Multilevel Monte Carlo method with applications to stochastic partial differential equations

In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo (MLMC) method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O(h ^?2) if h is the error of the approximation. The MLMC method is applied to functions of solutions of parabolic and hyperbolic stochastic partial differential equations as needed, for example, for option pricing. Simulations complete the paper.     

Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients

This paper proposes and analyses two numerical methods for solving elliptic partial differential equations with random coefficients, under the finite noise assumption. First, the stochastic discontinuous Galerkin method represents the stochastic solution in a Galerkin framework. Second, the Monte Carlo discontinuous Galerkin method samples the coefficients by a Monte Carlo approach. Both methods discretize the differential operators by the class of interior penalty discontinuous Galerkin methods. Error analysis is obtained. Numerical results show the sensitivity of the expected value and variance with respect to the penalty parameter of the spatial discretization.     

Testing random number generators for microcomputer applications of Monte Carlo simulation

Monte Carlo simulation is a venerable method of solving probabilistic modeling problems, but has historically been thought of as a mainframe technique because of its computational burden. With the great increases in the speed and power of microcomputers, there is increasing interest in microcomputer-based Monte Carlo techniques applied to probabilistic environmental modeling. Effective Monte Carlo simulation requires fast, accurate random number generators and generator performance is often software and hardware specific. This paper describes a package of diagnostic programs titled RANDALIZE that has been developed to interrogate proposed random number generators to determine if they yield sufficiently random variates with the desired probabilistic properties. The methods used to interrogate a random number sequence are defined and illustrated. Methods for interpreting the results are discussed. Example results are presented to illustrate program functions.     

Monte Carlo comparison of six hierarchical clustering methods on random data

There is mounting evidence to suggest that the complete linkage method does the best clustering job among all hierarchical agglomerative techniques, particularly with respect to misclassification in samples from known multivariate normal distributions. However, clustering methods are notorious for discovering clusters on random data sets also. We compare six agglomerative hierarchical methods on univariate random data from uniform and standard normal distributions and find that the complete linkage method generally is best in not discovering false clusters. The criterion is the ratio of number of within-cluster distances to number of all distances at most equal to the maximum within-cluster distance.     

An upscaling method using coefficient splitting and its applications to elliptic PDEs

In this paper, we develop an upscaling method using coefficient splitting techniques. Green’s function is constructed using the differential operator associated with the first part of the splitting. An effective upscaling coefficient is recursively calculated by Green’s function. The computation of the upscaling process involves some independent steps. Combining the proposed upscaling method with the stochastic collocation method, we present a stochastic space reduction collocation method, where the stochastic collocation method is performed on a lower dimension stochastic space than the full-dimension stochastic space. We thoroughly analyze the convergence of the proposed upscaling method for both deterministic and stochastic elliptic PDEs. Computation complexity is also addressed for the stochastic upscaling method. A number of numerical tests are presented to confirm the convergence analysis.     

Multigrid and sparse-grid schemes for elliptic control problems with random coefficients

A multigrid and sparse-grid computational approach to solving nonlinear elliptic optimal control problems with random coefficients is presented. The proposed scheme combines multigrid methods with sparse-grids collocation techniques. Within this framework the influence of randomness of problem’s coefficients on the control provided by the optimal control theory is investigated. Numerical results of computation of stochastic optimal control solutions and formulation of mean control functions are presented.     

Exploring pseudo- and chaotic random Monte Carlo simulations

Computer simulations are an increasingly important area of geoscience research and development. At the core of stochastic or Monte Carlo simulations are the random number sequences that are assumed to be distributed with specific characteristics. Computer-generated random numbers, uniformly distributed on (0, 1), can be very different depending on the selection of pseudo-random number (PRN) or chaotic random number (CRN) generators. In the evaluation of some definite integrals, the resulting error variances can even be of different orders of magnitude. Furthermore, practical techniques for variance reduction such as importance sampling and stratified sampling can be applied in most Monte Carlo simulations and significantly improve the results. A comparative analysis of these strategies has been carried out for computational applications in planar and spatial contexts. Based on these experiments, and on some practical examples of geodetic direct and inverse problems, conclusions and recommendations concerning their performance and general applicability are included.     

