Conventional Monte Carlo Simulation Research Articles

Efficient finite element reliability analysis employing sequentially-updated surrogate model for fragility curve derivation

As the threat of natural disasters to structures intensifies, risk assessment of infrastructure has gained much importance. Fragility curves are essential tools in predicting disaster-related losses and making disaster mitigation decisions. In this paper, we propose a new method to efficiently derive accurate fragility curves for structures with high levels of nonlinearity or complexity, addressing the computational challenges of conventional finite element reliability analysis (FERA). To reduce the computational cost for calculating probability of failure in FERA, the proposed method utilizes the first-order reliability method (FORM). However, even with this approach, the computational cost of deriving the fragility curve may remain high; therefore, a surrogate model is used to further reduce costs. By training the surrogate model using the initial structural damage probabilities for a few hazard intensities, an optimal starting point can be calculated for the subsequent FORM analysis. During the fragility analysis, the surrogate model can be updated sequentially to increase the efficiency of FORM analysis continuously. In particular, the training process of the surrogate model requires no separate or additional finite element analysis because it is constructed using previous FERA results. The accuracy and efficiency of the proposed method are tested using conventional FERA and Monte Carlo simulations through a hypothetical short-column example. In addition, fragility curves are derived through a bridge flood fragility assessment considering the scour and seismic vulnerability assessment of a buried gas pipeline considering soil-structure interactions. The derived fragility curves closely match those derived using the conventional FERA, and the computational costs are reduced by 36.54 % and 52.38 %, respectively, compared with the conventional FERA, confirming its cost-effectiveness.

Sensitivity and perturbation analysis of alpha-eigenvalue with time-dependent Monte Carlo perturbation techniques

Sensitivity analysis and perturbation calculation methods for prompt neutron decay constants α using the Monte Carlo method, such as iterative fission probability (IFP), differential operator sampling (DOS), and correlated sampling (CS), have been proposed thus far. However, these methods require a significant amount of memory storage. The method developed in this study applies DOS and CS techniques during a time-dependent Monte Carlo simulation for the pulsed neutron method in subcritical systems. The time variation of the neutron flux is fitted to an exponential function during the time range when the flux decays with the fundamental mode α. By applying the least squares method to the results of time-dependent DOS and CS, the sensitivity coefficients with respect to nuclear data and the change in α due to perturbation can be calculated. In contrast to the existing methods that require storing the total fission or time sources per generation over several (∼10) generations, the proposed method only needs to store the weighting coefficient of a single source particle until it dies. The additional memory storage per parameter is comparable to that of conventional time-dependent Monte Carlo simulations and much less than that of existing methods. Numerical tests for the proposed method have demonstrated the accuracy and usefulness of the time-dependent DOS and CS methods.

Robust Design Optimization of Supersonic Biplane Airfoil Using Efficient Uncertainty Analysis Method for Discontinuous Problem

Busemann’s supersonic biplane airfoil can reduce wave drag through shock interactions at its designed freestream Mach number. However, a choking phenomenon occurs with a decrease in the freestream Mach number, and the drag coefficient increases significantly, resulting in an aerodynamic problem with a discontinuous change in the performance function. In this study, an uncertainty analysis method, the divided inexpensive Monte Carlo simulation (IMCS), is proposed to solve discontinuous problems efficiently and is applied to Busemann’s biplane airfoil. In the divided IMCS, the discontinuity point is determined using a simple sampling method. The uncertainty input space is divided at the detected discontinuity point, and a surrogate model is constructed for each space. Uncertainty analysis was performed using the constructed surrogate models, and the results of the divided IMCS showed qualitative agreement with those of the conventional Monte Carlo simulation, which is the most straightforward uncertainty analysis method. Moreover, the divided IMCS significantly reduced the computational cost of the uncertainty analysis. A robust design optimization of the supersonic biplane airfoil was performed using the divided IMCS, yielding more robust designs than Busemann’s biplane airfoil. The usefulness of the divided IMCS for uncertainty analysis of discontinuous problems was confirmed.

