Traditional Monte Carlo Method Research Articles

A general purpose strategy for realizing the zero-variance importance sampling and calculating the unknown integration constant

A general purpose strategy is proposed to realize asymptotically the zero-variance importance sampling. The unknown integration constant can also be calculated simultaneously. This strategy can sample efficiently from multi-dimensional zero-variance importance function which is multi-modal by particular Markov Chain random walk. Sampling from this kind of distribution has been a challenge for a long time. Moreover, by using the probability density function reconstruction method, the unknown integration constant can be estimated. This feature is absent in traditional Markov Chain Monte Carlo method. Some multi-dimensional integrals are analyzed carefully. The results show this strategy is efficient.

Baseline Comparisons of Complementary Sampling Methods for Assembly Driven by Short-Ranged Pair Potentials toward Fast and Flexible Hybridization.

This work measures baseline sampling characteristics that highlight fundamental differences between sampling methods for assembly driven by short-ranged pair potentials. Such granular comparison is essential for fast, flexible, and accurate hybridization of complementary methods. Besides sampling speed, efficiency, and accuracy of uniform grid coverage, other sampling characteristics measured are (i) accuracy of covering narrow low energy regions that have low effective dimension (ii) ability to localize sampling to specific basins, and (iii) flexibility in sampling distributions. As a proof of concept, we compare a recently developed geometric methodology EASAL (Efficient Atlasing and Search of Assembly Landscapes) and the traditional Monte Carlo (MC) method for sampling the energy landscape of two assembling trans-membrane helices, driven by short-range pair potentials. By measuring the above-mentioned sampling characteristics, we demonstrate that EASAL provides localized and accurate coverage of crucial regions of the energy landscape of low effective dimension, under flexible sampling distributions, with much fewer samples and computational resources than MC sampling. EASAL's empirically validated theoretical guarantees permit credible extrapolation of these measurements and comparisons to arbitrary number and size of assembling units. Promising avenues for hybridizing the complementary advantages of the two methods are discussed.

An Iterative Response-Surface-Based Approach for Chance-Constrained AC Optimal Power Flow Considering Dependent Uncertainty

A modern power system is characterized by a stochastic variation of the loads and an increasing penetration of renewable energy generation, which results in large uncertainties in its states. These uncertainties bring formidable challenges to the power system planning and operation process. To address these challenges, we propose a cost-effective, iterative response-surface-based approach for the chance-constrained AC optimal power-flow problem that aims to ensure the secure operation of the power systems considering dependent uncertainties. Starting from a stochastic-sampling-based framework, we first utilize the copula theory to simulate the dependence among multivariate uncertain inputs. Then, to reduce the prohibitive computational time required in the traditional Monte-Carlo method, we propose, instead of using the original complicated power-system model, to rely on a polynomial-chaos-based response surface. This response surface allows us to efficiently evaluate the time-consuming power-system model at arbitrary distributed sampled values with a negligible computational cost. This further enables us to efficiently conduct an online stochastic testing for the system states that not only screens out the statistical active constraints, but also assists in a better design of the tightened bounds without using any Gaussian or symmetric assumption. Finally, an iterative procedure is executed to fine-tune the optimal solution that better satisfies a predefined probability. The simulations conducted in multiple test systems demonstrate the excellent performance of the proposed method.

Improving the computational efficiency of first arrival time uncertainty estimation using a connectivity-based ranking Monte Carlo method

The first arrival time of a solute plume migrating from its source to an environmentally sensitive target is one of the key quantities of interest when assessing the risks of groundwater contamination. First arrival times are correlated with the hydraulic connectivity properties of spatially heterogeneous porous formations. Hydraulic connectivity leads to the presence of preferential flow paths which in turn control the transport dynamics of the leading edge of the solute plume and therefore first arrival times. In applications, these arrival times are subject to uncertainty given the lack of a detailed site characterization. The Monte Carlo method is commonly adopted to estimate the uncertainty of solute arrival times, however it leads to a computational burden. In this work, we build upon the existing knowledge regarding the correlation between connectivity and first arrival times to propose an innovative connectivity-based ranking Monte Carlo approach to quantify the uncertainty of first arrival times. The proposed method is tailored to predict first arrival times and allows to alleviate the computational costs when compared to the traditional Monte Carlo method. Our method consists of ranking the randomly generated spatially heterogeneous hydraulic conductivity fields according to their connectivity. The connectivity metric adopted is based on the concept of the minimum hydraulic resistance and can be obtained at a very low computational cost through the use of graph theory. We illustrate the methodology by analyzing the convergence rate of the first arrival time means and standard deviations. We compare the convergence rate of the first arrival times statistics obtained through the proposed methodology with those computed through the traditional Monte Carlo method. Overall, our results indicate the that the proposed methodology ensures a faster convergence of the considered quantities, thus reducing the time required for their estimation and the associated computational burden.

