Importance Sampling Method Research Articles

Virial equation of state for a granular system.

The equation of state for an ideal gas is simple, which is . In the case of imperfect gases where mutual interactions among the constituents are important, pressure P can be expressed as the series expansion of density n with appropriate coefficients, known as virial coefficients . In this paper, we have obtained the first four virial coefficients for a model interaction potential using multidimensional Monte-Carlo integration and importance sampling methods. Next, we perform molecular dynamics simulations with the same for a many-particle system to obtain P as a function of T and n. We compare our numerical data with the virial equation of state.

Two-Phase Adaptive Kriging Model Based Importance Sampling Method for Estimating Time-Dependent Failure Probability

To improve the computational efficiency of the time-dependent failure probability (TDFP) to system with dynamic loads, materials degeneration and other time-varying factors, a two-phase adaptive Kriging model-based importance sampling (IS) method is proposed. The main contribution of this article is that a single Kriging model is established and adaptively refined to gradually approach the optimal IS probability density function (PDF) and estimate TDFP in two sequential phases. In the first phase, a generally approximate IS PDF is constructed by Kriging model based on its prediction characteristic and subsequently updated until convergence. Then, a simple rejection sampling technique is used to robustly generate IS samples for the estimation of TDFP. In the second phase, current Kriging model is subsequently refined for accurately identifying the classification of the IS samples generated in the last phase. Finally, the TDFP can be estimated using the products obtained in the above two phases. As the IS PDF established by Kriging model is used for directly approximating the optimal one, the proposed method has the capacity to deal with the complicated problem containing multiple failure regions. Additionally, since IS procedure can dramatically improve the sampling efficiency, the proposed method is an effective technique to the time-consuming TDFP analysis problems.

Cross-entropy based importance sampling for composite systems reliability evaluation with consideration of multivariate dependence

Multivariate kernel density estimation (KDE) is powerful for dependence modeling, but producing a non-sparse probability density estimation model. To address this issue, a sparse density estimation technique termed as generalized cross-entropy (GCE) density estimator, is proposed to estimate the joint probability density function (PDF) of correlated random variables (RVs). Moreover, to effectively accelerate the composite systems reliability evaluation, a unified cross-entropy based importance sampling (CEIS) method accommodating both GCE and KDE density estimators is presented, and a creative CE optimization approach is also developed based on the composition sampling algorithm, through which the analytical updating rules of IS-PDF parameters can be derived even though GCE and KDE estimators do not belong to the exponential distribution family (EF). Compared to the KDE estimator, the GCE estimator encounters much less dimensionality difficulty in CE optimization due to the model sparsity merit, and consequently achieves more accurate IS-PDF parameters and better CEIS efficiency. The superiority of GCE estimator over KDE estimator in correlation modeling and CEIS is verified by several numerical tests.

The methods of CADIS-NEE and CADIS-DXTRAN in NECP-MCX and their applications

This paper presents two new methods for variance reduction for shielding calculation in Monte Carlo radiation transport. One method is CADIS-NEE, which combines Consistent Adjoint Driven Importance Sampling (CADIS) and next-event estimator (NEE) methods to increase the calculation efficiency of tallies at points. The other is CADIS-deterministic transport (DXTRAN), which combines CADIS and DXTRAN to obtain higher performance than using CADIS and DXTRAN separately. The combination processes are derived and implemented in the hybrid Monte-Carlo-Deterministic particle-transport code NECP-MCX. Various problems are tested to demonstrate the effectiveness of the two methods. According to the results, the two combination methods have higher efficiency than using CADIS, NEE or DXTRAN separately. In a long-distance photon-transport problem, CADIS-NEE converges faster than NEE and the figure of merit (FOM) of CADIS-NEE is 75.6 times of NEE. In a labyrinthine problem, CADIS-DXTRAN's FOM surpasses that of DXTRAN and CADIS by a factor of 45.3 and 17.7, respectively. Therefore, it is advisable to employ these two novel methods selectively in appropriate scenarios to reduce variance.