Uncertainty Quantification and Sensitivity Analysis for the Self-Excited Vibration of a Spline-Shafting System

Abstract Self-excited vibrations can occur in the spline-shafting system due to internal friction of the tooth surface. However, due to manufacturing errors, design tolerances, and time-varying factors, the parameters that induce self-excited vibrations are always uncertain. This study provides new insights into the uncertainty quantification and sensitivity analysis of a spline-shaft system suffering from self-excited vibrations. The nonintrusive generalized polynomial chaos expansion (gPCE) with unknown deterministic coefficients is used to represent the propagation of uncertainties in the rotor dynamics, which allows rapid estimation of the statistics of the nonlinear responses. Furthermore, the global sensitivity analysis of the stochastic self-excited vibration response of the rotor system with probabilistic uncertain parameters is evaluated by Sobol indices. The relative influence of different random parameters on the vibration behavior and initial displacement conditions for the occurrence of self-excited vibration is investigated. The accuracy of the adopted method based on the gPCE metamodel is validated by conventional Monte Carlo simulation (MCS). Finally, the effects of parameter uncertainties considering random distribution characteristics on the stochastic vibration characteristics of the rotor system are discussed, which demonstrates the need to consider input uncertainties in analysis and design to ensure robust system performance.

Q deformed formulation of Hamiltonian SU(3) Yang-Mills theory

We study SU(3) Yang-Mills theory in (2 + 1) dimensions based on networks of Wilson lines. With the help of the q deformation, networks respect the (discretized) SU(3) gauge symmetry as a quantum group, i.e., SU(3)k, and may enable implementations of SU(3) Yang-Mills theory in quantum and classical algorithms by referring to those of the stringnet model. As a demonstration, we perform a mean-field computation of the groundstate of SU(3)k Yang-Mills theory, which is in good agreement with the conventional Monte Carlo simulation by taking sufficiently large k. The variational ansatz of the mean-field computation can be represented by the tensor networks called infinite projected entangled pair states. The success of the mean-field computation indicates that the essential features of Yang-Mills theory are well described by tensor networks, so that they may be useful in numerical simulations of Yang-Mills theory.

Short-Term EV Charging Demand Forecast with Feedforward Artificial Neural Network

The global increase in greenhouse gas emissions from automobiles has brought about the manufacture and usage of large quantities of electric vehicles (EVs). However, to ensure proper integration of EVs into the grid, there is a need to forecast the charging demand of EVs accurately. This paper presents a short-term electric vehicle charging demand forecast using a feedforward artificial neural network optimized with a modified local leader phase spider monkey optimization (MLLP-SMO) algorithm, a proposed variant of spider monkey optimization. A proportionate fitness selection is employed to improve the update process of the local leader phase of the spider monkey optimization. The proposed algorithm trains a feedforward neural network to forecast electric vehicle charging demand. The effectiveness of the proposed forecasting model was tested and validated with electric vehicle public charging data from the United Kingdom Power Networks Low Carbon London Project. The model's performance was compared to a feedforward neural network trained with particle swarm optimization, genetic algorithm, classical spider monkey optimization, and two conventional forecasting models, multi-linear regression and Monte Carlo simulation. The performance of the proposed forecasting model was assessed using the mean absolute percentage error of forecast and forecasting accuracy. The model produced a forecast accuracy and mean absolute percentage error of 99.88% and 3.384%, respectively. The results show that MLLP-SMO as a trainer predicted better than the other forecasting models and met industry standard forecast accuracy.

Analysis of relative error in perturbation Monte Carlo simulations of radiative transport.

