADC) enables exploitation of the sparsity of the effective ADC matrix, and it reduces the memory requirements by storing only a portion of configurations at each iteration. Furthermore, distributing memory and processing loads to different computing nodes enables the use of fast developing parallel computing resources. Here, the theory and implementation of QMCADC is reported and its viability is demonstrated by the first proof-of-principle calculations. The focus lies on the first excited state and the reproduction of the corresponding lowest vertical excitation energy of various molecular systems. QMCADC is shown to be a genuine stochastic solution of the ADC eigenvalue problem, and exact ADC values can be obtained with a marginal controllable error.

Path-Integral Calculation of the Second Dielectric and Refractivity Virial Coefficients of Helium, Neon, and Argon.

We present a method to calculate dielectric and refractivity virial coefficients using the path-integral Monte Carlo formulation of quantum statistical mechanics and validate it by comparing our results with equivalent calculations in the literature and with more traditional quantum calculations based on wavefunctions. We use state-of-the-art pair potentials and polarizabilities to calculate the second dielectric and refractivity virial coefficients of helium (both 3He and 4He), neon (both 20Ne and 22Ne), and argon. Our calculations extend to temperatures as low as 1 K for helium, 4 K for neon, and 50 K for argon. We estimate the contributions to the uncertainty of the calculated dielectric virial coefficients for helium and argon, finding that the uncertainty of the pair polarizability is by far the greatest contribution. Agreement with the limited experimental data available is generally good, but our results have smaller uncertainties, especially for helium. Our approach can be generalized in a straightforward manner to higher-order coefficients.

Influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates

Present study aims to investigate the influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates. Reddy's higher order shear deformation theory has been employed to model the displacement field. Variational approach has been used to derive the governing differential equations. Effect of material uncertainties in the formulation have been incorporated using first order perturbation technique (FOPT). An efficient stochastic finite element formulation (SFEM) have been used for the calculation of first and second order statics of natural frequency and transverse deflection. Validation of the results have been performed with the help of available literature and separately developed Monte Carlo formulation (MCS) algorithm. A large number of examples have been solved to quantify the effect of uncertainties on the vibration and bending characteristics of functionally graded skew sandwich plates.

Deviational simulation of phonon transport in graphene ribbons with ab initio scattering

We present a deviational Monte Carlo method for solving the Boltzmann-Peierls equation with ab initio 3-phonon scattering, for temporally and spatially dependent thermal transport problems in arbitrary geometries. Phonon dispersion relations and transition rates for graphene are obtained from density functional theory calculations. The ab initio scattering operator is simulated by an energy-conserving stochastic algorithm embedded within a deviational, low-variance Monte Carlo formulation. The deviational formulation ensures that simulations are computationally feasible for arbitrarily small temperature differences, while the stochastic treatment of the scattering operator is both efficient and exhibits no timestep error. The proposed method, in which geometry and phonon-boundary scattering are explicitly treated, is extensively validated by comparison to analytical results, previous numerical solutions and experiments. It is subsequently used to generate solutions for heat transport in graphene ribbons of various geometries and evaluate the validity of some common approximations found in the literature. Our results show that modeling transport in long ribbons of finite width using the homogeneous Boltzmann equation and approximating phonon-boundary scattering using an additional homogeneous scattering rate introduces an error on the order of 10% at room temperature, with the maximum deviation reaching 30% in the middle of the transition regime.

Residual Monte Carlo high-order solver for Moment-Based Accelerated Thermal Radiative Transfer equations

In this article we explore the possibility of replacing Standard Monte Carlo (SMC) transport sweeps within a Moment-Based Accelerated Thermal Radiative Transfer (TRT) algorithm with a Residual Monte Carlo (RMC) formulation. Previous Moment-Based Accelerated TRT implementations have encountered trouble when stochastic noise from SMC transport sweeps accumulates over several iterations and pollutes the low-order system. With RMC we hope to significantly lower the build-up of statistical error at a much lower cost. First, we display encouraging results for a zero-dimensional test problem. Then, we demonstrate that we can achieve a lower degree of error in two one-dimensional test problems by employing an RMC transport sweep with multiple orders of magnitude fewer particles per sweep. We find that by reformulating the high-order problem, we can compute more accurate solutions at a fraction of the cost.

