Standard Monte Carlo Method Research Articles

Microcanonical-ensemble computer simulation of the high-temperature expansion coefficients of the Helmholtz free energy of a square-well fluid

ABSTRACTThe microcanonical-ensemble computer simulation method (MCE) is used to evaluate the perturbation terms Ai of the Helmholtz free energy of a square-well (SW) fluid. The MCE method offers a very efficient and accurate procedure for the determination of perturbation terms of discrete-potential systems such as the SW fluid and surpass the standard NVT canonical ensemble Monte Carlo method, allowing the calculation of the first six expansion terms. Results are presented for the case of a SW potential with attractive ranges 1.1 ≤ λ ≤ 1.8. Using semi-empirical representation of the MCE values for Ai, we also discuss the accuracy in the determination of the phase diagram of this system.

Optimal multilevel randomized quasi-Monte-Carlo method for the stochastic drift–diffusion-Poisson system

In this paper, an optimal multilevel randomized quasi-Monte-Carlo method to solve the stationary stochastic drift–diffusion-Poisson system is developed. We calculate the optimal values of the parameters of the numerical method such as the mesh sizes of the spatial discretization and the numbers of quasi-points in order to minimize the overall computational cost for solving this system of stochastic partial differential equations. This system has a number of applications in various fields, wherever charged particles move in a random environment. It is shown that the computational cost of the optimal multilevel randomized quasi-Monte-Carlo method, which uses randomly shifted low-discrepancy sequences, is one order of magnitude smaller than that of the optimal multilevel Monte-Carlo method and five orders of magnitude smaller than that of the standard Monte-Carlo method. The method developed here is applied to a realistic transport problem, namely the calculation of random-dopant effects in nanoscale field-effect transistors.

The pricing of spread option using simulation

Monte Carlo simulation is one of the most common and popular method of options pricing. The advantages of this method are being easy to use, suitable for all kinds of standard and exotic options and also are suitable for higher dimensional problems. But on the other hand Monte Carlo variance convergence rate is which due to that it will have relatively slow convergence rate to answer the problems, as to achieve accuracy when it has been d-dimensions, complexity is . For this purpose, several methods are provided in quasi Monte Carlo simulation to increase variance convergence rate as variance reduction techniques, so far. One of the latest presented methods is multilevel Monte Carlo that is introduced by Giles in 2008. This method not only reduces the complexity of computing amount in use of Euler discretization scheme and the amount in use of Milstein discretization scheme, but also has the ability to combine with other variance reduction techniques. In this paper, using Multilevel Monte Carlo method by taking Milstein discretization scheme, pricing spread option and compared complexity of computing with standard Monte Carlo method. The results of Multilevel Monte Carlo method in pricing spread options are better than standard Monte Carlo simulation.

Quasi-Monte-Carlo-based probabilistic assessment of wall heat loss

In this paper, the potential of quasi-Monte Carlo methods for uncertainty propagation is assessed, via a case study of heat loss through a massive masonry wall. Four quasi-Monte Carlo sampling strategies – Optimized Latin hypercube, Sobol sequence, Niederreiter-Xing sequence and Good Lattice sequence – are applied and compared. Moreover, in order to terminate the quasi-Monte Carlo simulation when the desired accuracy is reached, an error estimation method is implemented. The outcomes show that all the four quasi-Monte Carlo methods outperform the standard Monte Carlo method; the Niederreiter-Xing sequence and Sobol sequence tend to be the best.

An efficient stochastic dynamic analysis of soil media using radial basis function artificial neural network

Since a lot of engineering problems are along with uncertain parameters, stochastic methods are of great importance for incorporating random nature of a system property or random nature of a system input. In this study, the stochastic dynamic analysis of soil mass is performed by finite element method in the frequency domain. Two methods are used for stochastic analysis of soil media which are spectral decomposition and Monte Carlo methods. Shear modulus of soil is considered as a random field and the seismic excitation is also imposed as a random process. In this research, artificial neural network is proposed and added to Monte Carlo method for sake of reducing computational effort of the random analysis. Then, the effects of the proposed artificial neural network are illustrated on decreasing computational time of Monte Carlo simulations in comparison with standard Monte Carlo and spectral decomposition methods. Numerical verifications are provided to indicate capabilities, accuracy and efficiency of the proposed strategy compared to the other techniques.

Quantifying uncertainty in the chemical master equation.

