Monte Carlo Solution Research Articles

Markov Chain Monte Carlo Solution of the Implicit Nonlinear Inverse Problem with Application to Curve Fitting and Filter Estimation

Markov Chain Monte Carlo Solution of the Implicit Nonlinear Inverse Problem with Application to Curve Fitting and Filter Estimation

Monte Carlo solution of the reflected systems’ criticality problems with Anlı–Güngör, Inönü and tetra-anisotropic kernels for both fission and scattered neutrons

Monte Carlo solution of the reflected systems’ criticality problems with Anlı–Güngör, Inönü and tetra-anisotropic kernels for both fission and scattered neutrons

Effect analysis of temperature and initial water content on the adsorption and separation of VOCs by ZIF‐8 based on molecular simulation

AbstractVolatile organic compounds (VOCs) have seriously polluted the atmospheric environment and caused harmful effects on human beings, so it is imperative to control oil vapor emissions. Metal organic framework materials, as one of the porous materials, have a good application prospect in the field of gas adsorption separation. In this paper, the effects of temperature and initial water content (IWC) on the adsorption properties of methane (CH4), ethane (C2H6), ethylene (C2H4), propane (C3H8), and propylene (C3H6) by ZIF‐8 were studied by combining the large gauge Monte Carlo (GCMC) and ideal adsorption solution theory. The results show that the higher the temperature is, the lower the adsorption capacity is, the higher the threshold pressure is, and the single molecule interaction energy changes little with temperature. But pore volume is still the main influencing factor. The IWC also reduces the saturated adsorption capacity. Preloaded water molecules can enhance the electrostatic interaction at low pressure, thus affecting the adsorption heat and interaction energy of single molecules. Overall, these findings provide valuable mechanistic insights into the effects of temperature and IWC on the adsorption properties of ZIF‐8 for VOCs.

Efficient path integral approach via analytical asymptotic expansion for nonlinear systems under Gaussian white noise

In this paper an efficient formulation of the Path integral (PI) approach is developed for determining the response probability density functions (PDFs) and first-passage statistics of nonlinear oscillators subject to stationary and time-modulated external Gaussian white noise excitations. Specifically, the evolution of the response PDF is obtained in short time steps, by using a discrete version of the Chapman-Kolmogorov equation and assuming a Gaussian form for the conditional response PDF. Next, the technique involves proceeding to treating the problem via an analytical asymptotic expansion procedure, namely the Laplace’s method of integration. In this manner, the repetitive double integrals involved in the standard implementation of the PI approach are evaluated in a closed form, while the response and first-passage PDFs are obtained by mundane step-by-step application of the derived approximate analytical expression. It is shown that the herein proposed formulation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard PI technique and Monte Carlo solution (MCS) approach. A number of nonlinear oscillators are considered in the numerical examples. Notably, for these systems both response PDFs and first-passage probabilities are presented, whereas comparisons with pertinent MCS data demonstrate the efficiency and accuracy of the technique.

A Eulerian Monte Carlo method for the numerical solution of the multivariate population balance equation

Statistical descriptions of particulate phases and granular media based on the notion of a property distribution are particularly advantageous for dense, many-particle dispersions since they support the direct evaluation of Eulerian property statistics, permit a kinematically simple treatment of particle collisions and feature a model complexity that is independent of the number of physical particles. On the minus side, the property distribution is governed by a high-dimensional population balance equation (PBE) whose moment reduction is accompanied by a challenging closure problem and engenders intricate dynamical nonlinearities. Circumventing both the curse of dimensionality and the moment closure problem, we present a Eulerian Monte Carlo (EMC) solution method that harnesses a representation of the property distribution in terms of an ensemble of property samples and is based on a statistical enforcement of the spatially and temporally discretized PBE. After application of a kinetic finite volume scheme, the PBE is subjected to a fractional steps decomposition that separates spatial transport from transport in property space. Conceptually, spatial transport is emulated by assembling the ensemble in one spatial grid cell from samples drawn from the ensembles in neighbouring cells at the previous time point. In this regard, we propose a two-level sampling scheme that is sensitive to large-scale features of the marginal velocity distribution and effectively reduces the instantaneous sampling error. The property transport step, by contrast, is based on the method of characteristics and involves the propagation of individual samples as property characteristics in time. While the sampling error afflicting the property statistics needs to be controlled using a top-level time or ensemble averaging, the EMC method remains well-conditioned for an arbitrary number of samples, is easy to implement and involves a computational expense that scales only linearly with the number of particle properties. Based on two benchmark examples, we analyze the spatial and statistical convergence properties of the EMC method and demonstrate its capacity to capture multiple crossings of particle trajectories. A third example involves the two-way coupled transport of size-polydispersed inertial particles in an inhomogeneous flow field.

