Research on a Monte Carlo global variance reduction method based on an automatic importance sampling method

Global variance reduction is a bottleneck in Monte Carlo shielding calculations.The global variance reduction problem requires that the statistical error of the entire space is uniform.This study proposed a grid-AIS method for the global variance reduction problem based on the AIS method,which was implemented in the Monte Carlo program MCShield.The proposed method was validated using the VENUS-Ⅲ international benchmark problem and a self-shielding calculation example.The results from the VENUS-Ⅲ benchmark problem showed that the grid-AIS method achieved a significant reduction in the variance of the statistical errors of the MESH grids,decreasing from 1.08×10^(-2) to 3.84×10^(-3),representing a 64.00% reduction.This demonstrates that the grid-AIS method is effective in addressing global issues.The results of the selfshielding calculation demonstrate that the grid-AIS method produced accurate computational results.Moreover,the grid-AIS method exhibited a computational efficiency approximately one order of magnitude higher than that of the AIS method and approximately two orders of magnitude higher than that of the conventional Monte Carlo method.

Active Kriging-Based Adaptive Importance Sampling for Reliability and Sensitivity Analyses of Stator Blade Regulator

The reliability and sensitivity analyses of stator blade regulator usually involve complex characteristics like highnonlinearity,multi-failure regions,and small failure probability,which brings in unacceptable computing efficiency and accuracy of the current analysismethods.In this case,by fitting the implicit limit state function(LSF)with active Kriging(AK)model and reducing candidate sample poolwith adaptive importance sampling(AIS),a novel AK-AIS method is proposed.Herein,theAKmodel andMarkov chainMonte Carlo(MCMC)are first established to identify the most probable failure region(s)(MPFRs),and the adaptive kernel density estimation(AKDE)importance sampling function is constructed to select the candidate samples.With the best samples sequentially attained in the reduced candidate samples and employed to update the Kriging-fitted LSF,the failure probability and sensitivity indices are acquired at a lower cost.The proposed method is verified by twomulti-failure numerical examples,and then applied to the reliability and sensitivity analyses of a typical stator blade regulator.Withmethods comparison,the proposed AK-AIS is proven to hold the computing advantages on accuracy and efficiency in complex reliability and sensitivity analysis problems.

Face tracking algorithm based on particle filter with mean shift importance sampling

The condensation tracking algorithm uses a prior transition probability as the proposal distribution, which does not make full use of the current observation. In order to overcome this shortcoming, a new face tracking algorithm based on particle filter with mean shift importance sampling is proposed. First, the coarse location of the face target is attained by the efficient mean shift tracker, and then the result is used to construct the proposal distribution for particle propagation. Because the particles obtained with this method can cluster around the true state region, particle efficiency is improved greatly. The experimental results show that the performance of the proposed algorithm is better than that of the standard condensation tracking algorithm.

Importance Sampling Method in V-Space

Based on the observation of importance sampling and second order information about the failure surface of a structure, an importance sampling region is defined in V-space which is obtained by rotating a U-space at the point of maximum likelihood. The sampling region is a hyper-ellipsoid that consists of the sampling ellipse on each plane of main curvature in V-space. Thus, the sampling probability density function can be constructed by the sampling region center and ellipsoid axes. Several examples have shown the efficiency and generality of this method.

DOA estimation of incoherently distributed sources using importance sampling maximum likelihood

In this paper, an importance sampling maximum likelihood(ISML) estimator for direction-of-arrival(DOA) of incoherently distributed(ID) sources is proposed. Starting from the maximum likelihood estimation description of the uniform linear array(ULA), a decoupled concentrated likelihood function(CLF) is presented. A new objective function based on CLF which can obtain a closed-form solution of global maximum is constructed according to Pincus theorem. To obtain the optimal value of the objective function which is a complex high-dimensional integral,we propose an importance sampling approach based on Monte Carlo random calculation. Next, an importance function is derived, which can simplify the problem of generating random vector from a high-dimensional probability density function(PDF) to generate random variable from a one-dimensional PDF. Compared with the existing maximum likelihood(ML) algorithms for DOA estimation of ID sources, the proposed algorithm does not require initial estimates, and its performance is closer to CramerRao lower bound(CRLB). The proposed algorithm performs better than the existing methods when the interval between sources to be estimated is small and in low signal to noise ratio(SNR)scenarios.