Perturbation and differential Monte Carlo (pMC/dMC) methods, used in conjunction with nonlinear optimization methods, have been successfully applied to solve inverse problems in diffuse optics. Application of pMC to systems over a large range of optical properties requires optimal "placement" of baseline conventional Monte Carlo (cMC) simulations to minimize the pMC variance. The inability to predict the growth in pMC solution uncertainty with perturbation size limits the application of pMC, especially for multispectral datasets where the variation of optical properties can be substantial. We aim to predict the variation of pMC variance with perturbation size without explicit computation of perturbed photon weights. Our proposed method can be used to determine the range of optical properties over which pMC predictions provide sufficient accuracy. This method can be used to specify the optical properties for the reference cMC simulations that pMC utilizes to provide accurate predictions over a desired optical property range. We utilize a conventional error propagation methodology to calculate changes in pMC relative error for Monte Carlo simulations. We demonstrate this methodology for spatially resolved diffuse reflectance measurements with ±20% scattering perturbations. We examine the performance of our method for reference simulations spanning a broad range of optical properties relevant for diffuse optical imaging of biological tissues. Our predictions are computed using the variance, covariance, and skewness of the photon weight, path length, and collision distributions generated by the reference simulation. We find that our methodology performs best when used in conjunction with reference cMC simulations that utilize Russian Roulette (RR) method. Specifically, we demonstrate that for a proximal detector placed immediately adjacent to the source, we can estimate the pMC relative error within 5% of the true value for scattering perturbations in the range of . For a distal detector placed at transport mean free paths relative to the source, our method provides relative error estimates within 20% for scattering perturbations in the range of . Moreover, reference simulations performed at lower values showed better performance for both proximal and distal detectors. These findings indicate that reference simulations utilizing continuous absorption weighting (CAW) with the Russian Roulette method and executed using optical properties with a low ratio spanning the desired range of values, are highly advantageous for the deployment of pMC to obtain radiative transport estimates over a wide range of optical properties.

A correlated sampling-based Monte Carlo simulation for fast CBCT iterative scatter correction.

In recent years, cone-beam computed tomography (CBCT) has played an important role in medical imaging. However, the applications of CBCT are limited due to the severe scatter contamination. Conventional Monte Carlo (MC) simulation can provide accurate scatter estimation for scatter correction, but the expensive computational cost has always been the bottleneck of MC method in clinical application. In this work, an MC simulation method combined with a variance reduction technique called correlated sampling is proposed for fast iterative scatter correction. Correlated sampling exploits correlation between similar simulation systems to reduce the variance of interest quantities. Specifically, conventional MC simulation is first performed on the scatter-contaminated CBCT to generate the initial scatter signal. In the subsequent correction iterations, scatter estimation is then updated by applying correlated MC sampling to the latest corrected CBCT images by reusing the random number sequences of the task-related photons in conventional MC. Afterward, the corrected projections obtained by subtracting the scatter estimation from raw projections are utilized for FDK reconstruction. These steps are repeated until an adequate scatter correction is obtained. The performance of the proposed framework is evaluated by the accuracy of the scatter estimation, the quality of corrected CBCT images and efficiency. Overall, the difference in mean absolute percentage error between scatter estimation with and without correlated sampling is 0.25% for full-fan case and 0.34% for half-fan case, respectively. In simulation studies, scatter artifacts are substantially eliminated, where the mean absolute error value is reduced from 15 to 2HU in full-fan case and from 53 to 13HU in half-fan case. Scatter-to-primary ratio is reduced to 0.02 for full-fan and 0.04 for half-fan, respectively. In phantom study, the contrast-to-noise ratio (CNR) is increased by a factor of 1.63, and the contrast is increased by a factor of 1.77. As for clinical studies, the CNR is improved by 11% and 14% for half-fan and full-fan, respectively. The contrast after correction is increased by 19% for half-fan and 44% for full-fan. Furthermore, root mean square error is also effectively reduced, especially from 78 to 4HU for full-fan. Experimental results demonstrate that the figure of merit is improved between 23 and 43 folds when using correlated sampling. The proposed method takes less than 25s for the whole iterative scatter correction process. The proposed correlated sampling-based MC simulation method can achieve fast and accurate scatter correction for CBCT, making it suitable for real-time clinical use.