Thermal rarefied gas flow simulations with moving boundaries based on discrete unified gas kinetic scheme and immersed boundary method

Rarefied gas flows in complex geometries with heat transfer and moving boundaries have received remarkable attention due to the prevailing of micro/nano science and engineering applications. This study presents an immersed boundary (IB)-based discrete unified gas kinetic scheme (DUGKS) for the simulation of complex moving boundary thermal rarefied flows. The fluid and temperature fields are solved according to the DUGKS with Shakhov collision model (SDUGKS), which is capable of modeling non-unit Prandtl number problems in all Knudsen regimes. The IB approach, featured by its superior performance to incorporate complicated boundary constraints on Cartesian grids, is applied here to handling the interaction between the micro gas and curved boundaries. Specifically, for the IB adopted in this study, we use the least square method for the reconstruction of the distribution functions near the boundary. The accuracy of the IB-SDUGKS is explored comprehensively in six benchmark problems, including the micro lid-driven cavity flow, thermal equilibrium gas in a circular domain, thermal transpiration in an enclosure, moving shuttle in a rarefied gas, movement of a piston with pressure differences and particle free motion in a lid-driven cavity. The present results agree well with the corresponding analytical and direct simulation Monte Carlo (DSMC) solutions, illustrating the versatility of the present IB-SDUGKS complex thermal rarefied flows.

Wigner transport in linear electromagnetic fields

Applying a Weyl–Stratonovich transform to the evolution equation of the Wigner function in an electromagnetic field yields a multidimensional gauge-invariant equation which is numerically very challenging to solve. In this work, we apply simplifying assumptions for linear electromagnetic fields and the evolution of an electron in a plane (two-dimensional transport), which reduces the complexity and enables to gain first experiences with a gauge-invariant Wigner equation. We present an equation analysis and show that a finite difference approach for solving the high-order derivatives allows for reformulation into a Fredholm integral equation. The resolvent expansion of the latter contains consecutive integrals, which is favorable for Monte Carlo solution approaches. To that end, we present two stochastic (Monte Carlo) algorithms that evaluate averages of generic physical quantities or directly the Wigner function. The algorithms give rise to a quantum particle model, which interprets quantum transport in heuristic terms.

Understanding the X-ray emission spectrum after excitation with a source of X-rays: From theory to experiment

The modified Boltzmann-Chandrasekhar equation of transport for photons is the proper framework for describing the photon radiation field with a complete description of the polarization state. The characterization of the radiation field requires a detailed knowledge of the interactions of photons with mater and comprises also the contribution of the secondary electrons to the photon field through mechanisms like inner impact ionization and bremsstrahlung. It will be shown a solution obtained without the need of solving the coupled transport electrons-photons. With all these interactions, the theoretical characterization of the X-ray spectrum of emission after excitation with a source of X-rays can be straightforwardly obtained from the albedo solution to the equation. In this work it will be privileged a Monte Carlo (MC) solution. However, this solution is still far from an experimental measurement modified by the radiation detection devices, comprised the pulse electronics. In this work we put together a MC simulation able to get a detailed transport solution and a complete characterization of the contributions of the detection chain. It is discussed the influence of the single contributions and how they combine to make that a simulated X-ray spectrum matches well a real measurement.

UBES: Unified scatter correction using ultrafast Boltzmann equation solver for conebeam CT

A semi-analytical solution to the unified Boltzmann equation is constructed to exactly describe the scatter distribution on a flat-panel detector for high-quality conebeam CT (CBCT) imaging. The solver consists of three parts, including the phase space distribution estimator, the effective source constructor and the detector signal extractor. Instead of the tedious Monte Carlo solution, the derived Boltzmann equation solver achieves ultrafast computational capability for scatter signal estimation by combining direct analytical derivation and time-efficient one-dimensional numerical integration over the trajectory along each momentum of the photon phase space distribution. The execution of scatter estimation using the proposed ultrafast Boltzmann equation solver (UBES) for a single projection is finalized in around 0.4 seconds. We compare the performance of the proposed method with the state-of-the-art schemes, including a time-expensive Monte Carlo (MC) method and a conventional kernel-based algorithm using the same dataset, which is acquired from the CBCT scans of a head phantom and an abdominal patient. The evaluation results demonstrate that the proposed UBES method achieves comparable correction accuracy compared with the MC method, while exhibits significant improvements in image quality over learning and kernel-based methods. With the advantages of MC equivalent quality and superfast computational efficiency, the UBES method has the potential to become a standard solution to scatter correction in high-quality CBCT reconstruction.