Static workspace analysis of a multi-robot collaborative towing system with floating base

The static workspace of the multi-robot collaborative towing system with floating base was analyzed. The spacial configuration of the system was built, and the system was divided into three types according to the actuation. The kinematic and dynamic models of the floating base and the towing system were established, respectively, by using the Newton-Euler equation. Then the static workspace of the system was analyzed, and the specific steps to solve the workspace with the improved Monte Carlo method were obtained. Finally, the static workspaces of three types of systems were solved though actual simulation calculation, the simulation results show that the improved Monte Carlo method can get a workspace with higher precision compare to the traditional Monte Carlo method. Results of this study lay a basis for the subsequent trajectory plan, stability analysis and configuration optimization of the system.

Bidirectional monte carlo method for thermal radiation transfer in participating medium

<abstract> A bidirectional Monte Carlo (BDMC) method based on reversibility of bundle trajectory and reciprocity of thermal radiative energy exchange was developed to solve radiative heat transfer in absorbing and scattering medium. Two types of sampling models were introduced into the Monte Carlo (MC) simulation, namely the equivalent sampling and the weight sampling, respectively. Mathematical formula for the sampling models and the statistical calculation of sampling bundles were derived. Furthermore, the reciprocity error correlation of radiative exchange factors between the BDMC method and the traditional Monte Carlo (TMC) method were demonstrated and analyzed. Radiative heat transfer in a two-dimensional rectangular domain with absorbing and scattering media was solved by using both the BDMC method and the TMC method. Radiative exchange factors and radiative equilibrium temperature profiles predicted by the BDMC method were compared with those predicted by the TMC method. The performance parameter P, defined to evaluate the performance of MC methods, was computed and compared between the BDMC and the TMC methods. The results showed the superiority of BDMC method compared with the TMC method for radiative heat transfer, in addition, the weight sampling was proved to be more flexible than the equivalent sampling in the BDMC method. </abstract>

Ising spin configurations with the deep learning method

In this work, we explore the statistical system's configuration generation with generative models from Deep Learning to go beyond conventional Monte Carlo methods. Specifically, we devise a conditional generative adversarial network (cGAN) for the Ising system spin configuration generation, and we demonstrate it's working outside the training range of temperature for the ensemble of configurations. Being different from the original GAN design, we add a further recognizer network for the constraint on the conditional parameters (in our case temperature) and also provide a diversity benefit for the generative model. We showed that the newly proposed cGAN here can learn the distribution of the Ising model for different temperatures, which can generate spin configuration efficiently with correct (within a probability distribution) temperature estimation for the microscopic configurations. Moreover, without information about criticality provided inside the training data set, it is also shown that the developed cGAN can generate Ising spin configurations around the phase transition point with the order parameter (mean magnetization) reasonably well-match to the conventional MCMC simulation but with parallel sampling advantage. We also compared typical spin configurations from cGAN with specified conditional temperature to be the critical temperature with samples simulated by MCMC, which visibly is not distinguishable. This thus can possibly help to avoid critical slowing-down as shown in the traditional Monte Carlo method.

An Efficient Anti-Optimization Approach for Uncertainty Analysis in Composite Laminates

Abstract This work presents an efficient approach to quantify uncertainties in composite laminates using the interval analysis, anti-optimization technique, and the α-cut procedure. The solutions are compared with the traditional and robust Monte Carlo method in 3 cases scenarios: natural frequencies, buckling, and strength safe factor. For natural frequencies and buckling loads, the presented Interval based methodology showed 2.5% to 4.5% larger error values when compared to the Monte Carlo method using the same number of function calls. This implies a larger uncertain area, and hence, a better solution. For the strength test using Tsai-Wu failure theory, the error values are even greater: 22% to 46%. A violation of the failure limit was detected by the proposed Interval based approach, but not detected by Monte Carlo method. The solutions show that the presented methodology yields a safer and more precise analysis when compared to the traditional Monte Carlo approach.