Modeling families of particle distributions with conditional GAN for Monte Carlo SPECT simulations

Objective. We propose a method to model families of distributions of particles exiting a phantom with a conditional generative adversarial network (condGAN) during Monte Carlo simulation of single photon emission computed tomography imaging devices. Approach. The proposed condGAN is trained on a low statistics dataset containing the energy, the time, the position and the direction of exiting particles. In addition, it also contains a vector of conditions composed of four dimensions: the initial energy and the position of emitted particles within the phantom (a total of 12 dimensions). The information related to the gammas absorbed within the phantom is also added in the dataset. At the end of the training process, one component of the condGAN, the generator (G), is obtained. Main results. Particles with specific energies and positions of emission within the phantom can then be generated with G to replace the tracking of particle within the phantom, allowing reduced computation time compared to conventional Monte Carlo simulation. Significance. The condGAN generator is trained only once for a given phantom but can generate particles from various activity source distributions.

Rapid detection of phase transitions from Monte Carlo samples before equilibrium

We found that Bidirectional LSTM and Transformer can classify different phases of condensed matter models and determine the phase transition points by learning features in the Monte Carlo raw data before equilibrium. Our method can significantly reduce the time and computational resources required for probing phase transitions as compared to the conventional Monte Carlo simulation. We also provide evidence that the method is robust and the performance of the deep learning model is insensitive to the type of input data (we tested spin configurations of classical models and green functions of a quantum model), and it also performs well in detecting Kosterlitz-Thouless phase transitions.

On the use of sparse Bayesian learning-based polynomial chaos expansion for global reliability sensitivity analysis

Global reliability sensitivity analysis determines the effects of input uncertain parameters on the failure probability of a system. Usually, the global reliability sensitivity analysis can be performed by the conventional Monte Carlo simulation (MCS) approach. However, the MCS approach requires a large number of model evaluations which limits MCS to apply for realistic problems. For that reason, a sparse polynomial chaos expansion (PCE) model is used in the present work based on a variational Bayesian (VB) inference. More specifically, the PCE coefficients are computed by the VB inference and the important terms in the PCE basis are selected by an automatic relevance determination (ARD) approach. Therefore, the VB inference is fully connected with the ARD approach. Global reliability sensitivity analysis is performed for some numerical examples using the sparse PCE model and all the results are compared with the MCS and the least angle regression (LARS)-based PCE model predicted results. The 95% confidence interval is also obtained by the VB approach to measure the prediction uncertainty. It is found that a very good result is obtained with the sparse PCE model using much less number of model evaluations as compared to the MCS approach. The accuracy of obtaining the PCE coefficients is higher by the VB inference than the LARS approach. Further, the required number of terms is small for the VB-PCE model and therefore, the number of PCE coefficients is also small.

Stochastic normalizing flows as non-equilibrium transformations

Normalizing flows are a class of deep generative models that provide a promising route to sample lattice field theories more efficiently than conventional Monte Carlo simulations. In this work we show that the theoretical framework of stochastic normalizing flows, in which neural-network layers are combined with Monte Carlo updates, is the same that underlies out-of-equilibrium simulations based on Jarzynski’s equality, which have been recently deployed to compute free-energy differences in lattice gauge theories. We lay out a strategy to optimize the efficiency of this extended class of generative models and present examples of applications.

