Basic Monte Carlo Method Research Articles

Combining Simple and Adaptive Monte Carlo Methods for Approximating Hypervolume

The computation of hypervolume is a key issue in multiobjective optimization, particularly, multiobjective evolutionary optimization. However, it is NP-hard to compute the exact hypervolume value. Monte Carlo methods have been widely used for approximating the hypervolume. Observing that the basic Monte Carlo method and the fully polynomial-time randomized approximation scheme (FPRAS) suit different solution sets, we propose a combination of these two methods and show that it performs very well on a number of solution sets.

How Adaptively Constructed Reduced Order Models Can Benefit Sampling-Based Methodsfor Reliability Analyses

Many sampling-based approaches are currently available for calculating the reliability of a design. The most efficient methods can achieve reductions in the computational cost by one to several orders of magnitude compared to the basic Monte Carlo method. This paper is specifically targeted at sampling-based approaches for reliability analysis, in which the samples represent calls to expensive finite element models. The aim of this paper is to illustrate how these methods can further benefit from reduced order modeling to achieve drastic additional computational cost reductions, in cases where the reliability analysis is carried out on finite element models. Standard Monte Carlo, importance sampling, separable Monte Carlo and a combined importance separable Monte Carlo approach are presented and coupled with reduced order modeling. An adaptive construction of the reduced basis models is proposed. The various approaches are compared on a thermal reliability design problem, where the coupling with the adaptively constructed reduced order models is shown to further increase the computational efficiency by up to a factor of six.

Multiple failure modes analysis of the dam system by means of line sampling simulation

As the number of aging dams is increasing, an efficient and feasible probabilistic method is required to analyze the uncertainties of a dam system with multiple failure modes. Due to the limitation of huge computation of Direct Simulation Monte Carlo, a new method is introduced to solve the reliability problem of a dam system. This new method determines the failure domain of a complex structure by resorting to lines rather than random points. It is used to calculate the failure probability and reliability sensitivity of a gravity dam, which is a large, complex structure with small probability; and the basic Monte Carlo method is also used for comparison. Results show that the new method is more efficient and feasible to analyze the reliability of a dam system with small failure probability and multiple dimensionalities.

Sequential Monte Carlo methods for Bayesian elliptic inverse problems

In this article, we consider a Bayesian inverse problem associated to elliptic partial differential equations in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic sequential Monte Carlo (SMC) method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method is established in a function space setting. We also develop an enhancement of the SMC methods for inverse problems which were introduced in Kantas et al. (SIAM/ASA J Uncertain Quantif 2:464---489, 2014); the enhancement is designed to deal with the additional complexity of this elliptic inverse problem. The efficacy of the methodology and its desirable theoretical properties, are demonstrated for numerical examples in both two and three dimensions.

Modeling the multimedia fate dynamics of γ-hexachlorocyclohexane in a large Chinese lake

Long-term annual dynamics from 1984 to 2020 (simulation #1) and seasonal variation from May 2010 to February 2011 (simulation #2) of γ-hexachlorocyclohexane (γ-HCH) in various environmental media in Lake Chaohu, China were simulated with an already developed fugacity-based level IV Quantitative Water Air Sediment Interaction (QWASI) model (Kong et al., 2012). The model was modified, as a fish sub-phase was included. Also the emission flux was added to study the impact of the lindane prohibition in simulation #1. Sensitivity analysis was conducted for both static and dynamic parameters, while in uncertainty analysis, both basic Monte Carlo and Bayesian Markov Chain Monte Carlo (MCMC) methods were undertaken for simulation #2 and the results were compared. Simulated data were consistent with the observations in simulation #1. Seasonal patterns in various media were also successfully modeled in simulation #2 and factors leading to this seasonality were discussed. Atmospheric advection input was the main source. In simulation #2, approximately 36 kilogram (kg) of γ-HCH in Lake Chaohu was removed per year. In addition, 31kg of γ-HCH was added to Lake Chaohu by air–water interface fluxes, and 13kg of γ-HCH was added to the sediment by water–sediment interface. Sensitivity of static and dynamic parameters was discussed. Uncertainty analysis by the basic Monte Carlo method for simulation #1 showed that the dispersion for each media was less than two orders of magnitude. Higher dispersions in fish and two sub-phases of the sediment were attributed to a larger variation in the relevant parameters. The MCMC method for simulation #2 eliminated 77% of the model true uncertainty in water ascertained by basic Monte Carlo method and significant elimination in other phases can be speculated. It is suggested that rather than calibrating the model, the main function of the MCMC for fugacity model should be to avoid overestimating uncertainty in model prediction.

A Partition Function Approximation Using Elementary Symmetric Functions

In statistical mechanics, the canonical partition function can be used to compute equilibrium properties of a physical system. Calculating however, is in general computationally intractable, since the computation scales exponentially with the number of particles in the system. A commonly used method for approximating equilibrium properties, is the Monte Carlo (MC) method. For some problems the MC method converges slowly, requiring a very large number of MC steps. For such problems the computational cost of the Monte Carlo method can be prohibitive. Presented here is a deterministic algorithm – the direct interaction algorithm (DIA) – for approximating the canonical partition function in operations. The DIA approximates the partition function as a combinatorial sum of products known as elementary symmetric functions (ESFs), which can be computed in operations. The DIA was used to compute equilibrium properties for the isotropic 2D Ising model, and the accuracy of the DIA was compared to that of the basic Metropolis Monte Carlo method. Our results show that the DIA may be a practical alternative for some problems where the Monte Carlo method converge slowly, and computational speed is a critical constraint, such as for very large systems or web-based applications.