A multi-objective-optimization based importance sampling method and its application in seismic stability analysis of gravity dam

The stable reliability analysis of dams is a challenging topic in geotechnical engineering safety. Because there are more complex system failure modes and high-dimensional parameters in large solid structures, it is difficult to calculate failure probability efficiently and accurately using general reliability analysis methods. Most methods suffer from complex iterative processes and the curse of dimensionality. In this study, an improved importance sampling method based on multi-objective optimization is proposed and applied to reliability analysis for dam sliding. The optimization function is constructed from the limit state surface and the original probability density. The optimal sampling center is selected from the optimization results by considering the dominant performance function as well as the coefficient of variation of the weight function. Four numerical examples and a case study of the sliding stability of a gravity dam demonstrate the accuracy and efficiency of the proposed method. The results show that the proposed method is not sensitive to the specific multi-objective optimization algorithm used. Simulations using multiple sampling centers perform better overall than simulations using a single sampling center. In particular, the proposed method remains applicable to parallel systems with 20 high-dimensional parameters that are virtually immune to the curse of dimensionality.

Probabilistic analysis of bolted joint anti-loosening under cyclic transverse load

The loosening of bolted joints under cyclic transverse load is a crucial failure mode, and the uncertainty of the relevant parameters of the bolted joint will affect the loosening state. However, there are few bolt-loosening probability analysis and evaluation methods based on parameter uncertainty in previous studies. In this paper, different loosening states are analyzed based on the relationship between pitch torque, thread friction torque, and bearing friction torque. Considering the uncertainty of process parameters, a probabilistic analysis method of the bolted joint anti-loosening under cyclic transverse load is proposed. In the proposed method, the finite element model verified by the fastening experiment is used to verify the accuracy of the bolted joints’ loosening criterion. Then, the loosening criterion is used as the Limit State Function (LSF). According to the LSF, an Adaptive Kriging-Kernel Density Importance Sampling (AK-KDIS) method is proposed for solving the anti-loosening reliability. This method combines adaptive Kriging model with the importance sampling method, effectively reducing the number of sampling samples and the number of LSF calls. The classical and robust Quasi-Monte Carlo (QMC) method is also used to evaluate the reliability of anti-loosening to verify the accuracy of the proposed AK-KDIS method. Finally, the practical application of probabilistic analysis of anti-loosening is presented, and the effectiveness of the proposed method is analyzed in detail. The results indicate that the AK-KDIS method has higher efficiency while ensuring calculation accuracy compared with the QMC method.

From Hubble to snap parameters: a Gaussian process reconstruction

ABSTRACT By using recent H(z) and supernovae Type Ia (SNe Ia) data, we reconstruct the evolution of kinematic parameters H(z), q(z), jerk, and snap, using a model-independent, non-parametric method, namely, the Gaussian processes. Throughout the present analysis, we have allowed for a spatial curvature prior, based on Planck 18 constraints. In the case of SNe Ia, we modify a python package (gapp) in order to obtain the reconstruction of the fourth derivative of a function, thereby allowing us to obtain the snap from comoving distances. Furthermore, using a method of importance sampling, we combine H(z) and SNe Ia reconstructions in order to find joint constraints for the kinematic parameters. We find for the current values of the parameters: H0 = 67.2 ± 6.2 km s−1 Mpc−1, $q_0 = -0.54^{+0.06}_{-0.05}$, $j_0=0.94^{+0.20}_{-0.18}$, and $s_0=-0.62^{+0.26}_{-0.25}$ at 1σ c.l. We find that these reconstructions are compatible with the predictions from flat lambda-cold dark matter model, at least for 2σ confidence intervals.

Bayesian Approximation Methods of the Estimation for Generalized Exponential Distribution and Neutrosophic Approximation Methods

In this article, we used approximate methods to obtain Bayes method estimations for the shape and scale parameters of the generalized exponential distribution, as three approximation methods were employed: Lindley approximation and neutrosophic Lindley approximation, Gibbs sampling and neutrosophic Gibbs sampling, the most important samples based on the gamma informative prior under the squared error loss function. Through different simulation experiments a comparison was made between those estimators of these three approximate methods, from the simulation results we found a relative preference for the important sampling method over the other two methods. The results of simulation experiments were also confirmed by applying these approximate methods to real data representing the operating times of one of the machines of the publishing, printing, and translation house in Baghdad. On the other hand, we apply the same method to the neutrosophic exponential distribution, and the results will be compared to the classical case.