We describe a novel approach to quantifying the uncertainty inherent in the chemical kinetic master equation with stochastic coefficients. A stochastic collocation method is coupled to an analytical expansion of the master equation to analyze the effects of both extrinsic and intrinsic noise. The method consists of an analytical moment-closure method resulting in a large set of differential equations with stochastic coefficients that are in turn solved via a Smolyak sparse grid collocation method. We discuss the error of the method relative to the dimension of the model and clarify which methods are most suitable for the problem. We apply the method to two typical problems arising in chemical kinetics with time-independent extrinsic noise. Additionally, we show agreement with classical Monte Carlo simulations and calculate the variance over time as the sum of two expectations. The method presented here has better convergence properties for low to moderate dimensions than standard Monte Carlo methods and is therefore a superior alternative in this regime.

Evaluating Wave Random Path Using Multilevel Monte Carlo

Wind waves are important due to their high energy and impact on marine activities. This phenomenon is affects directly or indirectly the construction of coastal infrastructure, shipping and recreational activities. Due to the issues presented, marine parameters are very important. In this study, we try to pay attention to wave as one of the most important marine parameters. As the movements of waves have high uncertainty, financial models can be used to simulate the wave's paths. We use the Monte Carlo method for this purpose. The Monte Carlo simulation is a flexible and simple tool that is widely used in the evaluation of random paths. To compute a random path, we require an integral discretization. In this paper, we study the valuation of European options using Monte Carlo simulation and then compare this result with multi-level Monte Carlo approach and other antithetic variables. Then, we use the multi-level Monte Carlo approach proposed by (M. B. Giles, 2008) for pricing under the two-factor stochastic volatility model. We show that the multi-level Monte Carlo method reduces the computational complexity and also cost of the two-factor stochastic volatility model when compared with the standard Monte Carlo method. Also, we compare the multi-level Monte Carlo method and standard Monte Carlo method using an Euler discretization scheme and then, analyze the numerical results.

A dynamical polynomial chaos approach for long-time evolution of SPDEs

We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the Markovian property of white noise, DgPC [1] implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen–Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier–Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.

A subset multicanonical Monte Carlo method for simulating rare failure events

Estimating failure probabilities of engineering systems is an important problem in many engineering fields. In this work we consider such problems where the failure probability is extremely small (e.g. ≤10−10). In this case, standard Monte Carlo methods are not feasible due to the extraordinarily large number of samples required. To address these problems, we propose an algorithm that combines the main ideas of two very powerful failure probability estimation approaches: the subset simulation (SS) and the multicanonical Monte Carlo (MMC) methods. Unlike the standard MMC which samples in the entire domain of the input parameter in each iteration, the proposed subset MMC algorithm adaptively performs MMC simulations in a subset of the state space, which improves the sampling efficiency. With numerical examples we demonstrate that the proposed method is significantly more efficient than both of the SS and the MMC methods. Moreover, like the standard MMC, the proposed algorithm can reconstruct the complete distribution function of the parameter of interest and thus can provide more information than just the failure probabilities of the systems.

Localized recursive spatial-temporal state quantification method for data assimilation of wildfire spread simulation

Data assimilation is a procedure to improve the state inference by assimilating the real-time observation data into dynamic systems, such as wildfire spread simulation. Various techniques are used for data assimilation, such as sequential Monte Carlo methods, also called particle filters. In the standard sequential Monte Carlo methods, the used number of particles is the same during the entire process and their convergence is not measured or characterized for runtime state estimation. Therefore, to guarantee the convergence, an abundance of particles is required so that they can be widely distributed in the state space to converge to the true posterior of the system state. In the application of wildfire spread simulation, the spatial states are large and the system behaves in a heterogeneous manner in different fire areas. The heterogeneous feature and the spatially and temporally dynamic behavior of the wildfire spread simulation cause the state inference uncertainty to be dynamically changed. In this paper, we propose the localized recursive spatial-temporal state quantification method to measure the convergence of particles at runtime and apply various approaches to improve the state inference. To show its effectiveness, we apply two different algorithms – the adaptively perturbing the localized state space algorithm and the adaptive particle filtering algorithm – to the localized recursive spatial-temporal state quantification method to improve the state estimation and enhance the performance respectively. The designed experiments are used to show the effectiveness of the adaptively perturbing the localized state space algorithm and the adaptive particle filtering algorithm in the improvement of the state estimation and the performance of data assimilation in wildfire spread simulation.