Application of Importance Sampling Method in Sliding Failure Simulation of Caisson Breakwaters

It is assumed that the storm wave takes place once a year during the design period, and Nhistories of storm waves are generated on the basis of wave spectrum corresponding to the N-year design period. The responses of the breakwater to the N histories of storm waves in the N-year design period are calculated by mass-spring-dashpot mode and taken as a set of samples. The failure probability of caisson breakwaters during the design period of N years is obtained by the statistical analysis of many sets of samples. It is the key issue to improve the efficiency of the common Monte Carlo simulation method in the failure probability estimation of caisson breakwaters in the complete life cycle. In this paper, the kernel method of importance sampling, which can greatly increase the efficiency of failure probability calculation of caisson breakwaters, is proposed to estimate the failure probability of caisson breakwaters in the complete life cycle. The effectiveness of the kernel method is investigated by an example. It is indicated that the calculation efficiency of the kernel method is over 10 times the common Monte Carlo simulation method.

Numerical Study of φ^4 Model by Potential Importance Sampling Method

We investigate the phenomena of spontaneous symmetry breaking for φ^4 model on a square lattice in the parameter space by using the potential importance samplingmethod, which was proposed by Milchev, Heermann, and Binder [J. Star. Phys. 44 （1986） 749]. The critical values of the parameters allow us to determine the phase diagram of the model. At the same time, some relevant quantifies such as susceptibility and specific heat are also obtained.

Importance Sampling Strategy for Oscillatory Stochastic Processes

This paper contributes to the structural reliability problem by presenting a novel approach that enables for identification of stochastic oscillatory processes as a critical input for given mechanical models. Identification development follows a transparent image processing paradigm completely independent of state-of-the-art structural dynamics, aiming at delivering a simple and wide purpose method. Validation of the proposed importance sampling strategy is based on multi-scale clusters of realizations of digitally generated non-stationary stochastic processes. Good agreement with the reference pure Monte Carlo results indicates a significant potential in reducing the computational task of first passage probabilities estimation, an important feature in the field of e.g., probabilistic seismic design or risk assessment generally.

A K-Means Clustering-Based Multiple Importance Sampling Algorithm for Integral Global Optimization

In this paper, we propose a K-means clustering-based integral level-value estimation algorithm to solve a kind of box-constrained global optimization problem. For this purpose, we introduce the generalized variance function associated with the level-value of the objective function to be minimized. The variance function has a good property when Newton’s method is used to solve a variance equation resulting by setting the variance function to zero. We prove that the largest root of the variance equation is equal to the global minimum value of the corresponding optimization problem. Based on the K-means clustering algorithm, the multiple importance sampling technique is proposed in the implementable algorithm. The main idea of the cross-entropy method is used to update the parameters of sampling density function. The asymptotic convergence of the algorithm is proved, and the validity of the algorithm is verified by numerical experiments.

ANNEALED IMPORTANCE SAMPLING FOR ISING MODELS WITH MIXED BOUNDARY CONDITIONS

This note introduces a method for sampling Ising models with mixed boundary conditions.As an application of annealed importance sampling and the Swendsen-Wang algorithm,the method adopts a sequence of intermediate distributions that keeps the temperature fixed but turns on the boundary condition gradually.The numerical results show that the variance of the sample weights is relatively small.

Neural Networks Combined with Importance Sampling Techniques for Reliability Evaluation of Explosive Initiating Device

Concerning the issue of high-dimensions and low-failure probabilities including implicit and highly nonlinear limit state function, reliability analysis based on the directional importance sampling in combination with the radial basis function （RBF） neural network is used, and the RBF neural network based on first-order reliability method （FORM） is to approximate the unknown implicit limit state functions and calculate the most probable point （MPP） with iterative algorithm. For good efficiency, based on the ideas that directional sampling reduces dimensionality and importance sampling focuses on the domain contributing to failure probability, the joint probability density function of importance sampling is constructed, and the sampling center is moved to MPP to ensure that more random sample points draw belong to the failure domain and the simulation efficiency is improved. Then the numerical example of initiating explosive devices for rocket booster explosive bolts demonstrates the applicability, versatility and accuracy of the approach compared with other reliability simulation algorithm.