Effective Error Floor Estimation Based on Importance Sampling with the Uniform Distribution

A key problem of low-density parity-check (LDPC) codes analysis is estimation of an extremely low error floor that occurs at a high level of the signal-to-noise ratio (SNR). The importance sampling (IS) method is a popular approach to address this problem. Existing works typically use a normal sampling probability density function (PDF) with shifted mean, which yields a large variance of the estimate. In contrast, uniform distribution has equally probable samples on the entire range and thus should reduce the variance, but results in a biased estimation. This paper proposes a modified IS approach (IS-U) that allows considering the uniform distribution as a sampling PDF, and shows that this estimation is better than the traditional one. Also, this paper demonstrates that the existing criteria cannot be applied to evaluate the accuracy of the IS-U on the whole SNR range. To address this issue, a new metric is proposed, which uses only the convergence rate and does not depend on the true data.

An improved adaptive Kriging model for importance sampling reliability and reliability global sensitivity analysis

An improved adaptive Kriging model for importance sampling (IS) reliability and reliability global sensitivity analysis is proposed by introducing the IS density function into learning function. Considering the variance information of Kriging prediction, the formula of traditional IS method is extended to the form considering the uncertainty of symbol function. The estimated variance of failure probability caused by the prediction uncertainty of Kriging model is obtained, and the corresponding coefficient of variation (COV) is defined. Based on the standard deviation information of failure probability, a novel learning function considering the characteristic of IS density function is proposed, which are used to minimize the prediction uncertainty of Kriging. The corresponding stopping criterion is defined based on the COV information. In order to increase the likelihood that the selected sample points fall around the limit state boundary, the penalty function method is introduced to improve the learning function. Once the failure probability is obtained, the variable global sensitivity index is calculated through the failed sample set and Bayes theorem. The results show that: By introducing IS density function and penalty function into learning function, the sample points which contribute more to the failure probability can be obtained more effectively in IS method. The proposed method has high accuracy and efficiency compared with traditional Kriging-based IS method.

On spatially correlated observations in importance sampling methods for subsidence estimation

AbstractThe particle filter is a data assimilation method based on importance sampling for state and parameter estimation. We apply a particle filter in two different quasi-static experiments with models of subsidence caused by a compacting reservoir. The first model considers uncorrelated model state variables and observations, with observed subsidence resulting from a single source of strain. In the second model, subsidence is a summation of subsidence contributions from multiple sources which causes spatial dependencies and correlations in the observed subsidence field. Assimilating these correlated subsidence fields may trigger weight collapse. With synthetic tests, we show in a model of subsidence with 50 independent state variables and spatially correlated subsidence a minimum of $$\varvec{10^{13}}$$ 10 13 particles are required to have information in the posterior distribution identical to that in a model with 50 independent and spatially uncorrelated observations. Spatial correlations cause an information loss which can be quantified with mutual information. We illustrate how a stronger spatial correlation results in lower information content in the posterior and we empirically derive the required ensemble size for the importance sampling to remain effective. We furthermore illustrate how this loss of information is reflected in the log likelihood, and how this depends on the number of model state variables. Based on these empirical results, we propose criteria to evaluate the required ensemble size in data assimilation of spatially correlated observation fields.

Accurate and Ultra-Efficient p-Value Calculation for Higher Criticism Tests

In modern data science, higher criticism (HC) method is effective for detecting rare and weak signals. The computation, however, has long been an issue when the number of p-values combined (K) and/or the number of repeated HC tests (N) are large. Some computing methods have been developed, but they all have significant shortcomings, especially when a stringent significance level is required. In this article, we propose an accurate and highly efficient computing strategy for four variations of HC. Specifically, we propose an unbiased cross-entropy-based importance sampling method ( IS C E ) to benchmark all existing computing methods, and develop a modified SetTest method (MST) that resolves numerical issues of the existing SetTest approach. We further develop an ultra-fast approach (UFI) combining pre-calculated statistical tables and cubic spline interpolation. Finally, following extensive simulations, we provide a computing strategy integrating MST, UFI, and other existing methods with R package “HCp” for virtually any K and small p-values ( ∼ 10 − 20 ). The method is applied to a COVID-19 disease surveillance example for spatio-temporal outbreak detection from case numbers of 804 days in 3342 counties in the United States. Results confirm viability of the computing strategy for large-scale inferences. Supplementary materials for this article are available online.