Quantum computing for chemistry and physics applications from a Monte Carlo perspective.

This Perspective focuses on the several overlaps between quantum algorithms and Monte Carlo methods in the domains of physics and chemistry. We will analyze the challenges and possibilities of integrating established quantum Monte Carlo solutions into quantum algorithms. These include refined energy estimators, parameter optimization, real and imaginary-time dynamics, and variational circuits. Conversely, we will review new ideas for utilizing quantum hardware to accelerate the sampling in statistical classical models, with applications in physics, chemistry, optimization, and machine learning. This review aims to be accessible to both communities and intends to foster further algorithmic developments at the intersection of quantum computing and Monte Carlo methods. Most of the works discussed in this Perspective have emerged within the last two years, indicating a rapidly growing interest in this promising area of research.

Analysis of loss of flow without scram test in the FFTF reactor – Part I: Preparation of neutronics data

This study presents a benchmark analysis of an unprotected loss of flow transient in a sodium-cooled fast reactor at the Fast Flux Test Facility (FFTF), carried out as part of an IAEA coordinated research project. Three codes, namely Serpent (Monte Carlo), DYN3D (3D nodal diffusion) and ATHLET (system thermal hydraulics), were employed in the benchmark exercise. Two distinct modeling approaches were utilized: 1) stand-alone ATHLET with point kinetics (PK) and system thermal hydraulics (TH); and 2) coupled DYN3D/ATHLET with spatial kinetics (SK) and system TH. Neutronics data essential for both approaches were generated using Serpent. The study is organized into three parts.Part I presents a summary of the preparation of neutronics data for PK or coupled SK/TH simulations and includes the outcomes of the static neutronics stage of the benchmark. The main focus lies on verifying the cross-section generation method for DYN3D by comparing its results against the reference Monte Carlo solutions obtained with Serpent.Part II will provide a detailed description of the ATHLET TH model of the system. This model will be thoroughly evaluated by comparing the results with the ANL benchmark solution and experimental data for the transient. Furthermore, a sensitivity analysis will be conducted to explore various modeling options and assess their impact on the simulation results.Part III will showcase the transient calculation results using the two modeling approaches. Additionally, an adaptive decay heat model for nodal codes will be introduced. The performance of both modeling approaches will be assessed by comparing their results to the available experimental data.

A novel numerical solution to lane-emden type equations using monte carlo technique

Monte Carlo (MC) methods are increasingly recognized as severe in many computational scientific fields and have diverse applications in many branches of science. This paper systematically provides two computational algorithms based on MC methods to solve different forms of Lane-Emden (LE) type equations. The proposed algorithms introduce solutions to 11 LE equations under various complex conditions. The performance and comparative study of numerical solutions based on the MC algorithms were computationally analyzed using other numerical/analytical methods available in the literature. We find that the MC solutions agree with the exact or Runge–Kutta solutions and different numerical methods applied to solve these equations.

Evolution of the Shape of a Gas Cloud during Pulsed Laser Evaporation into Vacuum: Direct Simulation Monte Carlo and the Solution of a Model Equation

The dynamics of gas expansion during nanosecond laser evaporation into vacuum is studied. The problem is considered in an axisymmetric formulation for a wide range of parameters: the number of evaporated monolayers and the size of the evaporation spot. To obtain a reliable numerical solution, two different kinetic approaches are used—the direct simulation Monte Carlo method and solution of the BGK model kinetic equation. The change in the shape of the cloud of evaporated substance during the expansion process is analyzed. The strong influence of the degree of rarefaction on the shape of the forming cloud is shown. When a large number of monolayers evaporate, good agreement with the continuum solution is observed.

A LÉVY-DRIVEN ORNSTEIN–UHLENBECK PROCESS FOR THE VALUATION OF CREDIT INDEX SWAPTIONS

A Lévy-driven Ornstein–Uhlenbeck process is proposed to model the evolution of the risk-free rate and default intensities for the purpose of evaluating option contracts on a credit index. Time evolution in credit markets is assumed to follow a gamma process in order to reflect the different speed at which credit products are exchanged with respect to securities, such as Treasuries, deemed risk-free. Formulas for the characteristic function, zero coupon bonds, moments of the process and its stationary distribution are derived. Numerical experiments showing convergence of standard numerical methods for the valuation PIDE to analytical and Monte Carlo solutions are shown. Calibration to market prices of options on a credit index is performed, and model- and market-implied summary statistics for the underlying credit spreads are estimated and compared.