Приближенное решение многомерных интегральных уравнений методами Монте Карло и квази-Монте Карло

The article considers an approach based on the random cubature method for solving both single and multidimensional singular integral equations, Volterra and Fredholm equations of the 1st kind, for ill-posed problems in the theory of integral equations, etc. A variant of the quasi-Monte Carlo method is studied. The integral in an integral equation is approximated using the traditional Monte Carlo method for calculating integrals. Multidimensional interpolation is applied on an arbitrary set of points. Examples of applying the method to a one-dimensional integral equation with a smooth kernel using both random and low-dispersed pseudo-random nodes are considered. A multidimensional linear integral equation with a polynomial kernel and a multidimensional nonlinear problem – the Hammerstein integral equation – are solved using the Newton method. The existence of several solutions is shown. Multidimensional integral equations of the first kind and their solution using regularization are considered. The Monte Carlo and quasi-Monte Carlo methods have not been used to solve such problems in the studied literature. The Lavrentiev regularization method was used, as well as random and pseudo-random nodes obtained using the Halton sequence. The problem of eigenvalues is solved. It is established that one of the best methods considered is the Leverrier-Faddeev method. The results of solving the problem for a different number of quadrature nodes are presented in the table. An approach based on parametric regularization of the core, an interpolation-projection method, and averaged adaptive densities are studied. The considered methods can be successfully applied in solving spatial boundary value problems for areas of complex shape. These approaches allow us to expand the range of problems in the theory of integral equations solved by Monte Carlo and quasi-Monte Carlo methods, since there are no restrictions on the value of the norm of the integral operator. A series of examples demonstrating the effectiveness of the method under study is considered.

Enhancing risk control ability of distribution network for improved renewable energy integration through flexible DC interconnection

The increased penetration of distributed renewable energy into modern distribution networks has resulted in a significant rise in the system operational risk. However, the risk control ability of traditional open-loop AC distribution networks is greatly limited due to the inherent restriction of the network structure, which hinders the integration of larger-scale renewable energy into the power grid. To address this issue, this paper developed an effective risk assessment model and risk averse method for fully utilizing flexible DC interconnections to enhance the risk control ability and to improve the penetration of renewable energy in distribution networks. Moreover, aiming at the complex coupling risk characteristics between the AC and DC system that are imposed by the DC interconnection and considering the inefficiency of traditional Monte Carlo sampling risk assessment methods, an effective point-estimation-based risk assessment method is proposed that conducts a limited number of sequential AC/DC power flow calculations with estimation points constructed from random input variables, thereby the operational risk of the system is effectively realized. On this basis, a risk-averse multi-objective optimization model is further established for minimizing both the operational risk and the expected network power loss, and an improved kriging-model-assisted multi-object gray wolf algorithm is proposed for quickly supporting the model solution. Finally, a fuzzy clustering method is presented for effectively identifying the optimal Pareto solution. A case study is conducted on an AC/DC hybrid system with three AC networks and one DC network. The results demonstrate that the proposed method can substantially improve the risk assessment efficiency compared with the traditional Monte Carlo method. In addition, the risk-averse optimization method can effectively reduce the system risk and improve the operational reliability as well as the penetration of renewable energy.

New method for fitting the low-energy constants in chiral perturbation theory

A new set of the next-to-leading order (NLO) and the next-to-next-to-leading order (NNLO) low-energy constants $L_i^r$ and $C_i^r$ in chiral perturbation theory is obtained. These values are computed using the new experimental data with a new calculation method. This method combines the traditional global fit and Monte Carlo method together. The higher order contributions are estimated with this method. The theoretical values of the observables provide good convergence at each chiral dimension, except for the NNLO values of the $\pi K$ scattering lengths $a_0^{3/2}$ and $a_0^{1/2}$. The fitted values for $L_i^r$ at NLO are close to their results with the new method at NNLO; i.e., these $L_i^r$ are nearly order-independent in this method. The estimated ranges for $C_i^r$ are consistent with those in the literature, and their possible upper or/and lower boundaries are given. The values of some linear combinations of $C_i^r$ are also given, and they are more reliable. If one knows a more exact value $C_i^r$, another $C_i^r$ can be obtained by these values.

Fast solver for uncertainty EM scattering problems by the perturbed-based MLFMA

A novel fast algorithm is proposed to analyze the EM scattering from electrically large conducting targets with varying geometrical shapes. Firstly, the target surface can be constructed with several control points by using the non-uniform rational B-spline surface modeling method. More specifically, the surface integral equation can be rewritten in terms of these control points, which contains the geometrical coordinate information. Moreover, it should be noted that the derivatives of both the L and K operates are needed to describe the perturbation of electric currents. Therefore, a perturbed multilevel fast multipole algorithm (P-MLFMA) is proposed to accelerate the matrix vector multiplication. At last, numerical results are given to verify the accuracy and efficiency of the proposed P-MLFMA. It can be seen that higher computational efficiency can be obtained when compared with both the traditional Monte Carlo method and the Perturbation Approach for MoM.

