Tomographic reconstruction using heuristic Monte Carlo methods

A tomographic reconstruction method based on Monte Carlo random searching guided by the information contained in the projections of radiographed objects is presented. In order to solve the optimization problem, a multiscale algorithm is proposed to reduce computation. The reconstruction is performed in a coarse-to-fine multigrid scale that initializes each resolution level with the reconstruction of the previous coarser level, which substantially improves the performance. The method was applied to a real case reconstructing the internal structure of a small metallic object with internal components, showing excellent results.     

Monte Carlo optimization

Monte Carlo optimization techniques for solving mathematical programming problems have been the focus of some debate. This note reviews the debate and puts these stochastic methods in their proper perspective.     

Stochastic Simulation and Monte Carlo methods applied to the assessment of hydro-thermal generating system operation

Summary Simulation can be defined as a numerical technique for conducting experiments on a digital computer, which involves certain types of mathematical and logical models that describe the behaviour of a system over extended periods of real time. Simulation is, in a wide sense, a technique for performing sampling experiments on a model of the system. Stochastic simulation implies experimenting with the model over time including sampling stochastic variates from probability distributions. This paper describes the main concepts of the application of Stochastic Simulation and Monte Carlo methods to the analysis of the operation of electric energy systems, in particular to hydro-thermal generating systems. These techniques can take into account virtually all contingencies inherent in the operation of the system. Also, the operating policies that have an important effect on the performance of these systems can be realistically represented.     

A Monte Carlo study of old and new frontier methods for efficiency measurement

This study presents the results of an extensive Monte Carlo experiment to compare different methods of efficiency analysis. In addition to traditional parametric–stochastic and nonparametric–deterministic methods recently developed robust nonparametric–stochastic methods are considered. The experimental design comprises a wide variety of situations with different returns-to-scale regimes, substitution elasticities and outlying observations. As the results show, the new robust nonparametric–stochastic methods should not be used without cross-checking by other methods like stochastic frontier analysis or data envelopment analysis. These latter methods appear quite robust in the experiments.     

Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints

In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints introduced by Birbil et al. (Math Oper Res 31:739–760, 2006). Firstly, by means of a Monte Carlo method, we obtain a nonsmooth discrete approximation of the original problem. Then, we propose a smoothing method together with a penalty technique to get a standard nonlinear programming problem. Some convergence results are established. Moreover, since quasi-Monte Carlo methods are generally faster than Monte Carlo methods, we discuss a quasi-Monte Carlo sampling approach as well. Furthermore, we give an example in economics to illustrate the model and show some numerical results with this example. The first author’s work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science and SRF for ROCS, SEM. The second author’s work was supported in part by the United Kingdom Engineering and Physical Sciences Research Council grant. The third author’s work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

Useful Monte Carlo optimization

This contribution to the debate on Monte Carlo optimization methods shows that there exist techniques that may be useful in many technical applications.     

On solving integral equations using Markov chain Monte Carlo methods

In this paper, we propose an original approach to the solution of Fredholm equations of the second kind. We interpret the standard Von Neumann expansion of the solution as an expectation with respect to a probability distribution defined on a union of subspaces of variable dimension. Based on this representation, it is possible to use trans-dimensional Markov chain Monte Carlo (MCMC) methods such as Reversible Jump MCMC to approximate the solution numerically. This can be an attractive alternative to standard Sequential Importance Sampling (SIS) methods routinely used in this context. To motivate our approach, we sketch an application to value function estimation for a Markov decision process. Two computational examples are also provided.     

拟蒙特卡罗法在亚洲期权定价中的应用

亚洲期权是场外交易中几种最受欢迎的新型期权之一,但它的价格却没有解析表达式,到目前为止,亚洲期权的定价仍是个公开问题.本文采用拟蒙特卡罗法中的Halton序列来估计它的价格,数值结果表明当观察点的个数N13时,它比蒙特卡罗法要好.本文还利用MATLAB程序生成了随机Halton序列,并将它与控制变量法结合起来估计亚洲期权的价格,估计值标准差的比较表明它在大多情况下比相应的蒙特卡罗法的估计效果要好.     

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.     

Discrete stochastic consistent estimators of the Monte Carlo method

The efficiency of discrete stochastic consistent estimators (the weighted uniform sampling and estimator with a correcting multiplier) of the Monte Carlo method is investigated. Confidence intervals and upper bounds on the variances are obtained, and the computational cost of the corresponding discrete stochastic numerical scheme is estimated.     

The information-based complexity of approximation problem by adaptive Monte Carlo methods

In this paper, we study the complexity of information of approximation problem on the multivariate Sobolev space with bounded mixed derivative MWpr,α(Td), 1 < p < ∞, in the norm of Lq(Td), 1 < q < ∞, by adaptive Monte Carlo methods. Applying the discretization technique and some properties of pseudo-s-scale, we determine the exact asymptotic orders of this problem.     

Bayesian Monte Carlo estimation for profile hidden Markov models

Hidden Markov models are used as tools for pattern recognition in a number of areas, ranging from speech processing to biological sequence analysis. Profile hidden Markov models represent a class of so-called “left–right” models that have an architecture that is specifically relevant to classification of proteins into structural families based on their amino acid sequences. Standard learning methods for such models employ a variety of heuristics applied to the expectation-maximization implementation of the maximum likelihood estimation procedure in order to find the global maximum of the likelihood function. Here, we compare maximum likelihood estimation to fully Bayesian estimation of parameters for profile hidden Markov models with a small number of parameters. We find that, relative to maximum likelihood methods, Bayesian methods assign higher scores to data sequences that are distantly related to the pattern consensus, show better performance in classifying these sequences correctly, and continue to perform robustly with regard to misspecification of the number of model parameters. Though our study is limited in scope, we expect our results to remain relevant for models with a large number of parameters and other types of left–right hidden Markov models.     

Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation

We construct weighted modifications of statistical modeling of an ensemble of interacting particles which is connected with approximate solution of a nonlinear Boltzmann equation.     

基于蒙特卡罗方法的中国中长期煤炭需求预测

为避免传统预测方法的参数取值主观性问题,采用参数随机产生的蒙特卡罗方法预测中国中长期煤炭需求。首先分析了经济增长、能源结构和产业结构三个主要煤炭需求影响因素,并基于1980~2015年间各影响因素及煤炭消费的历史数据和最小二乘法的多元线性回归拟合煤炭需求方程。在此基础上,构建各影响因素的概率分布,采用蒙特卡罗方法模拟1981~2015年的煤炭需求,发现仿真结果可以较好拟合现实,可作为仿真预测的有效工具。结合经济新常态和能源结构调整的现状,控制参数取值范围进行蒙特卡罗仿真预测,结果显示,2016~2025年的煤炭需求呈先上升后下降趋势,并于2020年达到需求的峰值40.25亿吨,这些结果对于煤炭产业的科学决策有重要的作用。     

Coarse-Grained Langevin Approximations and Spatiotemporal Acceleration for Kinetic Monte Carlo Simulations of Diffusion of Interacting Particles

Kinetic Monte Carlo methods provide a powerful computational tool for the simulation of microscopic processes such as the diffusion of interacting particles on a surface, at a detailed atomistic level. However such algorithms are typically computationatly expensive and are restricted to fairly small spatiotemporal scales. One approach towards overcoming this problem was the development of coarse-grained Monte Carlo algorithms. In recent literature, these methods were shown to be capable of efficiently describing much larger length scales while still incorporating information on microscopic interactions and fluctuations. In this paper, a coarse-grained Langevin system of stochastic differential equations as approximations of diffusion of interacting particles is derived, based on these earlier coarse-grained models. The authors demonstrate the asymptotic equivalence of transient and long time behavior of the Langevin approximation and the underlying microscopic process, using asymptotics methods such as large deviations for interacting particles systems, and furthermore, present corresponding numerical simulations, comparing statistical quantities like mean paths, auto correlations and power spectra of the microscopic and the approximating Langevin processes. Finally, it is shown that the Langevin approximations presented here are much more computationally efficient than conventional Kinetic Monte Carlo methods, since in addition to the reduction in the number of spatial degrees of freedom in coarse-grained Monte Carlo methods, the Langevin system of stochastic differential equations allows for multiple particle moves in a single timestep.     

Unbiased multi-index Monte Carlo

We introduce a new class of Monte Carlo-based approximations of expectations of random variables such that their laws are only available via certain discretizations. Sampling from the discretized versions of these laws can typically introduce a bias. In this paper, we show how to remove that bias, by introducing a new version of multi-index Monte Carlo (MIMC) that has the added advantage of reducing the computational effort, relative to i.i.d. sampling from the most precise discretization, for a given level of error. We cover extensions of results regarding variance and optimality criteria for the new approach. We apply the methodology to the problem of computing an unbiased mollified version of the solution of a partial differential equation with random coefficients. A second application concerns the Bayesian inference (the smoothing problem) of an infinite-dimensional signal modeled by the solution of a stochastic partial differential equation that is observed on a discrete space grid and at discrete times. Both applications are complemented by numerical simulations.     

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.     

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.     

Lightning modeling and protection zone of conducting rod using Monte Carlo technique

Lightning strike is a harmful process and protection from lightning using conducting rods has been a subject of discussion for decades. In particular, there have been a lot of researches regarding the protection zone of a single conducting rod. This is important for the purpose of installing them to protect complex building structures. In the present article, the protection zone of a conducting rod has been obtained by Monte Carlo simulation method. The lightning process has been modeled for computer simulation. The authors have presented a new Monte Carlo modeling of lightning path. The origin of the downward stepped leader and the magnitude of peak current of the return stroke have appropriate random distributions. The obtained results are compared with already available experimental results present in various literatures. It is shown that this technique is reliable and in fair agreement with the established theories for finding protection zone of conducting rods.     

Dynamically Rescaled Hamiltonian Monte Carlo for Bayesian Hierarchical Models

Dynamically rescaled Hamiltonian Monte Carlo is introduced as a computationally fast and easily implemented method for performing full Bayesian analysis in hierarchical statistical models. The method relies on introducing a modified parameterization so that the reparameterized target distribution has close to constant scaling properties, and thus is easily sampled using standard (Euclidian metric) Hamiltonian Monte Carlo. Provided that the parameterizations of the conditional distributions specifying the hierarchical model are “constant information parameterizations” (CIPs), the relation between the modified- and original parameterization is bijective, explicitly computed, and admit exploitation of sparsity in the numerical linear algebra involved. CIPs for a large catalogue of statistical models are presented, and from the catalogue, it is clear that many CIPs are currently routinely used in statistical computing. A relation between the proposed methodology and a class of explicitly integrated Riemann manifold Hamiltonian Monte Carlo methods is discussed. The methodology is illustrated on several example models, including a model for inflation rates with multiple levels of nonlinearly dependent latent variables. Supplementary materials for this article are available online.