Multi-hazard stochastic response assessment of base-isolated buildings

The stochastic response of multi-story buildings isolated by laminated rubber bearing (LRB), lead-rubber bearing (N-Z system), friction pendulum system (FPS), and resilient-friction base isolator (R-FBI) is investigated under the independent multi-hazard scenario of earthquake ground motion (EQGM), wind load (WL), and blast-induced ground motion (BIGM), which are considered as uncertain inputs. A polynomial regression-based Monte Carlo simulation (PRBMCS) is proposed to enable computationally efficient fragility assessment of base-isolated buildings. The suitability of the PRBMCS for stochastic assessment of the base-isolated buildings is compared with the conventional Monte Carlo simulation (MCS), response surface method (RSM) and Cloud analysis. The fragility curves obtained using the PRBMCS are in good agreement with that of the MCS relatively. The study includes: the effects of the characteristic parameters of the four base isolators on the fragility of the base-isolated buildings under the three hazards; influence of the extent of uncertainty in the excitations; and comparison of fragilities of the base-isolated and fixed-base buildings. Notably, the influences of characteristic parameters of the isolators on the fragility of the base-isolated buildings are different under the three hazards. The differences in the stochastic behaviour under the different hazards could have contradictory influences on the selection of isolation parameters.

An efficient Bi-level hybrid multi-objective reliability-based design optimization of composite structures

This paper presents computationally an efficient multi-objective reliability-based design optimization (RBDO) method for composite structures with uncertain design parameters. The approach adopted is based on a novel hybrid decomposing-based multi-objective particle swarm optimization and sequential quadratic programming algorithms, which is coupled with bi-level modelling and optimization strategy of composite structures and RBDO using Cross entropy method. The reliability analysis considers the rare-event corresponding to the limit states, which is based on Tsai-Wu failure index. In addition, the growing requirements for composite structural design with a minimum weight, introduces significant research challenges in the RBDO approach as the weight and reliability are the two important opposing design requirements. To address this research challenges, the proposed approach involves both reliability and weight as objective functions. A laminated composite plate and benchmark problems were used to demonstrate the efficiency and accuracy of the proposed method. Results show that the proposed method provides computationally efficient and accurate evaluations of the performance of the design of composite structures in comparison to the conventional Monte Carlo simulation method and can provide a potential for solving large-scale multi-objective RBDO of composite structures design problems.

A Novel Machine Learning Method for Accelerated Modeling of the Downwelling Irradiance Field in the Upper Ocean

The downwelling irradiance is a crucial part of the air-sea heat flux and a major energy source for the photosynthesis processes in the upper ocean. The modeling of the irradiance field is often time-consuming in conventional Monte Carlo (MC) simulations. We propose an accelerated model based on machine learning to significantly reduce the computational cost. By introducing a generalized beam spreading function, we transform the raw data generated with the MC simulation into training data of reduced dimensions, which is then used to develop an artificial neural network. To validate the machine learning model, we use the MC model to simulate the irradiance field under a broadband wave field. For both clear and turbid seawater, the irradiance field predicted by the machine learning model agrees well with the MC simulation result. Compared with the MC simulation that is computationally expensive, our machine learning model is O(1,000) times faster.

An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems

In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In particular, the random diffusivity coefficient is described by a finite-dimensional random variable or a truncated combination of random variables with the Karhunen-Loève decomposition, then a probabilistic collocation method (PCM) with sparse grids is employed to sample the stochastic process. On each sample, the deterministic nonlocal diffusion problem is discretized with an optimization-based meshfree quadrature rule. In terms of analysis, we first show the analytic regularity of solutions with respect to parameters in the diffusion coefficients and then present the rigorous convergence analysis for the proposed numerical scheme, i.e., the scheme is asymptotic compatible spatially and achieves an algebraic or sub-exponential convergence rate in the random coefficients space as the number of collocation points grows. These results are further confirmed by a number of benchmark problems. Finally, to validate the applicability of this approach we consider a randomly heterogeneous nonlocal problem with a given spatial correlation structure, demonstrating that the proposed PCM approach achieves substantial speed-up compared to conventional Monte Carlo simulations.