Reliability Sensitivity Algorithm Based on Stratified Importance Sampling Method for Multiple Failure Modes Systems

Combining the advantages of the stratified sampling and the importance sampling, a stratified importance sampling method （SISM） is presented to analyze the reliability sensitivity for structure with multiple failure modes. In the presented method, the variable space is divided into several disjoint subspace by n-dimensional coordinate planes at the mean point of the random vec- tor, and the importance sampling functions in the subspaces are constructed by keeping the sampling center at the mean point and augmenting the standard deviation by a factor of 2. The sample size generated from the importance sampling function in each subspace is determined by the contribution of the subspace to the reliability sensitivity, which can be estimated by iterative simulation in the sampling process. The formulae of the reliability sensitivity estimation, the variance and the coefficient of variation are derived for the presented SISM. Comparing with the Monte Carlo method, the stratified sampling method and the importance sampling method, the presented SISM has wider applicability and higher calculation efficiency, which is demonstrated by numerical examples. Finally, the reliability sensitivity analysis of flap structure is illustrated that the SISM can be applied to engineering structure.

Passive source localization using importance sampling based on TOA and FOA measurements

Passive source localization via a maximum likelihood （ML） estimator can achieve a high accuracy but involves high calculation burdens, especially when based on time-of-arrival and frequency-of-arrival measurements for its internal nonlinearity and nonconvex nature. In this paper, we use the Pincus theorem and Monte Carlo importance sampling （MCIS） to achieve an approximate global solution to the ML problem in a computationally efficient manner. The main contribution is that we construct a probability density function （PDF） of Gaussian distribution, which is called an important function for efficient sampling, to approximate the ML estimation related to complicated distributions. The improved performance of the proposed method is at- tributed to the optimal selection of the important function and also the guaranteed convergence to a global maximum. This process greatly reduces the amount of calculation, but an initial solution estimation is required resulting from Taylor series expansion. However, the MCIS method is robust to this prior knowledge for point sampling and correction of importance weights. Simulation results show that the proposed method can achieve the Cram6r-Rao lower bound at a moderate Gaussian noise level and outper- forms the existing methods.

Reliability and reliability sensitivity analysis of structure by combining adaptive linked importance sampling and Kriging reliability method

The application of reliability analysis and reliability sensitivity analysis methods to complicated structures faces two main challenges:small failure probability(typical less than 10-5)and time-demanding mechanical models.This paper proposes an improved active learning surrogate model method,which combines the advantages of the classical Active Kriging–Monte Carlo Simulation(AK-MCS)procedure and the Adaptive Linked Importance Sampling(ALIS)procedure.The proposed procedure can,on the one hand,adaptively produce a series of intermediate sampling density approaching the quasi-optimal Importance Sampling(IS)density,on the other hand,adaptively generate a set of intermediate surrogate models approaching the true failure surface of the rare failure event.Then,the small failure probability and the corresponding reliability sensitivity indices are efficiently estimated by their IS estimators based on the quasi-optimal IS density and the surrogate models.Compared with the classical AK-MCS and Active Kriging–Importance Sampling(AK-IS)procedure,the proposed method neither need to build very large sample pool even when the failure probability is extremely small,nor need to estimate the Most Probable Points(MPPs),thus it is computationally more efficient and more applicable especially for problems with multiple MPPs.The effectiveness and engineering applicability of the proposed method are demonstrated by one numerical test example and two engineering applications.

A robust and efficient structural reliability method combining radial-based importance sampling and Kriging

Simulation based structural reliability analysis suffers from a heavy computational burden, as each sample needs to be evaluated on the performance function, where structural analysis is performed. To alleviate the computational burden, related research focuses mainly on reduction of samples and application of surrogate model, which substitutes the performance function. However,the reduction of samples is achieved commonly at the expense of loss of robustness, and the construction of surrogate model is computationally expensive. In view of this, this paper presents a robust and efficient method in the same direction. The present method uses radial-based importance sampling (RBIS) to reduce samples without loss of robustness. Importantly, Kriging is fully used to efficiently implement RBIS. It not only serves as a surrogate to classify samples as we all know, but also guides the procedure to determine the optimal radius, with which RBIS would reduce samples to the highest degree. When used as a surrogate, Kriging is established through active learning, where the previously evaluated points to determine the optimal radius are reused. The robustness and efficiency of the present method are validated by five representative examples, where the present method is compared mainly with two fundamental reliability methods based on active learning Kriging.