Application of new Monte Carlo algorithms to random spin systems

We explain the idea of the probability-changing cluster (PCC) algorithm, which is an extended version of the Swendsen-Wang algorithm. With this algorithm, we can tune the critical point automatically. We show the effectiveness of the PCC algorithm for the case of the three-dimensional (3D) Ising model. We also apply this new algorithm to the study of the 3D diluted Ising model. Since we tune the critical point of each random sample automatically with the PCC algorithm, we can investigate the sample-dependent critical temperature and the sample average of physical quantities at each critical temperature, systematically. We have also applied another newly proposed algorithm, the Wang-Landau algorithm, to the study of the spin glass problem.     

Monte Carlo simulations of microstrip lines with random substrate impurity

Monte Carlo simulations of microstrip lines with random substrate impurity are performed. Each realization of current distribution on the microstrip line with substrate impurity is calculated by the hybrid finite element method/multilevel fast multipole algorithm. After taking the ensemble average of all the realizations, the super‐resolution estimation of signal parameters via rotational invariance technique algorithm is employed to extract the characteristic parameters of the transmission line from the current distribution. Comparisons of our numerical results and previously published results for the conventional microstrip line with homogeneous substrate confirm the validity, accuracy, and efficiency of our simulation tool. Our Monte Carlo simulations demonstrate that the effect of substrate impurity on the characteristics of a microstrip line can be quite accurately estimated by replacing the inhomogeneous substrate with a homogeneous one whose dielectric constant is the averaged value for different materials weighted by their volume ratios. This simple estimation approach, however, does not apply to the attenuation constant of the line due to the effect of finite substrate width. © 2001 John Wiley & Sons, Inc. Int J RF and Microwave CAE 11: 177–187 (2001)     

Optimization of decentralized random field estimation networks under communication constraints through Monte Carlo methods

We propose a new methodology for designing decentralized random field estimation schemes that takes the tradeoff between the estimation accuracy and the cost of communications into account. We consider a sensor network in which nodes perform bandwidth limited two-way communications with other nodes located in a certain range. The in-network processing starts with each node measuring its local variable and sending messages to its immediate neighbors followed by evaluating its local estimation rule based on the received messages and measurements. Local rule design for this two-stage strategy can be cast as a constrained optimization problem with a Bayesian risk capturing the cost of transmissions and penalty for the estimation errors. A similar problem has been previously studied for decentralized detection. We adopt that framework for estimation, however, the corresponding optimization schemes involve integral operators that are impossible to evaluate exactly, in general. We employ an approximation framework using Monte Carlo methods and obtain an optimization procedure based on particle representations and approximate computations. The procedure operates in a message-passing fashion and generates results for any distributions if samples can be produced from, e.g., the marginals. We demonstrate graceful degradation of the estimation accuracy as communication becomes more costly.     

Multilevel Schwarz methods for elliptic partial differential equations

We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method. In our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary value problems, including a convection–diffusion problem when suitable stabilization becomes necessary.     

Error and complexity of random walk Monte Carlo radiosity

The author studies the error and complexity of the discrete random walk Monte Carlo technique for radiosity, using both the shooting and gathering methods. The author shows that the shooting method exhibits a lower complexity than the gathering one, and under some constraints, it has a linear complexity. This is an improvement over a previous result that pointed to an O(n log n) complexity. The author gives and compares three unbiased estimators for each method, and obtains closed forms and bounds for their variances. The author also bounds the expected value of the mean square error (MSE). Some of the results obtained are also shown to be valid for the nondiscrete gathering case. The author also gives bounds for the variances and MSE for the infinite path length estimators; these bounds might be useful in the study of biased estimators resulting from cutting off the infinite path     

Global‐view coefficients: a data management solution for parallel quantum Monte Carlo applications

Quantum Monte Carlo (QMC) applications perform simulation with respect to an initial state of the quantum mechanical system, which is often captured by using a cubic B‐spline basis. This representation is stored as a read‐only table of coefficients and accesses to the table are generated at random as part of the Monte Carlo simulation. Current QMC applications, such as QWalk and QMCPACK, replicate this table at every process or node, which limits scalability because increasing the number of processors does not enable larger systems to be run. We present a partitioned global address space approach to transparently managing this data using Global Arrays in a manner that allows the memory of multiple nodes to be aggregated. We develop an automated data management system that significantly reduces communication overheads, enabling new capabilities for QMC codes. Experimental results with QWalk and QMCPACK demonstrate the effectiveness of the data management system. Copyright © 2016 John Wiley & Sons, Ltd.     