Optimising an output of a slip line field using a chaotic swamp optimisation algorithm

In solving earth pressure problems, many input parameters are needed to account for more realistic soil properties and complex boundary conditions. Finding an optimal output while considering many inputs can be costly and time consuming. The input parameters themselves may be significantly uncertain due to lack of laboratory test data and variations in design and construction stages. Adopting an optimisation technique can be useful in such a problem. In this study, a chaotic particle swamp optimisation algorithm is applied to three different earth pressure problems – that is, a retaining wall problem, a plane strain tunnel problem and a suffosion sinkhole problem, to determine their global optimal outputs and corresponding input parameters. The problems are solved using a slip line theory satisfying static equilibrium and the Mohr–Coulomb failure criterion. This improved particle swamp optimisation algorithm is found to show better performances, in accuracy and computational effort, compared to conventional Monte Carlo simulations.

Phase structure of the CP(1) model in the presence of a topological θ -term

We numerically study the phase structure of the $CP(1)$ model in the presence of a topological $\ensuremath{\theta}$-term, a regime afflicted by the sign problem for conventional lattice Monte Carlo simulations. Using a bond-weighted tensor renormalization group method, we compute the free energy for inverse couplings ranging from $0\ensuremath{\le}\ensuremath{\beta}\ensuremath{\le}1.1$ and find a $CP$-violating, first-order phase transition at $\ensuremath{\theta}=\ensuremath{\pi}$. In contrast to previous findings, our numerical results provide no evidence for a critical coupling ${\ensuremath{\beta}}_{c}<1.1$ above which a second-order phase transition emerges at $\ensuremath{\theta}=\ensuremath{\pi}$ and/or the first-order transition line bifurcates at $\ensuremath{\theta}\ensuremath{\ne}\ensuremath{\pi}$. If such a critical coupling exists, as suggested by Haldane's conjecture, our study indicates that is larger than ${\ensuremath{\beta}}_{c}>1.1$.

On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model

The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical arbitrages prevention, liquidity asymmetry, and metaorders. Unlike stochastic differential equation, the stochastic Volterra equation is extremely computationally expensive to simulate. In other words, it is difficult to compute option prices under the rough Heston model by conventional Monte Carlo simulation. In this paper, we prove that Euler’s discretization method for the stochastic Volterra equation with non-Lipschitz diffusion coefficient E[|Vt−Vtn|p] is finitely bounded by an exponential function of t. Furthermore, the weak error |E[Vt−Vtn]| and convergence for the stochastic Volterra equation are proven at the rate of O(n−H). In addition, we propose a mixed Monte Carlo method, using the control variate and multilevel methods. The numerical experiments indicate that the proposed method is capable of achieving a substantial cost-adjusted variance reduction up to 17 times, and it is better than its predecessor individual methods in terms of cost-adjusted performance. Due to the cost-adjusted basis for our numerical experiment, the result also indicates a high possibility of potential use in practice.

Aggregation of superparamagnetic colloids strongly confined in spherical cavities under magnetic fields

Aggregates of superparamagnetic colloids confined in liposomes in the presence of a controllable external magnetic field are studied here by computer simulation. A model of a hard sphere with a central dipole has been used to simulate secondary particles of colloids confined to move within a liposome, which is modelled as a hard spherical cavity. We focus our study on the structure of aggregates created by secondary particles, which in turn are formed by superparamagnetic particles. This study was performed by using the conventional Monte Carlo simulations and the cluster-moving Monte Carlo method. The results obtained with both methods show a dependence on the number of particles and on the ratio cavity-colloid diameters. However, the usual Monte Carlo method does not allow the collective movement of the aggregates. The simulation results show, that for the particular case of the liposome diameter obtained from the experiments , the formation of a single ribbons located at the centre of the cavity for N = 30 and N = 60. In addition, we analyse the structure formation for different values of , the equilibrium structures are chains for systems for lower , whereas ribbons formation are structures of equilibrium for .