A fully Bayesian approach to sparse reduced-rank multivariate regression

In the context of high-dimensional multivariate linear regression, sparse reduced-rank regression (SRRR) provides a way to handle both variable selection and low-rank estimation problems. Although there has been extensive research on SRRR, statistical inference procedures that deal with the uncertainty due to variable selection and rank reduction are still limited. To fill this research gap, we develop a fully Bayesian approach to SRRR. A major difficulty that occurs in a fully Bayesian framework is that the dimension of parameter space varies with the selected variables and the reduced-rank. Due to the varying-dimensional problems, traditional Markov chain Monte Carlo (MCMC) methods such as Gibbs sampler and Metropolis-Hastings algorithm are inapplicable in our Bayesian framework. To address this issue, we propose a new posterior computation procedure based on the Laplace approximation within the collapsed Gibbs sampler. A key feature of our fully Bayesian method is that the model uncertainty is automatically integrated out by the proposed MCMC computation. The proposed method is examined via simulation study and real data analysis.

The time-varying asymmetry of exchange rate returns: A stochastic volatility – stochastic skewness model

Abstract While the time-varying volatility of financial returns has been extensively modelled, most existing stochastic volatility models either assume a constant degree of return shock asymmetry or impose symmetric model innovations. However, accounting for time-varying asymmetry as a measure of crash risk is important for both investors and policy makers. This paper extends a standard stochastic volatility model to allow for time-varying skewness of the return innovations. We estimate the model by extensions of traditional Markov Chain Monte Carlo (MCMC) methods for stochastic volatility models. When applying this model to the returns of four major exchange rates, skewness is found to vary substantially over time. In addition, stochastic skewness can help to improve forecasts of risk measures. Finally, the results support a potential link between carry trading and crash risk.

Uncertainty RCS Computation for Multiple and Multilayer Thin Medium-Coated Conductors by an Improved TDS Approximation

An efficient uncertainty RCS computation technology is proposed to analyze the EM scattering from 3-D medium-coated targets with varying geometrical shapes. First, the scattering target is described by a nonuniform rational B-spline surface, where the uncertain units are described by several independent random variables. Then, an improved thin dielectric sheet-based method of moments is presented to model the scattering from the multiple and multilayer thin medium-coated objects. As a result, only unknowns exist on the target’s outer surface, and the computational resources can significantly be saved in comparison with solvers that associate unknowns with all the surfaces and interfaces. It should also be noted that the half RWG basis functions are introduced at the interface of different media for multiple medium-coated conductors. Finally, both the mean value and variance of the scattering electric current are derived by applying a perturbation approach to account for the target’s uncertain variable geometry. Moreover, the validity of the proposed method is verified in comparison with the traditional Monte Carlo (MC) method, which is used to solve several certain problems, and it is found that the proposed method can provide higher efficiency compared with the MC method.

Study in Network Stability based on MC

Monte Carlo is a convenient method for statistics sampling theory of mathematical and physical. It is widely applied to the field of staff performance in geological, electric, medical, optical, water conservancy and so on. In order to effectively assess the stability and reliability of the network, this paper put forward an improved Monte Carlo evaluation method on the basis of the traditional Monte Carlo method. First of all, the paper elaborated the principle of improved Monte Carlo method and the specific method of how to calculate the stability of the network. The improved Monte Carlo method employed the recursive variance reduction method to get a smaller sample in the original sample, to ensure the estimate is unbiased. Secondly, the paper applied the ideology of permutation and combination to the network nodes and links in the model, and combined with the upper and lower boundary of network performance, to calculate the connected stability. Finally, the paper carried out experiments between the improved and traditional Monte Carlo method in Matlab simulation to contrast and analyze. The results show that the improved Monte Carlo method to evaluate the stability of the network has higher accuracy and smaller variance. So the experiment proved that it has the high feasibility and validity.

