Standard Monte Carlo Method Research Articles

Feasibility of the analytical dose calculation method for Au-198 brachytherapy

PurposeA dose calculation algorithm Computed Tomography (CT)-based analytical dose calculation method (CTanly), which can correct for subject inhomogeneity and size-dependent scatter doses, was applied to the 198Au seed. In this study, we evaluated the effectiveness of the CTanly method by comparing the gold standard Monte Carlo (MC) method and the conventional TG43 method on two virtual phantoms and patient CT images simulating oral cancer. MethodsAs virtual phantoms, a water phantom and a heterogeneous phantom with soft tissue inserted cubic fat, lung, and bone were used. A 2-mm-thick lead plate was also inserted into the heterogeneous phantom as a dose attenuator. Virtual 198Au seeds and a 2-mm-thick lead plate were placed on the patient CT images. Dose distributions obtained via the TG43 and CTanly methods were compared with those of the MC by gamma analysis with 2%/2-mm thresholds. The computation durations were also compared. ResultsIn the water phantom, dose distributions comparable to those obtained via the MC method were obtained regardless of the algorithm. For the inhomogeneity phantom and patient case, the CTanly method showed an improvement in the gamma passing rate and dose distributions similar to those of the MC method were obtained. The computation time, which was days with the MC method, was reduced to minutes with the CTanly method. ConclusionsThe CTanly method is effective for 198Au seed dose calculations and takes a shorter time to obtain the dose distributions than the MC method.

A multilevel Monte Carlo algorithm for stochastic differential equations driven by countably dimensional Wiener process and Poisson random measure

In this paper, we investigate properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which depends on two parameters i.e., grid density n∈N and truncation dimension parameter M∈N, is of the order n−1/2+δ(M) such that δ(⋅) is positive and decreasing to 0. We derive a complexity model and provide proof for the complexity upper bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both n and M. The complexity is measured in terms of upper bound for mean-squared error and is compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation details are also reported.

Enhancing standard CFD based practices for site assessment through polynomial surrogates for estimating uncertainty in wind speed

One of the challenges in digitizing the wind energy sector is to have reliable, accurate and efficient data assessment and prediction tools. We believe that accurate surrogates are key towards improving efficiency and reliability for informed decision making. For a quantitative understanding of wind and energy in complex terrains, a detailed site assessment is done, most often using mesoscale and/or microscale computational fluid dynamics (CFD) based approach. Typically, one would require multiple scenarios/case for a CFD simulator to incorporate the influence of a parameter on variability of wind speed and energy production. These simulations are computationally expensive, and the industry is constantly on a look out for more efficient methods. Towards this end, we propose using a polynomial surrogate based on Polynomial Chaos Expansion (PCE) for uncertainty quantification that requires only a few CFD simulations to assess the variability of wind speed with respect to variability in an input parameter. The variability estimates from PCE were first validated against the standard Monte Carlo method on simple benchmarks and then tested for complex terrains. In this work, we present results that help us understand how canopy foliage, particularly its height, impacts the variability in wind speed in complex terrains. Since the polynomial surrogate relies only on a limited set of input variations (forest height) to estimate the variability distributions for wind speed locally (for specified points of interests), it has proven to be a low-cost and accurate method for estimating uncertainty. In this paper, we discuss the results for wind speed variability for five sites with varying complexity using the polynomial surrogate model. Furthermore, the singular value decomposition (SVD) entropy value is used to quantify terrain complexity for the sites considered. We show how the variability in wind speed can potentially be correlated to terrain complexity. This is our first attempt in providing an empirically (including multiple sites) derived relationship that can be used as a basis for fast and reliable variability estimator for wind resource planners.

