Efficiency of Monte Carlo computations in very high dimensional spaces

A standard measure for comparing different Monte Carlo estimators is the efficiency, which generally thought to be declining with increasing the number of dimensions. Here we give some numerical examples, ranging from one-hundred to one-thousand dimensional integration problems, that contradict this belief. Monte Carlo integrations carried out in one-thousand dimensional spaces is the other nontrivial result reported here. The examples concern the computation of the probabilities of convex sets (polyhedra and hyperellipsoids) in case of multidimensional normal probabilities.     

A Monte Carlo method for an objective Bayesian procedure

This paper describes a method for an objective selection of the optimal prior distribution, or for adjusting its hyper-parameter, among the competing priors for a variety of Bayesian models. In order to implement this method, the integration of very high dimensional functions is required to get the normalizing constants of the posterior and even of the prior distribution. The logarithm of the high dimensional integral is reduced to the one-dimensional integration of a cerain function with respect to the scalar parameter over the range of the unit interval. Having decided the prior, the Bayes estimate or the posterior mean is used mainly here in addition to the posterior mode. All of these are based on the simulation of Gibbs distributions such as Metropolis' Monte Carlo algorithm. The improvement of the integration's accuracy is substantial in comparison with the conventional crude Monte Carlo integration. In the present method, we have essentially no practical restrictions in modeling the prior and the likelihood. Illustrative artificial data of the lattice system are given to show the practicability of the present procedure.     

资产跳跃下CM策略多期收入保证价格模拟

恒定混合策略(CM策略)多期收入保证价格是保本基金发行方采取设置止损的CM\linebreak策略作为投资策略时收取保 本费的理论依据, 其中标的资产由复合泊松过程和维纳过程共同驱动, 这一定价问题内嵌奇异期权, 蒙特卡罗模拟方法擅长处理这种高维数量金融问题. 基于风险中性测度推导出多期收入保证价格的现值表达式, 用条件蒙特卡罗推导出这一现值表达式的模拟公式. 在给定参数下分别用普通蒙特卡罗和条件蒙特卡罗计算CM策略多期收入保证价格的数值解, 结果显示两种蒙特卡罗方法均能有效计算其数值解, 之后通过给定显著性水平下的置信区间长度评价两种方法的精确度, 结果显示条件蒙特卡罗比普通蒙特卡罗有很大改进. 接着运用条件蒙特卡罗模拟研究多期收入保证价格对不同参数范围的变化情况.     

An iterative computation of approximations on Korobov-like spaces

This paper treats the multidimensional application of a previous iterative Monte Carlo algorithm that enables the computation of approximations in L². The case of regular functions is studied using a Fourier basis on periodised functions, Legendre and Tchebychef polynomial bases. The dimensional effect is reduced by computing these approximations on Korobov-like spaces. Numerical results show the efficiency of the algorithm for both approximation and numerical integration.     

Subregion-Adaptive Integration of Functions Having a Dominant Peak

Abstract

Many statistical multiple integration problems involve integrands that have a dominant peak. In applying numerical methods to solve these problems, statisticians have paid relatively little attention to existing quadrature methods and available software developed in the numerical analysis literature. One reason these methods have been largely overlooked, even though they are known to be more efficient than Monte Carlo for well-behaved problems of low dimensionality, may be that when applied naively they are poorly suited for peaked-integrand problems. In this article we use transformations based on “split t” distributions to allow the integrals to be efficiently computed using a subregion-adaptive numerical integration algorithm. Our split t distributions are modifications of those suggested by Geweke and may also be used to define Monte Carlo importance functions. We then compare our approach to Monte Carlo. In the several examples we examine here, we find subregion-adaptive integration to be substantially more efficient than importance sampling.     

Automatic Differentiation to Facilitate Maximum Likelihood Estimation in Nonlinear Random Effects Models

Maximum likelihood estimation in random effects models for non-Gaussian data is a computationally challenging task that currently receives much attention. This article shows that the estimation process can be facilitated by the use of automatic differentiation, which is a technique for exact numerical differentiation of functions represented as computer programs. Automatic differentiation is applied to an approximation of the likelihood function, obtained by using either Laplace's method of integration or importance sampling. The approach is applied to generalized linear mixed models. The computational speed is high compared to the Monte Carlo EM algorithm and the Monte Carlo Newton–Raphson method.     