Bayesian Model Calibration with Interpolating Polynomials based on Adaptively Weighted Leja Nodes

An efficient algorithm is proposed for Bayesian model calibration, which is commonly used to estimate the model parameters of non-linear, computationally expensive models using measurement data. The approach is based on Bayesian statistics: using a prior distribution and a likelihood, the posterior distribution is obtained through application of Bayes' law. Our novel algorithm to accurately determine this posterior requires significantly fewer discrete model evaluations than traditional Monte Carlo methods. The key idea is to replace the expensive model by an interpolating surrogate model and to construct the interpolating nodal set maximizing the accuracy of the posterior. To determine such a nodal set an extension to weighted Leja nodes is introduced, based on a new weighting function. We prove that the convergence of the posterior has the same rate as the convergence of the model. If the convergence of the posterior is measured in the Kullback-Leibler divergence, the rate doubles. The algorithm and its theoretical properties are verified in three different test cases: analytical cases that confirm the correctness of the theoretical findings, Burgers' equation to show its applicability in implicit problems, and finally the calibration of the closure parameters of a turbulence model to show the effectiveness for computationally expensive problems.

Calculating the Expected Value of Sample Information in Practice: Considerations from 3 Case Studies.

Background. Investing efficiently in future research to improve policy decisions is an important goal. Expected value of sample information (EVSI) can be used to select the specific design and sample size of a proposed study by assessing the benefit of a range of different studies. Estimating EVSI with the standard nested Monte Carlo algorithm has a notoriously high computational burden, especially when using a complex decision model or when optimizing over study sample sizes and designs. Recently, several more efficient EVSI approximation methods have been developed. However, these approximation methods have not been compared, and therefore their comparative performance across different examples has not been explored. Methods. We compared 4 EVSI methods using 3 previously published health economic models. The examples were chosen to represent a range of real-world contexts, including situations with multiple study outcomes, missing data, and data from an observational rather than a randomized study. The computational speed and accuracy of each method were compared. Results. In each example, the approximation methods took minutes or hours to achieve reasonably accurate EVSI estimates, whereas the traditional Monte Carlo method took weeks. Specific methods are particularly suited to problems where we wish to compare multiple proposed sample sizes, when the proposed sample size is large, or when the health economic model is computationally expensive. Conclusions. As all the evaluated methods gave estimates similar to those given by traditional Monte Carlo, we suggest that EVSI can now be efficiently computed with confidence in realistic examples. No systematically superior EVSI computation method exists as the properties of the different methods depend on the underlying health economic model, data generation process, and user expertise.

MOCMC: Method of Characteristics Moment Closure, a Numerical Method for Covariant Radiation Magnetohydrodynamics

Abstract We present a conservative numerical method for radiation magnetohydrodynamics with frequency-dependent full transport in stationary spacetimes. This method is stable and accurate for both large and small optical depths and radiation pressures. The radiation stress–energy tensor is evolved in flux-conservative form, and closed with a swarm of samples that each transport a multigroup representation of the invariant specific intensity along a null geodesic. In each zone, the enclosed samples are used to efficiently construct a Delaunay triangulation of the unit sphere in the comoving frame, which in turn is used to calculate the Eddington tensor, average source terms, and adaptively refine the sample swarm. Radiation four-forces are evaluated in the moment sector in a semi-implicit fashion. The radiative transfer equation is solved in invariant form deterministically for each sample. Since each sample carries a discrete representation of the full spectrum, the cost of evaluating the transport operator is independent of the number of frequency groups, representing a significant reduction of algorithmic complexity for transport in frequency-dependent problems. The major approximation we make in this work is performing scattering in an angle-averaged way. Local adaptivity in samples also makes this scheme more amenable to nonuniform meshes than a traditional Monte Carlo method. We describe the method and present results on a suite of test problems. We find that Method of Characteristics Moment Closure converges at least as ∼N −1, rather than the canonical Monte Carlo N −1/2, where N is the number of samples per zone.

Bayesian empirical likelihood for ridge and lasso regressions

Ridge and lasso regression models, which are also known as regularization methods, are widely used methods in machine learning and inverse problems that introduce additional information to solve ill-posed problems and/or perform feature selection. The ridge and lasso estimates for linear regression parameters can be interpreted as Bayesian posterior estimates when the regression parameters have Normal and independent Laplace (i.e., double-exponential) priors, respectively. A significant challenge in regularization problems is that these approaches assume that data are normally distributed, which makes them not robust to model misspecification. A Bayesian approach for ridge and lasso models based on empirical likelihood is proposed. This method is semiparametric because it combines a nonparametric model and a parametric model. Hence, problems with model misspecification are avoided. Under the Bayesian empirical likelihood approach, the resulting posterior distribution lacks a closed form and has a nonconvex support, which makes the implementation of traditional Markov chain Monte Carlo (MCMC) methods such as Gibbs sampling and Metropolis–Hastings very challenging. To solve the nonconvex optimization and nonconvergence problems, the tailored Metropolis–Hastings approach is implemented. The asymptotic Bayesian credible intervals are derived.