Exponential Enhancement of the Efficiency of Quantum Annealing by Non-Stoquastic Hamiltonians

Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like $XX$ and $XXXXX$ with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.

Random Features for Shift-Invariant Kernels with Moment Matching

In order to grapple with the conundrum in the scalability of kernel-based learning algorithms, the method of approximating nonlinear kernels via random feature maps has attracted wide attention in large-scale learning systems. Specifically, the associated sampling procedure is one critical component that dictates the quality of random feature maps. However, for high-dimensional features, the standard Monte Carlo sampling method has been shown to be less effective in producing low-variance random samples. In consequence, it demands constructing a large number of features to attain the desired accuracy for downstream use. In this paper, we present a novel sampling algorithm powered by moment matching techniques to reduce the variance of random features. Our extensive empirical studies and comparisons with several highly competitive peer methods verify the superiority of the proposed algorithm in Gram matrix approximation and generalization errors in regression. Our rigorous theoretical proofs justify that the proposed algorithm is guaranteed achieving lower variance than the standard Monte Carlo method in high dimensional settings.

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of derivative information amounts to a correction to the standard Monte Carlo estimation procedure that reduces the variance under certain conditions. We express the finite element models in variational form using the high-level Unified Form Language (UFL). We derive the tangent linear model automatically from this high-level description and use it to efficiently calculate the required derivative information. To study the effectiveness of the derivative-driven method we consider two stochastic PDEs; a one-dimensional Burgers equation with stochastic viscosity and a three-dimensional geometrically non-linear Mooney–Rivlin hyperelastic equation with stochastic density and volumetric material parameter. Our results show that for these problems the first-order derivative-driven Monte Carlo method is around one order of magnitude faster than the standard Monte Carlo method and at the cost of only one extra tangent linear solution per estimation problem. We find similar trends when comparing with a modern non-intrusive multi-level polynomial chaos expansion method. We parallelise the task of the repeated forward model evaluations across a cluster using the ipyparallel and mpi4py software tools. A complete working example showing the solution of the stochastic viscous Burgers equation is included as supplementary material.

Bootstrap Based Uncertainty Propagation for Data Quality Estimation in Crowdsensing Systems

The diffusion of mobile devices equipped with sensing, computation, and communication capabilities is opening unprecedented possibilities for high-resolution, spatio-temporal mapping of several phenomena. This novel data generation, collection, and processing paradigm, termed crowdsensing, lays upon complex, distributed cyberphysical systems. Collective data gathering from heterogeneous, spatially distributed devices inherently raises the question of how to manage different quality levels of contributed data. In order to extract meaningful information, it is, therefore, desirable to the introduction of effective methods for evaluating the quality of data. In this paper, we propose an approach aimed at systematic accuracy estimation of quantities provided by end-user devices of a crowd-based sensing system. This is obtained thanks to the combination of statistical bootstrap with uncertainty propagation techniques, leading to a consistent and technically sound methodology. Uncertainty propagation provides a formal framework for combining uncertainties, resulting from different quantities influencing a given measurement activity. Statistical bootstrap enables the characterization of the sampling distribution of a given statistics without any prior assumption on the type of statistical distributions behind the data generation process. The proposed approach is evaluated on synthetic benchmarks and on a real world case study. Cross-validation experiments show that confidence intervals computed by means of the presented technique show a maximum 1.5% variation with respect to interval widths computed by means of controlled standard Monte Carlo methods, under a wide range of operating conditions. In general, experimental results confirm the suitability and validity of the introduced methodology.

A Cholesky-Based SGM-MLFMM for Stochastic Full-Wave Problems Described by Correlated Random Variables

In this letter, the multilevel fast multipole method (MLFMM) is combined with the polynomial chaos expansion (PCE)-based stochastic Galerkin method (SGM) to stochastically model scatterers with geometrical variations that need to be described by a set of correlated random variables (RVs). It is demonstrated how Cholesky decomposition is the appropriate choice for the RVs transformation, leading to an efficient SGM-MLFMM algorithm. The novel method is applied to the uncertainty quantification of the currents induced on a rough surface, being a classic example of a scatterer described by means of correlated RVs, and the results clearly demonstrate its superiority compared to the nonintrusive PCE methods and to the standard Monte Carlo method.

