Probabilistic sensitivity of limit states of structures. The Monte Carlo simulation

The main issue of the paper is the probabilistic sensitivity of the limit states of structures with respect to selected input design variables. Attempt to the problem is done by the dedicated Monte Carlo simulation procedure. Basic design variables are random variables of given probability distributions, presented in the form of random numbers. Uni-parametrical increment of the dominant basic variable (basic variables) is done, finally achieving the structural limit state. The simulation procedure restuls in a set of limit multipliers. Statistical analysis leads to the estimate of the probability density function of the limit state. Thus the numerical image is presented of the probabilistic sensitivity of the structural limit state. Reliability or the probability of failure are to be estimated, as statistical parameters of the histogram. Numerical examples of engineering structures illustrate the method introduced in the paper, conclusions are formulated eventually. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Combinatorics of reliability Monte Carlo

We study one class of unbiased Monte Carlo estimators for system reliability, avoiding the rare event difficulty. This class is closely related to the system combinatorics and contains unique “extreme” members, having the minimum and maximum relative error. Some known Monte Carlo heuristics for network reliability, including fully polynomial cases, are of this type. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14: 329–343, 1999     

Sampling efficiency in Monte Carlo analyses

Stochastic processes that are sampled in Monte Carlo analyses can be so complex that sampling efficiency is difficult to attain. To handle these difficulties we introduce a model of the elements of a stochastic process which are relevant to the problem of sampling efficiency. From this model we derive a multistage estimating procedure which automatically adjusts the parameters of an efficient sampling design.     

Conditioning in Markov Chain Monte Carlo

Abstract

The so-called “Rao-Blackwellized” estimators proposed by Gelfand and Smith do not always reduce variance in Markov chain Monte Carlo when the dependence in the Markov chain is taken into account. An illustrative example is given, and a theorem characterizing the necessary and sufficient condition for such an estimator to always reduce variance is proved.     

Quantum Monte Carlo study of magnetic and superconducting properties of graphene

We apply quantum Monte Carlo technique to address the issue of high controllability of ferromagnetism in graphene and the issue of electron correlation driven superconductivity in graphene, by simulating the t − U − V Hubbard model on a honeycomb lattice. In the region of low electron band filling below the Van Hove singularity, the system shows a short‐range ferromagnetic correlation, which is strengthened slightly by the on‐site Coulomb interaction and markedly by the next nearest‐neighbor hoping integral. The strong dependence of ferromagnetism on the electron band filling can be manipulated by applying electric gate voltage. For V=0 and close to the half filling, we find that pairing with d + id symmetry dominates pairing with extended‐s symmetry. However, as the system size increases the long‐range part of the d + id pairing correlation decreases and tends to vanish in the thermodynamic limit. An inclusion of nearest‐neighbor interaction V, either repulsive or attractive, has a small effect on the extended‐s pairing correlation, but strongly suppresses the d + id pairing correlation. Copyright © 2013 John Wiley & Sons, Ltd.     

Monte Carlo study of time asymptotics of the polarized radiation intensity

The parameters of time asymptotics of the polarized radiation intensity are estimated. Precision Monte Carlo estimates of these parameters are derived for finite medium layers by iterating the resolvent of the corresponding transfer operator with a given scattering matrix and by evaluating parametric time derivatives. The computations are performed for two versions of the problem: with a Rayleigh scattering matrix and an aerosol scattering matrix. It is shown that the asymptotics of the radiation intensity are affected by polarization, except for the spatially homogeneous problem, for which the results are obtained analytically.     

Monte Carlo EM加速算法

EM算法是近年来常用的求后验众数的估计的一种数据增广算法, 但由于求出其E步中积分的显示表达式有时很困难, 甚至不可能, 限制了其应用的广泛性. 而Monte Carlo EM算法很好地解决了这个问题, 将EM算法中E步的积分用Monte Carlo模拟来有效实现, 使其适用性大大增强. 但无论是EM算法, 还是Monte Carlo EM算法, 其收敛速度都是线性的, 被缺损信息的倒数所控制, 当缺损数据的比例很高时, 收敛速度就非常缓慢. 而Newton-Raphson算法在后验众数的附近具有二次收敛速率. 本文提出Monte Carlo EM加速算法, 将Monte Carlo EM算法与Newton-Raphson算法结合, 既使得EM算法中的E步用Monte Carlo模拟得以实现, 又证明了该算法在后验众数附近具有二次收敛速度. 从而使其保留了Monte Carlo EM算法的优点, 并改进了Monte Carlo EM算法的收敛速度. 本文通过数值例子, 将Monte Carlo EM加速算法的结果与EM算法、Monte Carlo EM算法的结果进行比较, 进一步说明了Monte Carlo EM加速算法的优良性.     

Monte Carlo Maximum Likelihood in Model-Based Geostatistics

When using a model-based approach to geostatistical problems, often, due to the complexity of the models, inference relies on Markov chain Monte Carlo methods. This article focuses on the generalized linear spatial models, and demonstrates that parameter estimation and model selection using Markov chain Monte Carlo maximum likelihood is a feasible and very useful technique. A dataset of radionuclide concentrations on Rongelap Island is used to illustrate the techniques. For this dataset we demonstrate that the log-link function is not a good choice, and that there exists additional nonspatial variation which cannot be attributed to the Poisson error distribution. We also show that the interpretation of this additional variation as either micro-scale variation or measurement error has a significant impact on predictions. The techniques presented in this article would also be useful for other types of geostatistical models.     