Premerger Sky Localization of Gravitational Waves from Binary Neutron Star Mergers Using Deep Learning

The simultaneous observation of gravitational waves (GW) and prompt electromagnetic counterparts from the merger of two neutron stars can help reveal the properties of extreme matter and gravity during and immediately after the final plunge. Rapid sky localization of these sources is crucial to facilitate such multimessenger observations. As GWs from binary neutron star (BNS) mergers can spend up to 10–15 minutes in the frequency bands of the detectors at design sensitivity, early-warning alerts and premerger sky localization can be achieved for sufficiently bright sources, as demonstrated in recent studies. In this work, we present premerger BNS sky localization results using GW-SkyLocator, a deep-learning model capable of inferring sky location posterior distributions of GW sources at orders of magnitude faster speeds than standard Markov Chain Monte Carlo methods. We test our model’s performance on a catalog of simulated injections from Sachdev, recovered at 0–60 s before the merger, and obtain comparable sky localization areas to the rapid localization tool BAYESTAR. These results show the feasibility of our model for premerger sky localization and the possibility of follow-up observations for precursor emissions from BNS mergers.

A predictor–corrector Monte Carlo method for thermal radiative transfer equations

In this paper, we develop a predictor–corrector Monte Carlo method for solving the thermal radiative transfer equations. A two-step numerical method is employed when the material temperature violates the maximum principle. It utilizes the effective scattering technique in the standard implicit Monte Carlo (IMC) method in each step. Then a linear combination of the prediction and correction solutions is obtained as the final solution, where the coefficient is determined by solving an optimization problem. The new method is more implicit than the IMC method in that it uses a more implicit estimate for the material temperature dependent quantities. Moreover, we note that the last modification step is necessary because using the iterative solution directly may overestimate the material temperature when large time steps are used. Numerical simulations show that the new method can significantly mitigate the violation of maximum principle in comparison with the standard IMC method, and results in more accurate solutions than the original predictor–corrector method when the time step is large.

Monte Carlo convergence rates for kth moments in Banach spaces

We formulate standard and multilevel Monte Carlo methods for the kth moment Mεk[ξ] of a Banach space valued random variable ξ:Ω→E, interpreted as an element of the k-fold injective tensor product space ⊗εkE. For the standard Monte Carlo estimator of Mεk[ξ], we prove the k-independent convergence rate 1−1p in the Lq(Ω;⊗εkE)-norm, provided that (i) ξ∈Lkq(Ω;E) and (ii) q∈[p,∞), where p∈[1,2] is the Rademacher type of E. By using the fact that Rademacher averages are dominated by Gaussian sums combined with a version of Slepian's inequality for Gaussian processes due to Fernique, we moreover derive corresponding results for multilevel Monte Carlo methods, including a rigorous error estimate in the Lq(Ω;⊗εkE)-norm and the optimization of the computational cost for a given accuracy. Whenever the type of the Banach space E is p=2, our findings coincide with known results for Hilbert space valued random variables.We illustrate the abstract results by three model problems: second-order elliptic PDEs with random forcing or random coefficient, and stochastic evolution equations. In these cases, the solution processes naturally take values in non-Hilbertian Banach spaces. Further applications, where physical modeling constraints impose a setting in Banach spaces of type p<2, are indicated.

Randomized vector iterative linear solvers of high precision for large dense system

Abstract In this paper we suggest randomized linear solvers with a focus on refinement issue to achieve a high precision while maintaining all the advantages of the Monte Carlo method for solving systems of large dimension with dense matrices. It is shown that each iterative refinement step reduces the error by one order of magnitude. The crucial point of the suggested method is, in contrast to the standard Monte Carlo method, that the randomized vector algorithm computes the entire solution column at once, rather than a single component. This makes it possible to efficiently construct the iterative refinement method. We apply the developed method for solving a system of elasticity equations.

Influence of Random Errors in Element Positions on Performance of Antenna Arrays Considering Mutual Coupling Effect

In engineering, random errors in radiating element positions caused by the manufacturing and assembly process could lead to degradation of the radiation performance of an antenna array. In order to accurately evaluate the deterioration of the antenna performance in the design phase, a statistical analysis method considering the mutual coupling effect between array elements is developed herein. The mutual coupling effect is described in terms of modal coupling and is governed by the modal coupling equilibrium equation. The statistical characteristics such as the mean and variance of the gain pattern are obtained based on the second-order Taylor expansion, and the expansion coefficients are obtained by the adjoint method. The correctness and effectiveness of the proposed method are verified through a comparison with the standard Monte Carlo method.