Edward Epstein's Stochastic–Dynamic Approach to Ensemble Weather Prediction

In the late 1960s, well before the availability of computer power to produce ensemble weather forecasts, Edward Epstein (1931–2008) developed a stochastic–dynamic prediction (SDP) method for calculating the temporal evolution of mean value, variance, and covariance of the model variables: the statistical moments of a time-varying probability density function that define an ensemble forecast. This statistical–dynamical approach to ensemble forecasting is an alternative to the Monte Carlo formulation that is currently used in operations. The stages of Epstein's career that led to his development of this methodology are presented with the benefit of his oral history and supporting documentation that describes the retreat of strict deterministic weather forecasting. The important follow-on research by two of Epstein's proteges, Rex Fleming and Eric Pitcher, is also presented. A low-order nonlinear dynamical system is used to discuss the rudiments of SDP and Monte Carlo and to compare these approximate methods...

A model Monte Carlo collision operator for toroidal plasmas

In order to simulate radio refquency (RF)-heating in toroidal plasmas in the banana regime a model collision operator has been developed, which relaxes the distribution function towards a prescribed local Maxwellian either determined by experiments or transport codes. The pitch angle scattering by Coulomb collisions gives rise to spatial diffusion in toroidal plasmas because of the coupling between spatial and velocity coordinates. The coupling between the spatial and velocity components results in drift terms in the Monte Carlo formulation of the Fokker–Planck equation due to spatial derivatives of the Jacobian, the fraction of the trapped particles, the density and the temperature profiles. A simple RF operator is used to test the collision operator in conjunction with RF heating. The formation of a high-energy tail on the distribution function during RF heating leads to reduction of the density of the thermal ions as the tail builds up. For central heating this reduction can lead to hollow density profiles of thermal ions. The spatial diffusion caused by the relaxation of the thermal ions towards a prescribed density profile then gives rise to an increase of the density of resonant ions in regions with strong heating where the thermal ions diffuse towards higher energies.

Monte Carlo methods for radiative transfer in quasi-isothermal participating media

Based on the superposition principle, we propose in this study a Monte Carlo (MC) formulation for radiative transfer in quasi-isothermal media which consists in directly computing the difference between the actual radiative field and the equilibrium radiative field at the minimum temperature in the medium. This shift formulation is implemented for the prediction of radiative fluxes and volumetric powers in a combined free convection–radiation problem where a differentially heated cubical cavity is filled with air with a small amount of H2O and CO2. High resolution spectra are used to describe radiative properties of the gas in this 3D configuration. We show that, compared to the standard analog MC method, the shift approach leads to a huge saving of required computational times to reach a given convergence level. In addition, this approach is compared to reciprocal MC formulations and is shown to be more efficient for the prediction of wall fluxes but slightly less efficient for volumetric powers.

Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations

We present a new Monte Carlo method for obtaining solutions of the Boltzmann equation for describing phonon transport in micro and nanoscale devices. The proposed method can resolve arbitrarily small signals (e.g. temperature differences) at small constant cost and thus represents a considerable improvement compared to traditional Monte Carlo methods whose cost increases quadratically with decreasing signal. This is achieved via a control-variate variance reduction formulation in which the stochastic particle description only solves for the deviation from a nearby equilibrium, while the latter is described analytically. We also show that simulating an energy-based Boltzmann equation results in an algorithm that lends itself naturally to exact energy conservation thereby considerably improving the simulation fidelity. Simulations using the proposed method are used to investigate the effect of porosity on the effective thermal conductivity of silicon. We also present simulations of a recently developed thermal conductivity spectroscopy process. The latter simulations demonstrate how the computational gains introduced by the proposed method enable the simulation of otherwise intractable multiscale phenomena.

Error Estimate for the Ensemble Kalman Filter Analysis Step

The ensemble Kalman filter (EnKF) is an ensemble-based Monte Carlo formulation of the Kalman filter. In most practical cases it is based on a low-rank approximation of a covariance matrix from a moderately sized ensemble. Sampling errors lead to artificial effects, such as spurious correlations, deteriorating the estimates and the forecasts of the system states. Using random matrix theory, we derive the distribution of an energy norm of the EnKF sampling error for the estimate of the mean, assuming noiseless data. Despite this restriction, the obtained distribution should improve the understanding of the EnKF reliability. The distribution depends on ensemble size, model dimension, and observation locations. We demonstrate the use of the distribution on an example.

Effect of Cross Section Uncertainties on Criticality Benchmark Problem Analysis by MCCARD

A new Monte Carlo (MC) formulation aimed at quantifying eects of both statistical and nuclear cross section uncertainties on neutronics design parameters on nuclear systems is presented. The eectiveness of the formulation is examined through computing the contribution of the cross section uncertainties to the variance of the MC estimate on ke of two fast spectrum criticality benchmark problems; GODIVA and JEZEBEL using McCARD as a MC neutronics code and JENDL 3.3 for needed cross section and covariance data. It is shown that the MC formulation is capable of identifying the nuclide and the type of the cross sections that contribute signicantly to, and to elicit the necessary improvement in the existing covariance data to reduce, the uncertainties of the criticality calculations.