Counter-checking uncertainty calculations in Bayesian operational modal analysis with EM techniques

Bayesian operational modal analysis makes inference about the modal properties (e.g., natural frequency, damping ratio) of a structure using ‘output-only’ ambient vibration data. With sufficient data in applications, the posterior probability density function (PDF) of modal properties can be approximated by a Gaussian PDF, whose covariance matrix is given by the inverse of the Hessian of negative log-likelihood function (NLLF) at the most probable value. Existing methodologies for computing the Hessian are based on semi-analytical formulae that offer an efficient and reliable means for applications. Inevitably, their computer coding can be involved, e.g., a mix of variables with different sensitivities, singularity of Hessian due to constraints. In the absence of analytical or numerically ‘exact’ result for benchmarking, computer code verification during development stage is also non-trivial. Currently, finite difference method is often used as the only and last resort for verification, although there are also difficulties in, e.g., the choice of step size, and criterion for comparison/convergence. Motivated by these, this work explores an identity in the theory of Expectation-Maximisation (EM) algorithm to provide an alternative means for evaluating the Hessian of NLLF. Such identity allows one to evaluate the Hessian by means of Monte Carlo simulation, averaging over random samples of hidden variables. While the existing semi-analytical approach is still preferred for Hessian calculations in applications for its high definitive accuracy and speed, the proposed Monte Carlo solution offers a convenient means for counter-checking during code development. Theoretical implications of the identity will be discussed and numerical examples will be given to illustrate implementation aspects.

On a Data-Driven Optimization Approach to the PID-Based Algorithmic Trading

This paper proposes an optimal trading algorithm based on a novel application of conventional control engineering (CE). We consider a fundamental CE concept, namely, the feedback control, and apply it to algorithmic trading (AT). The concrete feedback control strategy is designed in a form of the celebrated proportional–integral–derivative (PID) model. The highly fluctuating nature of the modern financial markets has led to the adoption of a model-free realization of the generic PID framework. The control theoretical methodology we propose is combined with the advanced statistics for the historical market data. We obtain a specific log-normal probability distribution function (pdf) associated with the specific quantities associated with the available stock data. The empirical log-normal pdf mentioned above enables the necessary PID gains optimization. For this aim, we apply the data-driven optimization approaches and consider the corresponding Monte Carlo solution procedure. The optimized PID trading algorithm we propose is also studied in the Fourier analysis framework. This equivalent frequency domain representation involves a new concept in financial engineering, namely, the “stock market energy” concept. For the evaluation, we implement the proposed PID optimal trading algorithm and develop a Python-based prototype software. We finally apply the corresponding prototype software to a data set from the Binance BTC/USDT (Bitcoin/Tether) stock market. The experimental result illustrates the implementability of the proposed optimal PID trading scheme and also shows the effectiveness of the proposed CE methods in the modern AT.

Variance reduced particle solution of the Fokker-Planck equation with application to rarefied gas and plasma dynamics

We develop an importance-weight-based variance reduction method for direct Monte Carlo simulations of the Fokker-Planck equation with applications to rarefied gas and plasma dynamics. We show that there exists a class of Fokker-Planck equations that admit stable weight evolution processes along particle trajectories, including those with drift and diffusion coefficients independent of moments higher than the kinetic energy, such as the linearized Landau equation. When drift or diffusion coefficients depend on higher-order moments, such as in the cubic Fokker-Planck equation, maintaining stability and accuracy of the weight evolution becomes challenging. In this work, stability and conservation for the cubic FP model are achieved using the maximum cross-entropy formulation introduced in recent work Sadr and Hadjiconstantinou (2023) [30]. Several test cases show that significant speed-up is obtained using the proposed variance reduced method compared to the standard Monte Carlo solution in the low-signal limit.