Parametrically splitting algorithms

The notion of parametrically splitting algorithms is introduced, which are characterized by their capability of solving a given problem on a large number of processors with few data transfers. These algorithms include many numerical integration algorithms, Monte Carlo and quasi-Monte Carlo methods, and so on.     

An implementation of the method of Ermakov and Zolotukhin for multidimensional integration and interpolation

Summary An interactive procedure is discussed for generating samples from the density function of Ermakov and Zolotukhin for application to Monte Carlo multiple integration and interpolation. The computational details of the implementation are described together with a numerical example.     

DSMC Modeling of a Single Hot Spot Evolution Using the Landau-Fokker-Planck Equation

Numerical solution of a fully nonlinear one dimensional in space and three dimensional in velocity space electron kinetic equation is presented. Direct Simulation Monte Carlo (DSMC) method used for the nonlinear Landau-Fokker-Planck (LFP) collision operator is combined with Particle-in-Cell (PiC) simulations. An assumption of a self-consistent ambipolar electric field is used. The illustrative simulation results for the relaxation of the initial temperature perturbation are compared with the antecedent analytical and numerical results.     

利用随机模拟方法求解第二类积分方程

给出一种求解第二类Fredholm和Volterra积分方程的数值算法,算法在数值积分技术的基础上使用Monte Carlo随机模拟方法求积分方程的近似解.通过数值例子证明了该算法是有效的.     

Efficient deterministic numerical simulation of stochastic asset-liability management models in life insurance

New regulations, stronger competitions and more volatile capital markets have increased the demand for stochastic asset-liability management (ALM) models for insurance companies in recent years. The numerical simulation of such models is usually performed by Monte Carlo methods which suffer from a slow and erratic convergence, though. As alternatives to Monte Carlo simulation, we propose and investigate in this article the use of deterministic integration schemes, such as quasi-Monte Carlo and sparse grid quadrature methods. Numerical experiments with different ALM models for portfolios of participating life insurance products demonstrate that these deterministic methods often converge faster, are less erratic and produce more accurate results than Monte Carlo simulation even for small sample sizes and complex models if the methods are combined with adaptivity and dimension reduction techniques. In addition, we show by an analysis of variance (ANOVA) that ALM problems are often of very low effective dimension which provides a theoretical explanation for the success of the deterministic quadrature methods.     

Noisy Hamiltonian Monte Carlo for Doubly Intractable Distributions

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the statistician’s toolbox as an alternative sampling method in settings when standard Metropolis–Hastings is inefficient. HMC generates a Markov chain on an augmented state space with transitions based on a deterministic differential flow derived from Hamiltonian mechanics. In practice, the evolution of Hamiltonian systems cannot be solved analytically, requiring numerical integration schemes. Under numerical integration, the resulting approximate solution no longer preserves the measure of the target distribution, therefore an accept–reject step is used to correct the bias. For doubly intractable distributions—such as posterior distributions based on Gibbs random fields—HMC suffers from some computational difficulties: computation of gradients in the differential flow and computation of the accept–reject proposals poses difficulty. In this article, we study the behavior of HMC when these quantities are replaced by Monte Carlo estimates. Supplemental codes for implementing methods used in the article are available online.     

Quadrature Methods for Bayesian Optimal Design of Experiments With Nonnormal Prior Distributions

Many optimal experimental designs depend on one or more unknown model parameters. In such cases, it is common to use Bayesian optimal design procedures to seek designs that perform well over an entire prior distribution of the unknown model parameter(s). Generally, Bayesian optimal design procedures are viewed as computationally intensive. This is because they require numerical integration techniques to approximate the Bayesian optimality criterion at hand. The most common numerical integration technique involves pseudo Monte Carlo draws from the prior distribution(s). For a good approximation of the Bayesian optimality criterion, a large number of pseudo Monte Carlo draws is required. This results in long computation times. As an alternative to the pseudo Monte Carlo approach, we propose using computationally efficient Gaussian quadrature techniques. Since, for normal prior distributions, suitable quadrature techniques have already been used in the context of optimal experimental design, we focus on quadrature techniques for nonnormal prior distributions. Such prior distributions are appropriate for variance components, correlation coefficients, and any other parameters that are strictly positive or have upper and lower bounds. In this article, we demonstrate the added value of the quadrature techniques we advocate by means of the Bayesian D-optimality criterion in the context of split-plot experiments, but we want to stress that the techniques can be applied to other optimality criteria and other types of experimental designs as well. Supplementary materials for this article are available online.     