Iterative Monte Carlo formulation of real-time correlation functions

We present an iterative Monte Carlo path integral methodology for evaluating thermally averaged real-time correlation functions. Standard path integral Monte Carlo methods are used to sample paths along the imaginary time contour. Propagation of the density matrix is performed iteratively on a grid composed of the end points of the sampled paths. Minimally oscillatory propagators are constructed using energy filtering techniques. A single propagation yields the values of the correlation function at all intermediate time points. Model calculations suggest that the method yields accurate results over several oscillation periods and the statistical error grows slowly with increasing propagation time.

Longitudinal functional principal component modelling via Stochastic Approximation Monte Carlo

The authors consider the analysis of hierarchical longitudinal functional data based upon a functional principal components approach. In contrast to standard frequentist approaches to selecting the number of principal components, the authors do model averaging using a Bayesian formulation. A relatively straightforward reversible jump Markov Chain Monte Carlo formulation has poor mixing properties and in simulated data often becomes trapped at the wrong number of principal components. In order to overcome this, the authors show how to apply Stochastic Approximation Monte Carlo (SAMC) to this problem, a method that has the potential to explore the entire space and does not become trapped in local extrema. The combination of reversible jump methods and SAMC in hierarchical longitudinal functional data is simplified by a polar coordinate representation of the principal components. The approach is easy to implement and does well in simulated data in determining the distribution of the number of principal components, and in terms of its frequentist estimation properties. Empirical applications are also presented.

Delayed-gamma signature calculation for neutron-induced fission and activation using MCNPX, Part III: Transport theory

In Parts I and II, we presented the MCNPX Monte Carlo delayed-gamma theory and illustrative simulation results. The complexity and diversity of the relevant physics items can render the comprehension of their scope and interrelationship difficult. In Part III we augment the Monte Carlo formulation of Part I using transport theory as a means of showing the physics content for the MCNPX delayed-gamma feature.

Integral form of nuclide generation and depletion equations for Monte Carlo simulation. Application to perturbation calculations

The general integral form of the nuclide generation/depletion equation suitable for a resolution by the Monte Carlo method is presented. A formal analogy between particle transport and nuclide generation/depletion is shown. Moreover, the Monte Carlo formulation of the nuclide generation/depletion problem allows to do perturbation calculation by using the correlated sampling method.

Bose atoms in a trap: A variational Monte Carlo formulation for the universal behavior at the van der Waals length scale

We present a variational Monte Carlo formulation for the universal equations of state at the van der Waals length scale [B. Gao, J. Phys. B 37, L227 (2004)] for $N$ Bose atoms in a trap. The theory illustrates both how such equations of state can be computed exactly, and the existence and the importance of long-range atom-atom correlation under strong confinement. Explicit numerical results are presented for $N=3$ and 5, and used to provide a quantitative understanding of the shape-dependent confinement correction that is important for few atoms under strong confinement.

Computing reactive flows with a field Monte Carlo formulation and multi-scale methods

Two novel techniques are merged together to compute efficiently flows involving chemical reaction or combustion. In order to predict the evolution of the reacting flow, a probability density function approach is applied to the chemical species, which is described by a set of pure Eulerian transport equations with stochastic source terms. The proposed model, which has been extended in this case to a multi-scalar situation, has the advantage that the chemistry terms are closed, requiring no modeling. The formulation is solved by a stabilized finite element method based on the variational multi-scale theory, which enables an accurate and stable treatment of the difficult and stiff source terms employing iterative techniques. The combination of these ingredients results in a robust and accurate tool for the computation of reactive flows, as will be shown in the numerical examples.

Application of ensemble-based techniques in fish stock assessment

Abstract A basic fish-stock assessment system requires an integrated use of a model for the time evolution of a fish stock and information about the fish stock from catch data. Typical for common fish stock assessment systems has been the use of fairly simplistic data assimilation methodologies for the integration of observations with the dynamical models. On the other hand, there has been a fast development of assimilation techniques which can be used with highly nonlinear and complex dynamical models. In this paper some of these methods, which are based on Monte Carlo formulations, will be examined with a simple fish stock model. The methods used are the ensemble Kalman filter, the ensemble Kalman smoother and the simpler ensemble smoother.