Monte Carlo methods in fuzzy queuing theory

In this paper, we consider an optimization problem in fuzzy queuing theory that was first used in web planning. This fuzzy optimization problem has no solution algorithm and approximate solutions were first produced by computing the fuzzy value of the objective function for only sixteen values of the fuzzy variables. We introduce our fuzzy Monte Carlo method, using a quasi-random number generator, to produce 100,000 random sequences of fuzzy vectors for the fuzzy variables, which will present a much better approximate solution.     

Monte Carlo Simulation methods for slope stability

The use of the Monte Carlo Simulation method is discussed, as a sensitivity-testing tool for slope stability and also as a method of calculating P (probability of sliding failure of a given earth slope) rather than the conventional F (factor of safety against sliding failure). Probability of failure values are obtained as a result of putting the Department of Transport's program CIRCA (for slope stability) into simulation mode and are compared with values obtained analytically. The Bishop simplified formula used to determine the most critical slip circle, is the mechanical model adopted in CIRCA. Familiarity with the Bishop formula itself is assumed by the reader. Some meaning is given also to the concept of setting allowable criteria for P values, for comparison with calculated values, in order to determine whether a slope is adequately safe.     

Discrete range clustering using Monte Carlo methods

For automatic obstacle avoidance guidance during rotorcraft low altitude flight a reliable model of the nearby environment is needed. Such a model may be constructed by applying surface fitting techniques to the dense range map obtained by active sensing using radars. However, for covertness passive sensing techniques using electro-optic sensors is desirable. As opposed to the dense range map obtained via active sensing, passive sensing algorithms produce reliable range at sparse locations and, therefore, surface fitting techniques to fill the gaps in the range measurement are not directly applicable. Both, for automatic guidance and as a display for aiding the pilot, these discrete ranges need to be grouped into sets which correspond to objects in the nearby environment. The focus of this paper is on using Monte Carlo methods for clustering range points into meaningful groups. We compare three different approaches and present results of application of these algorithms to an image sequence acquired by onboard cameras during a helicopter flight. Starting with an initial grouping, these algorithms are iteratively applied with a new group creation algorithm to determine the optimal number of groups and the optimal group membership. The results indicate that the simulated annealing methods do not offer any significant advantage over the basic Monte Carlo method for this discrete optimization problem     

Monte Carlo methods in fuzzy linear regression

We apply our new fuzzy Monte Carlo method to a certain fuzzy linear regression problem to estimate the best solution. The best solution is a vector of triangular fuzzy numbers, for the fuzzy coefficients in the model, which minimizes one of two error measures. We use a quasi-random number generator to produce random sequences of these fuzzy vectors which uniformly fill the search space. We consider an example problem and show this Monte Carlo method obtains the best solution for one error measure and is approximately best for the other error measure.     

Monte Carlo随机数在图像加密中的应用

随着图像加密技术的发展,出现了各种各样的加密方法,其中有基于Arnold变换、仿射变换、Hilbert曲线、幻方、骑士巡游、Gray码、混沌序列和基于频域的置乱加密技术。讨论了Monte Carlo随机数的性质,构造了一种新的基于Monte Carlo随机数的图像加密算法。实验结果表明该方法计算简单,运行速度快,并具有一定的安全性。     

Model replication techniques for parameter-influence studies and Monte Carlo simulation with random parameters

Modern computers produce large volumes of simulation results so quickly that their management becomes a formidable task. We describe interactive computer software for replicating simulation models with different parameters. A single simulation run then produces results for hundreds of models with different parameter values without the loop overhead imposed by repeated simulations. One can

• 1.arrange corresponding values of model-parameter values and model performance measures in corresponding arrays suitable for use in commercially available spreadsheet and relational-database programs for further processing and archival storage, or
• 2.produce a Monte Carlo sample of model runs with random parameter values and compute statistics such as various averages or probability estimates as functions of simulation time in a single simulation run.

     

Monte Carlo methods for derivatives of options with discontinuous payoffs

Various Monte Carlo methods have been proposed to estimate the derivatives of contingent claims prices. The Monte Carlo approximate likelihood ratio estimator is studied. Recent convergence results are extended in order to show that the Monte Carlo approximate likelihood ratio derivative estimator is asymptotically equivalent, up to a second-order bias component, to an estimator based on a covariation approximation, the Monte Carlo Covariation estimator. Both converge slower than the Monte Carlo Malliavin derivative estimators. Theoretical convergence results are illustrated in a numerical experiment dealing with the risk management of digital options in a CEV model.