A novel single-loop Kriging importance sampling method for estimating failure probability upper bound under random-interval mixed uncertainties

For reliability analysis under wide existing random and interval mixed uncertainties (RIMU), the failure probability upper bound (FPUB), a conservative index to quantify the reliability level of the structure, is the basis for structural safety assessment under RIMU. To accurately and efficiently estimate FPUB, this paper proposes a novel importance sampling method based single-loop Kriging model (IS-SLK), in which the IS variance reduction is combined with the SLK of the performance function, and a new most reduction learning function under RIMU (MRLF-RI) is established. By introducing the IS probability density function (PDF), the proposed IS-SLK firstly expresses the FPUB as the product of the approximate FPUB estimated by current SLK and the correction term, then it designs a strategy of two-step adaptive updating SLK to calibrate FPUB. The innovation of the IS-SLK includes two aspects. Firstly, it constructs a quasi optimal IS PDF to approach the optimal one based on adaptively updating SLK, and the simple reject sampling method is employed to extract the sample of IS PDF, on which the required size of candidate sample pool for obtaining the convergent FPUB can be reduced and the robustness of the algorithm is improved. Secondly, based on the comprehensive consideration of multiple factors, which include the misclassification probability of structural state, joint PDF and Kriging prediction at the IS candidate sample, as well as the logical relationship between the states of random input candidate samples and those at different realizations of interval input, the new learning function named as MRLF-RI is established to adaptively select the training sample contributing most to improve the estimation accuracy of FPUB. Moreover, by deriving the upper bound of the expected relative error, a convergence criterion of MRLF-RI is constructed to further improve the efficiency. Compared with the up-to-date methods, the proposed IS-SLK is more efficient, and its superiority is verified by four examples.

Probability-Based Design of Reinforced Rock Slopes Using Coupled FORM and Monte Carlo Methods

The efficiency of the first-order reliability method (FORM) and the accuracy of Monte Carlo simulations (MCS) are coupled in probability-based designs of reinforced rock slopes, including a Hong Kong slope with exfoliation joints. Load–resistance duality is demonstrated and resolved automatically in a foundation on rock with a discontinuity plane. Other examples include the lengthy Hoek and Bray deterministic vectorial procedure for comprehensive pentahedral blocks with external load and bolt force, which is made efficient and more succinct before extending it to probability-based design via MCS-enhanced FORM. The FORM–MCS–FORM design procedure is proposed for cases with multiple failure modes. For cases with a dominant single failure mode, the time-saving importance sampling (IS) and the fast second-order reliability method (SORM) can be used in lieu of MCS. Two cases of 3D reinforced blocks (pentahedral and tetrahedral, respectively) with the possibility of multiple sliding modes are investigated. In the case of the reinforced pentahedral block, direct MCS shows that there is only one dominant failure mode, for which the efficient method of importance sampling at the FORM design point provides fast verification of the revised design. In the case of the reinforced tetrahedral block, there are multiple failure modes contributing to the total failure probability, for which the proposed MCS-enhanced FORM procedure is demonstrated to be essential. Comparisons are made between Excel MCS and MATLAB MCS.

Importance sampling and Bayesian model comparison in ecology and evolution

Abstract Bayesian approaches to the modelling of ecological systems are increasingly popular, but there are competing methods for formal model comparisons. Here, we focus on the task of performing multimodel inference through estimating posterior model weights, which encompasses uncertainties in the choice of competing model structure into the inference outputs. Model‐based approaches such as reversible‐jump Markov chain Monte Carlo (RJ‐MCMC) are flexible and allow multimodel inference, but can be complex to implement and optimise, and so we translate a model‐based approach for ecological applications using Importance Sampling to estimate the marginal likelihood of the data given a particular model. This approach allows for model comparison through the estimation of Bayes' Factors or interpretable posterior model probabilities, yielding model weights that facilitate multimodel inference through Bayesian model averaging. We demonstrate Importance Sampling with two case study investigations in animal demography: censused analysis of banded mongoose (Mungos mungo) survival where missing data are uncommon, and capture–mark–recapture analysis of European badger (Meles meles) survival where data are commonly missing. We compare outcomes of the model comparison using the Importance Sampling approach to those obtained through single‐model inference approaches using Deviance information criteria and the Watanabe–Akaike information criteria. The results of the Importance Sampling method aligns with RJ‐MCMC model comparisons while often being more straightforward to fit and optimise, particularly if the competing models are non‐nested.