Sampled Monte Carlo uncertainty analysis for photochemical grid models

This study reports on the development and testing of a method of quantifying the uncertainties in concentration predictions by a complex photochemical grid model (PGM), using a modification of the basic Monte Carlo method (MCM). The computationally intensive aspects of applying a full MCM to hundreds of PGM inputs and model parameters is replaced by a highly restricted sampling approach that exploits the spatial persistence found in predicted concentration fields. The sampling approach to the MCM is being explored as an efficient approach to assess the uncertainty in the differences in predicted maximum ozone concentration between base case and control scenarios. The MCM is applied to several dozen surface cells, with the goal of sampling the spatial pattern of uncertainty in the PGM-predicted differences in surface ozone concentration fields between a pair of base and control scenarios. The uncertainty in model inputs and parameters is simulated using several types of stochastic models. These stochastic models are driven using Latin hypercube sampling (LHS) to generate a non-redundant ensemble of alternative model inputs. Preliminary testing of the sampled MCM approach was conducted using the UAM-IV PGM on the New York ozone attainment modeling domain for the 6–8 July 1988 ozone episode. One hundred alternative concentration estimates were generated for a base scenario and for control scenarios representing 50%, 10% and 5% reduction of NO x emissions. The upper and lower bounds of the concentration difference ensemble that define a 95% confidence range were spatially interpolated from 27 monitoring sites to the full (surface) modeling domain, using the field of zero uncertainty (ZU) concentration differences. For the 50% NO x control scenario, predicted increases in peak ozone concentration smaller than 20 ppb were generally not significant from zero. By contrast, predicted decreases in peak ozone greater than 10 ppb were usually significant. For a control scenario with a small 5% NO x reduction, predicted concentration differences and confidence intervals were much smaller, but predicted changes in peak ozone were significant at a number of sample cells.

Investigation of amplified spontaneous X-ray laser radiation using the equation for the transverse correlation function of the field

A physical model, based on the solution of the quasi-optics equation for the transverse correlation function (TCF) of the field amplitude, is developed for investigating the brightness, angular divergence, and spatial coherence of the amplified spontaneous emission in a laboratory X-ray laser. The model takes account of the spontaneous source of radiation, diffraction, regular refraction, regular amplification taking account of saturation, nonresonance absorption, scattering by small-scale fluctuations of the electron density and the gain, and scattering by random hose-like deviations of the extended plasma medium of the X-ray laser. It is established that the TCF method makes it possible to obtain the final result much more quickly than the basic Monte Carlo method for the parabolic equation for the field amplitude. As a result of the statistical linearization of the equation for the transverse correlation function in the presence of gain saturation, this method overestimates the absolute values of the average intensity of the amplified spontaneous radiation, but the maximum overestimation does not exceed 10%. It is found that fluctuations of the optical parameters of the medium of the X-ray laser degrade the quality of the amplified spontaneous radiation beam, and they are the analog of the nonresonance absorption from the standpoint of the effect on the brightness of the laser and therefore decrease the observed gain. For the characteristic conditions of an X-ray laser with a quasistationary generation scheme, the contribution of small-scale gain fluctuations and random hose-like deviations of the plasma filament of the laser to the scattering of the amplified spontaneous radiation is much smaller than the contribution of small-scale density fluctuations. Calculations of the amplified spontaneous radiation in an X-ray laser, which is produced by unilateral irradiation of a curved target and possesses an asymmetric plasma electron density profile in the gain zone, are performed. It is shown that in the gain saturation regime the coherence length and the coherent power of the amplified spontaneous radiation can be substantially increased, realizing in the gain zone a convex electron density profile instead of a typical concave profile. It is found that this improvement of the coherence occurs only under conditions such that the characteristic depth of the small-scale density fluctuations does not exceed several percent of the typical regular values of the density in the gain zone.

Discrete range clustering using Monte Carlo methods

For automatic obstacle avoidance guidance during rotorcraft low altitude flight a reliable model of the nearby environment is needed. Such a model may be constructed by applying surface fitting techniques to the dense range map obtained by active sensing using radars. However, for covertness passive sensing techniques using electro-optic sensors is desirable. As opposed to the dense range map obtained via active sensing, passive sensing algorithms produce reliable range at sparse locations and, therefore, surface fitting techniques to fill the gaps in the range measurement are not directly applicable. Both, for automatic guidance and as a display for aiding the pilot, these discrete ranges need to be grouped into sets which correspond to objects in the nearby environment. The focus of this paper is on using Monte Carlo methods for clustering range points into meaningful groups. We compare three different approaches and present results of application of these algorithms to an image sequence acquired by onboard cameras during a helicopter flight. Starting with an initial grouping, these algorithms are iteratively applied with a new group creation algorithm to determine the optimal number of groups and the optimal group membership. The results indicate that the simulated annealing methods do not offer any significant advantage over the basic Monte Carlo method for this discrete optimization problem.

Optimal Importance‐Sampling Density Estimator

Importance‐sampling technique has been used in recent years in conjunction with Monte Carlo simulation method to evaluate the reliability of structural systems. Since the efficiency of the importance‐sampling method depends primarily on the choice of the importance‐sampling density, the use of the kernel method to estimate the optimal importance‐sampling density is proposed. This method deviates from the current practice of prescribing the importance‐sampling density from a given parametric family of density functions. Instead, the data obtained from an initial Monte Carlo run are utilized to determine the required importance‐sampling density. The kernel method yields unbiased estimates of the probability of failure. Two measures are developed to quantify the efficiency of the kernel method relative to the basic Monte Carlo method. The first measure, called the marginal efficiency, is used as an indicator of the effectiveness of the kernel method, whereas the second measure, the overall efficiency, define...