Analysis of estimation accuracy of the first moments of a Monte Carlo solution to an SDE with Wiener and Poisson components

In this paper, the estimation accuracy of the first moments of a numerical solution to an SDE with Wiener and Poisson components is investigated by a generalized explicit Euler method. Exact expressions for the mathematical expectation and variance of a test SDE solution are obtained. These expressions allow us to investigate the estimation accuracy obtained by a Monte Carlo method versus the SDE parameters, the integration step, and the size of the ensemble of simulated trajectories of the solution. The results of test numerical experiments are presented.     

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.     

Quasi-Monte Carlo Sampling for Solving Partial Differential Equations by Deep Neural Networks

Solving partial differential equations in high dimensions by deep neural networks has brought significant attentions in recent years. In many scenarios, the loss function is defined as an integral over a high-dimensional domain. Monte-Carlo method, together with a deep neural network, is used to overcome the curse of dimensionality, while classical methods fail. Often, a neural network outperforms classical numerical methods in terms of both accuracy and efficiency. In this paper, we propose to use quasi-Monte Carlo sampling, instead of Monte-Carlo method to approximate the loss function. To demonstrate the idea, we conduct numerical experiments in the framework of deep Ritz method. For the same accuracy requirement, it is observed that quasi-Monte Carlo sampling reduces the size of training data set by more than two orders of magnitude compared to that of Monte-Carlo method. Under some assumptions, we can prove that quasi-Monte Carlo sampling together with the deep neural network generates a convergent series with rate proportional to the approximation accuracy of quasi-Monte Carlo method for numerical integration. Numerically the fitted convergence rate is a bit smaller, but the proposed approach always outperforms Monte Carlo method.     

金融中的Levy模型及其仿真

近20年来,金融中Levy模型与蒙特卡洛仿真技术日益受到重视. 在连续时间过程的金融建模中带跳跃的Levy模型相比于连续轨道的布朗运动模型能很好地刻画市场的跳跃,更好地拟合金融数据的统计特征,更准确地对衍生品定价. 但是,相较于经典的Black-Scholes模型,用Levy模型对衍生品定价以及求解对冲策略的计算复杂度大大增加. 蒙特卡洛仿真成为Levy模型计算中最重要的方法之一. 首先详细地介绍了Levy模型引入的背景,并引出仿真方法在其中重要的应用价值. 最后,简要地给出了Levy过程仿真及其梯度估计的基本方法.     

Playdoh: A lightweight Python library for distributed computing and optimisation

Parallel computing is now an essential paradigm for high performance scientific computing. Most existing hardware and software solutions are expensive or difficult to use. We developed Playdoh, a Python library for distributing computations across the free computing units available in a small network of multicore computers. Playdoh supports independent and loosely coupled parallel problems such as global optimisations, Monte Carlo simulations and numerical integration of partial differential equations. It is designed to be lightweight and easy to use and should be of interest to scientists wanting to turn their lab computers into a small cluster at no cost.     

Adaptive integration for multi-factor portfolio credit loss models

We propose algorithms of adaptive integration for calculation of the tail probability in multi-factor credit portfolio loss models. We first modify the classical Genz-Malik rule, a deterministic multiple integration rule suitable for portfolio credit models with number of factors less than 8. Later on we arrive at the adaptive Monte Carlo integration, which essentially replaces the deterministic integration rule by antithetic random numbers. The latter can not only handle higher-dimensional models but is also able to provide reliable probabilistic error bounds. Both algorithms are asymptotic convergent and consistently outperform the plain Monte Carlo method.