Section optimization design of UHPC beam bridges based on improved particle swarm optimization

Ultra-high performance concrete has excellent mechanical properties such as ultra-high strength and high durability, and has a broad application prospect in the field of bridge engineering. To make full use of the superior mechanical properties of UHPC and reduce the cost of the bridge, the particle swarm optimization algorithm is used to optimize the structural design of the UHPC beam bridge; Aiming at the problem that the traditional particle swarm optimization algorithm is easy to fall into local optimum, the nonlinear adaptive weight update method is used to improve the particle swarm optimization algorithm. Based on the above-mentioned improved particle swarm optimization algorithm, the optimal design method of ordinary reinforced UHPC beams and prestressed UHPC beams with commonly used highway spans is studied. The reliability of the optimized structural limit state is analyzed by Monte Carlo (MC) importance sampling method. The results show that compared with the traditional particle swarm optimization algorithm, the improved particle swarm optimization algorithm has higher convergence speed and convergence accuracy. The optimal height-span ratio of ordinary reinforced UHPC beams decreases with the increase of the span, and the construction cost gradually increases; The optimal height-span ratio of the prestressed beam decreases first and then increases with the increase of the span, and the construction cost gradually increases. The calculation results of load capacity reliability indexes of optimization results are all higher than the target reliability indexes of similar components stipulated in China’s “Uniform Standard for Structural Reliability Design of Highway Engineering”.

A novel second‐order mixture importance sampling strategy for system reliability analysis

AbstractReliability analysis and evaluation of systems with multiple failure modes is extremely challenging. In this paper, a novel, efficient, and accurate system reliability method is proposed based on the second‐order approximation and the mixture important sampling (mixture‐IS) method. The proposed method does not require the construction of surrogate models and integrates the advantages of second‐order approximation and mixture‐IS strategies, which can greatly reduce the computational effort of estimating the system failure probability as well as ensuring the accuracy. The accuracy and efficiency of the proposed method are verified by several numerical and engineering problems, and comparisons are made against other existing reliability methods. The results indicate that the method can effectively reduce the number of limit state function calls and achieve high computational accuracy.

Multi-Angle Reliability Evaluation of Grid-Connected Wind Farms with Energy Storage Based on Latin Hypercube Important Sampling

Aiming to combat the problems of slow speed and poor accuracy of reliability evaluation of the power system in wind farms with energy storage, this paper proposes a method of reliability evaluation based on Latin hypercube important sampling (LHIS). Firstly, we aimed to establish the Latin hypercube important sampling evaluation model by combining the Latin hypercube sampling method with the important sampling method. Secondly, we aimed to optimize the sample probability distribution of the components and conduct hierarchical sampling of the system. Then, the comprehensive risk indicator (CRI) was proposed to evaluate the operational risk and the wind storage generation interrupted energy benefit (WSGIEB) was proposed to evaluate the contribution of the reliability. Finally, simulation experiments were carried out through various power system operation scenarios. The simulation results show that the proposed method is 47% higher than the improving importance sampling method (IM-IS) in evaluation speed and 33% higher than the improving importance sampling method in calculation accuracy.

Deep spatio-temporal 3D dilated dense neural network for traffic flow prediction

Traffic flow prediction is increasingly vital for the administration of metropolitan areas. Many research on spatio-temporal networks have been explored but the impacts of both spatial and temporal flexibility, complex spatial correlation has not been considered simultaneously. We present the Spatio-Temporal 3D Multiscale Dilated Dense Network (ST-3DMDDN), a novel 3D Convolutional Neural Network (3DCNN) deep learning neural network for the road level and region level traffic flow prediction. It uses autocorrelation analysis’ early fusion method for importance sampling, a 3D multiscale dilated convolutional network to capture nearby and remote correlations simultaneously, and a densely connected network for deeper feature extraction. Considering traffic flow’s heterogeneity, a new block called the “Spatial and Channel Recalibrate” (SCR) is designed to accurately analyze the correlation contributions. The ST-3DMDDN model is evaluated using three real traffic flows, and the findings indicate that our approach surpasses the performance of the baseline approaches.

Power distribution network reliability calculation method based on collective multi-label classifier and energy entropy

Reliable power supply from the grid is vital to life and work, production activities, and national security. At present, the simulation calculation method of distribution network reliability needs a lot of time. In view of this shortcoming, this paper proposes a multi-label classifier model using the importance sampling method, which can shorten the calculation time by reasonably extracting samples. At the same time, the multi-label classifier can evaluate the failure probability of the electric power distribution network at the bus level. Finally, we use the IEEE RTS model to test the proposed method, and the effectiveness of the proposed method is verified.