A Eulerian population balance/Monte Carlo approach for simulating laminar aluminum dust flames

Recently, metal powders have been conceptualized as carbon-free recyclable energy carriers that may form a cornerstone of a sustainable energy economy. The combustion of metal dusts in oxidizing atmospheres is exothermal and yields oxide particles that could, potentially, be retrieved and, subsequently, recharged by conversion to pure metals using green primary energy sources. As a step towards a predictive tool for designing metal dust combustors, we present a fully Eulerian modelling approach for laminar particle-laden reactive flows that is, conceptually, based on a population balance description of the dispersed particles and relies on a stochastic Eulerian solution strategy. While the population balance equation (PBE) is formulated for the number-weighted distribution of characteristic properties among all particles near a spatial location, it is kinetically informed by the rates at which mass, momentum and heat are exchanged between the carrier gas and the particulate phase on the single particle level. Within the scope of the Eulerian Monte Carlo solution scheme, the property distribution is discretely represented in terms of the total number density and a finite number of property samples and the computational work is channelled towards the Eulerian estimation of mean particle properties.For the case of reactive aluminum particles, we combine a kinetic description of the gas-particle heat and mass transfer with a transport-limited continuum formulation to obtain rate expressions that are valid across the entire particle size range from the free molecular through the continuum regime. Besides velocity, the particle properties include only the particle mass, temperature and oxide mass fraction. This set of thermochemical degrees of freedom is retained also as phase transitions due to melting occur, drawing on a smooth blend of the solid and liquid thermodynamic and material properties. The particle-level formulation encompasses aluminum evaporation, surface oxidation, scavenging of oxide smoke, oxide evaporation/dissociation and radiation. After investigating how these effects translate, through the PBE, to the particle population level and affect the combustion in a homogeneous dust reactor, we analyze the combustion of an aluminum dust in a counterflow flame and validate predictions of the particles’ centerline velocity profile and the flame speed by comparison with available experimental data. Concomitantly, nitrogen oxide emissions are investigated along with the particle burnout and outlet size distribution.

Inversion in binary gas mixtures in rarefied flow conditions: Direct simulation Monte Carlo solution and comparison with the analytical solutions at free molecular regime

This paper analyzes the mixing of gases in a plane channel at rarefied conditions. The direct simulation Monte Carlo method is employed to simulate gas mixing in parallel mixers working at different Knudsen numbers and having different values of wall accommodation coefficient. Results show that the normal-to-wall component of the mole fraction gradient may have the same sign as the corresponding component of the diffusive mass flux vector near the diffuse solid walls in contrast to the predictions of Fick's law for continuum conditions. This non-continuum behavior, which is called “inversion” in the present study, will become more pronounced at higher Knudsen numbers, whereas it will become less evident for smaller wall accommodation coefficients. To confirm that the observed phenomenon is consistent with the basic physical laws governing the rarefied gas dynamics and it is not an artifact of the numerical method, a new analytical model based on the kinetic theory of gases is developed for the parallel mixers that have diffuse walls and are working in the free-molecular regime. Excellent agreement is observed between the analytical and direct simulation Monte Carlo results in the free molecular flow regime. Both methods predict the occurrence of inversion near the diffuse walls at highly rarefied flow conditions.

Stochastic estimation of Green’s functions with application to diffusion and advection-diffusion-reaction problems

A stochastic method is described for estimating Green’s functions (GF’s), appropriate to linear advection-diffusion-reaction transport problems, evolving in arbitrary geometries. By allowing straightforward construction of approximate, though high-accuracy GF’s, within any geometry, the technique solves the central challenge in obtaining Green’s function solutions. In contrast to Monte Carlo solutions of individual transport problems, subject to specific sets of conditions and forcing, the proposed technique produces approximate GF’s that can be used: a) to obtain (infinite) sets of solutions, subject to any combination of (random and deterministic) boundary, initial, and internal forcing, b) as high fidelity direct models in inverse problems, and c) as high quality process models in thermal and mass transport design, optimization, and process control problems. The technique exploits an equivalence between the adjoint problem governing the transport problem Green’s function, G(x,t|x′,t′), and the backward Kolmogorov problem governing the transition density, p(x,t|x′,t′), of the stochastic process used in Green’s function construction. We address nonspecialists and report four contributions. First, a recipe is outlined for diagnosing when stochastic Green’s function estimation can be used, and for subsequently estimating the transition density and associated Green’s function. Second, a naive estimator for the transition density is proposed and tested. Third, Green’s function estimation error produced by random walker absorption at Dirichlet boundaries is suppressed using a simple random walker splitting technique. Last, spatial discontinuity in estimated GF’s, produced by the naive estimator, is suppressed using a simple area averaging method. The paper provides guidance on choosing key numerical parameters, and the technique is tested against two simple unsteady, linear heat conduction problems, and an unsteady groundwater dispersion problem, each having known, exact GF’s.