Population Monte Carlo Algorithm in High Dimensions

The population Monte Carlo algorithm is an iterative importance sampling scheme for solving static problems. We examine the population Monte Carlo algorithm in a simplified setting, a single step of the general algorithm, and study a fundamental problem that occurs in applying importance sampling to high-dimensional problem. The precision of the computed estimate from the simplified setting is measured by the asymptotic variance of estimate under conditions on the importance function. We demonstrate the exponential growth of the asymptotic variance with the dimension and show that the optimal covariance matrix for the importance function can be estimated in special cases.     

Multilevel Monte Carlo in approximate Bayesian computation

In the following article, we consider approximate Bayesian computation (ABC) inference. We introduce a method for numerically approximating ABC posteriors using the multilevel Monte Carlo (MLMC). A sequential Monte Carlo version of the approach is developed and it is shown under some assumptions that for a given level of mean square error, this method for ABC has a lower cost than i.i.d. sampling from the most accurate ABC approximation. Several numerical examples are given.     

Implementations of the Monte Carlo EM Algorithm

The Monte Carlo EM (MCEM) algorithm is a modification of the EM algorithm where the expectation in the E-step is computed numerically through Monte Carlo simulations. The most exible and generally applicable approach to obtaining a Monte Carlo sample in each iteration of an MCEM algorithm is through Markov chain Monte Carlo (MCMC) routines such as the Gibbs and Metropolis–Hastings samplers. Although MCMC estimation presents a tractable solution to problems where the E-step is not available in closed form, two issues arise when implementing this MCEM routine: (1) how do we minimize the computational cost in obtaining an MCMC sample? and (2) how do we choose the Monte Carlo sample size? We address the first question through an application of importance sampling whereby samples drawn during previous EM iterations are recycled rather than running an MCMC sampler each MCEM iteration. The second question is addressed through an application of regenerative simulation. We obtain approximate independent and identical samples by subsampling the generated MCMC sample during different renewal periods. Standard central limit theorems may thus be used to gauge Monte Carlo error. In particular, we apply an automated rule for increasing the Monte Carlo sample size when the Monte Carlo error overwhelms the EM estimate at any given iteration. We illustrate our MCEM algorithm through analyses of two datasets fit by generalized linear mixed models. As a part of these applications, we demonstrate the improvement in computational cost and efficiency of our routine over alternative MCEM strategies.     

二重积分的蒙特卡罗数值算法

研究了二重积分的蒙特卡罗高精度求解算法,理论上对比了它们基于相同样本量的计算精度.结合MATLAB语言进行了算例分析,给出了可推广的一般化的蒙特卡罗算法程序.     

Method of Moments Using Monte Carlo Simulation

Abstract

We present a computational approach to the method of moments using Monte Carlo simulation. Simple algebraic identities are used so that all computations can be performed directly using simulation draws and computation of the derivative of the log-likelihood. We present a simple implementation using the Newton-Raphson algorithm with the understanding that other optimization methods may be used in more complicated problems. The method can be applied to families of distributions with unknown normalizing constants and can be extended to least squares fitting in the case that the number of moments observed exceeds the number of parameters in the model. The method can be further generalized to allow “moments” that are any function of data and parameters, including as a special case maximum likelihood for models with unknown normalizing constants or missing data. In addition to being used for estimation, our method may be useful for setting the parameters of a Bayes prior distribution by specifying moments of a distribution using prior information. We present two examples—specification of a multivariate prior distribution in a constrained-parameter family and estimation of parameters in an image model. The former example, used for an application in pharmacokinetics, motivated this work. This work is similar to Ruppert's method in stochastic approximation, combines Monte Carlo simulation and the Newton-Raphson algorithm as in Penttinen, uses computational ideas and importance sampling identities of Gelfand and Carlin, Geyer, and Geyer and Thompson developed for Monte Carlo maximum likelihood, and has some similarities to the maximum likelihood methods of Wei and Tanner.     

Monte Carlo solution of combinatorial optimization problems

Doklady Mathematics - A general method for solving combinatorial optimization problems based on the Metropolis algorithm is developed. The method is easy to implement, efficient, and universal. It...     

The Optimal Error of Monte Carlo Integration

The study of optimal errors of Monte Carlo methods has gained interest in recent years. Since presently no general means are available, the investigation of model problems may help one to understand the mechanisms behind them. The author provides the optimal error for the Monte Carlo integration for input data from a ball of continuous functions. As it turns out, a slight modification of the "crude Monte Carlo method" with fixed cardinality is strictly optimal even among possibly nonlinear Monte Carlo rules with varying cardinality.     

Monte Carlo simulation of boundary layer transition

A statistical method for simulating a boundary layer transition flow is proposed as based on experimental data on the kinematics and dynamics of turbulent spots (Emmons spots) on a flat plate placed in an incompressible fluid. The method determines intermittency with allowance for overlapping spots, which makes it possible to determine the forces on the plate surface and the flow field near the transition region if the mean streamwise velocity field in a developed turbulent boundary layer is known as a function of the Reynolds number. In contrast to multiparameter transition models, this approach avoids the use of nonphysical parameter values.