A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

In this paper we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomial chaos expansions (PCE) to quantify uncertainty in models subject to uncertain input parameters. The standard sampling approach for recovering sparse polynomials is to use Monte Carlo (MC) sampling of the density of orthogonality. However MC methods result in poor function recovery when the polynomial degree is high. Here we propose a general algorithm that can be applied to any admissible weight function on a bounded domain and a wide class of exponential weight functions defined on unbounded domains. Our proposed algorithm samples with respect to the weighted equilibrium measure of the parametric domain, and subsequently solves a preconditioned $\ell^1$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. Numerical examples are also provided that demonstrate that our proposed Christoffel Sparse Approximation algorithm leads to comparable or improved accuracy even when compared with Legendre and Hermite specific algorithms.

A Christoffel function weighted least squares algorithm for collocation approximations

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial Chaos approximation in uncertainty quantification where a polynomial approximation is formed from a combination of orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density of orthogonality. Our proposed algorithm samples with respect to the equilibrium measure of the parametric domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

Numerical inversion of a characteristic function: An alternative tool to form the probability distribution of output quantity in linear measurement models

<p><span>Measurement uncertainty analysis based on combining the state-of-knowledge distributions requires evaluation of the probability density function (PDF), the cumulative distribution function (CDF), and/or the quantile function (QF) of a random variable reasonably associated with the measurand. This can be derived from the characteristic function (CF), which is defined as a Fourier transform of its probability distribution function. Working with CFs provides an alternative and frequently much simpler route than working directly with PDFs and/or CDFs. In particular, derivation of the CF of a weighted sum of independent random variables is a simple and trivial task. However, the analytical derivation of the PDF and/or CDF by using the inverse Fourier transform is available only in special cases. Thus, in most practical situations, a numerical derivation of the PDF/CDF from the CF is an indispensable tool. In metrological applications, such approach can be used to form the probability distribution for the output quantity of a measurement model of additive, linear or generalized linear form. In this paper we propose new original algorithmic implementations of methods for numerical inversion of the characteristic function which are especially suitable for typical metrological applications. The suggested numerical approaches are based on the Gil-Pelaez inverse formulae and on using the approximation by discrete Fourier transform and the fast Fourier transform (FFT) algorithm for computing PDF/CDF of the univariate continuous random variables. As illustrated here, for typical metrological applications based on linear measurement models the suggested methods are an efficient alternative to the standard Monte Carlo methods.</span></p>

Valuation of barrier options using sequential Monte Carlo

Sequential Monte Carlo (SMC) methods have successfully been used in many applications in engineering, statistics and physics. However, these are seldom used in financial option pricing literature and practice. This paper presents SMC method for pricing barrier options with continuous and discrete monitoring of the barrier condition. Under the SMC method, simulated asset values rejected due to barrier condition are re-sampled from asset samples that do not breach the barrier condition improving the efficiency of the option price estimator; while under the standard Monte Carlo many simulated asset paths can be rejected by the barrier condition making it harder to estimate option price accurately. We compare SMC with the standard Monte Carlo method and demonstrate that the extra effort to implement SMC when compared with the standard Monte Carlo is very little while improvement in price estimate can be significant. Both methods result in unbiased estimators for the price converging to the true value as $1/\sqrt{M}$, where $M$ is the number of simulations (asset paths). However, the variance of SMC estimator is smaller and does not grow with the number of time steps when compared to the standard Monte Carlo. In this paper we demonstrate that SMC can successfully be used for pricing barrier options. SMC can also be used for pricing other exotic options and also for cases with many underlying assets and additional stochastic factors such as stochastic volatility; we provide general formulas and references.

On bayesian inference for almost periodic in mean autoregressive models

The goal of the paper is to discuss Bayesian estimation of a class of univariate time-series models being able to represent complicated patterns of “cyclical” fluctuations in mean function. We highlight problems that arise in Bayesian estimation of parametric time-series model using the Flexible Fourier Form of Gallant (1981). We demonstrate that the resulting posterior is likely to be highly multimodal, therefore standard Markov Chain Monte Carlo (MCMC in short) methods might fail to explore the whole posterior, especially when the modes are separated. We show that the multimodality is actually an issue using the exact solution (i.e. an analytical marginal posterior) in an approximate model. We address that problem using two essential steps. Firstly, we integrate the posterior with respect to amplitude parameters, which can be carried out analytically. Secondly, we propose a non-parametrically motivated proposal for the frequency parameters. This allows for construction of an improved MCMC sampler that effectively explores the space of all the model parameters, with the amplitudes sampled by the direct approach outside the MCMC chain. We illustrate the problem using simulations and demonstrate our solution using two real-data examples.