Accelerated testing for automated vehicles safety evaluation in cut-in scenarios based on importance sampling,genetic algorithm and simulation applications

Purpose–It would take billions of miles’field road testing to demonstrate that the safety of automated vehicle is statistically significantly higher than the safety of human driving because that the accident of vehicle is rare event.Design/methodology/approach–This paper proposes an accelerated testing method for automated vehicles safety evaluation based on improved importance sampling(IS)techniques.Taking the typical cut-in scenario as example,the proposed method extracts the critical variables of the scenario.Then,the distributions of critical variables are statistically fitted.The genetic algorithm is used to calculate the optimal IS parameters by solving an optimization problem.Considering the error of distribution fitting,the result is modified so that it can accurately reveal the safety benefits of automated vehicles in the real world.Findings–Based on the naturalistic driving data in Shanghai,the proposed method is validated by simulation.The result shows that compared with the existing methods,the proposed method improves the test efficiency by 35 per cent,and the accuracy of accelerated test result is increased by 23 per cent.Originality/value–This paper has three contributions.First,the genetic algorithm is used to calculate IS parameters,which improves the efficiency of test.Second,the result of test is modified by the error correction parameter,which improves the accuracy of test result.Third,typical high-risk cut-in scenarios in China are analyzed,and the proposed method is validated by simulation.

Modified filtered importance sampling for virtual spherical Gaussian lights

This paper proposes a modification of the filtered importance sampling method, and improves the quality of virtual spherical Gaussian light(VSGL)-based real-time glossy indirect illumination using this modification. The original filtered importance sampling method produces large overlaps of and gaps between filtering kernels for high-frequency probability density functions(PDFs). This is because the size of the filtering kernel is determined using the PDF at the sampled center of the kernel. To reduce those overlaps and gaps, this paper determines the kernel size using the integral of the PDF within the filtering kernel. Our key insight is that these integrals are approximately constant, if kernel centers are sampled using stratified sampling. Therefore, an appropriate kernel size can be obtained by solving this integral equation. Using the proposed kernel size for filtered importance samplingbased VSGL generation, undesirable artifacts are significantly reduced with a negligibly small overhead.

B-splines smoothed rejection sampling method and its applications in quasi-Monte Carlo integration

The rejection sampling method is one of the most popular methods used in Monte Carlo methods. It turns out that the standard rejection method is closely related to the problem of quasi-Monte Carlo integration of characteristic functions, whose accuracy may be lost due to the discontinuity of the characteristic functions. We proposed a B-splines smoothed rejection sampling method, which smoothed the characteristic function by B-splines smoothing technique without changing the integral quantity. Numerical experiments showed that the convergence rate of nearly O( N^-1 ) is regained by using the B-splines smoothed rejection method in importance sampling.

Sensitivity of Sample for Simulation-Based Reliability Analysis Methods

In structural reliability analysis,simulation methods are widely used.The statistical characteristics of failure probability estimate of these methods have been well investigated.In this study,the sensitivities of the failure probability estimate and its statistical characteristics with regard to sample,called‘contribution indexes’,are proposed to measure the contribution of sample.The contribution indexes in four widely simulation methods,i.e.,Monte Carlo simulation(MCS),importance sampling(IS),line sampling(LS)and subset simulation(SS)are derived and analyzed.The proposed contribution indexes of sample can provide valuable information understanding the methods deeply,and enlighten potential improvement of methods.It is found that the main differences between these investigated methods lie in the contribution indexes of the safety samples,which are the main factors to the efficiency of the methods.Moreover,numerical examples are used to validate these findings.

Simulation method for reliability of TT&C mission with high redundancy and small time horizon

The tracking, telemetry and command （TT＆C） mission is extremely reliable for its characters of small time horizon and high redundancy. The combined forcing and failure biasing （CFFB） method that is usually used for simulating the unreliability of the highly dependable mission system seems not so efficient for the TT＆C mission. The concept about the importance of failure transition is proposed based on the logical relationship between TT＆C mission and its involved resources. Then, the importance is used for readjusting the transition rate of the failure transition when using the forcing and failure biasing during the simulation. Examples show that the improved CFFB method can evidently increase the occurrence of the TT＆C mission failure event and decrease the sample variance. More redundancy of the TT＆C mission leads to the improved CFFB method more efficient.