Reliability analysis of the stress intensity factor using multilevel Monte Carlo methods

This paper presents a reliability analysis for fracture toughness using a multilevel refinement on a hierarchy of computational models. A 2D finite element model discretized by quadrilateral elements is developed to analyze the stress intensity with the presence of an initial edge crack. The multilevel simulations are obtained considering a non-uniform sequence of mesh refinement in the vicinity of the crack tip. We set the probabilistic problem accounting for applied stress and crack size uncertainties. We analyze several error tolerances using the standard and multilevel Monte Carlo methods combined with the selective refinement procedure. The probability of failure is estimated by expanding it in a telescoping sum of an initial approximation at the coarsest mesh and a series of incremental corrections between the subsequent levels. In our analysis, we take on two common fracture problems; a single-edge notched tension to investigate the pure mode-I and an asymmetric four-points bending to consider the mixed mode-I/II. The results show significant savings in the computation cost.

Density-extrapolation Global Variance Reduction (DeGVR) method for large-scale radiation field calculation

The lack of a universal and effective Global Variance Reduction (GVR) method for the deep-penetration Monte Carlo calculation makes it difficult or impossible to calculate the radiation field of large nuclear facilities. We propose a Density-extrapolation Global Variance Reduction (DeGVR) method for the large-scale radiation field calculation. After reducing the density of all materials to avoid the deep penetration problem, two global flux distributions of the two different material densities are obtained with two fixed-source calculations, and then the global flux distribution of the original material density is obtained by extrapolating the two global flux distributions. The global flux distribution of the original material density is then used for global variance reduction. In a large-scale model with a flux attenuation of 1080, the DeGVR method only takes 40.3 minutes to build the radiation field, improves the Average Figure-of-Merit (AV.FOM) up to 85652 times and the counting rate per time (CRPT) up to 88107 times compared with the standard Monte Carlo method. In addition to the high efficiency, the DeGVR method has high universality and robustness because it constructs the global information in a clever way that does not rely on the complex mathematical derivation and geometric modeling. The DeGVR method shows excellent application potential in large-scale radiation analysis.

Tensor renormalization group study of orientational ordering in simple models of adsorption monolayers.

A simple lattice model of the orientational ordering in organic adsorption layers that considers the directionality of intermolecular interactions is proposed. The symmetry and the number of rotational states of the adsorbed molecule are the main parameters of the model. The model takes into account both the isotropic and directional contributions to the molecule-molecule interaction potential. Using several special cases of this model, we have shown that the tensor renormalization group (TRG) approach can be successfully used for the analysis of orientational ordering in organic adsorption layers with directed intermolecular interactions. Adsorption isotherms, potential energy, and entropy have been calculated for the model adsorption layers differing in the molecule symmetry and the number of rotational states. The calculated thermodynamic characteristics show that entropy effects play a significant role in the self-assembly of dense phases of the molecular layers. All the results obtained with the TRG have been verified by the standard Monte Carlo method. The proposed model reproduces the main features of the phase behavior of the real adsorption layers of benzoic, terephthalic, and trimesic acids on a homogeneous surface of metal single crystals and graphite.

Complex Langevin method on rotating matrix quantum mechanics at thermal equilibrium

Abstract Rotating systems in thermal equilibrium are ubiquitous in our world. In the context of high-energy physics, rotations would affect the phase structure of quantum chromodynamics (QCD). However, the standard Monte Carlo methods in rotating systems are problematic because the chemical potentials for the angular momenta (angular velocities) cause sign problems even for bosonic variables. In this article, we demonstrate that the complex Langevin method (CLM) may overcome this issue. We apply the CLM to the Yang–Mills (YM)-type one-dimensional matrix model (matrix quantum mechanics) that is a large-N reduction (or dimensional reduction) of the (D + 1)-dimensional U(N) pure YM theory [bosonic Banks–Fischler–Shenker–Susskind (BFSS) model]. This model shows a large-N phase transition at finite temperature, which is analogous to the confinement/deconfinement transition of the original YM theory, and our CLM predicts that the transition temperature decreases as the angular momentum chemical potential increases. In order to verify our results, we compute several quantities via the minimum sensitivity method and find good quantitative agreements. Hence, the CLM works properly in this rotating system. We also argue that our results are qualitatively consistent with a holography and the recent studies of the imaginary angular velocity in QCD. As a byproduct, we develop an analytic approximation to treat the so-called “small black hole” phase in the matrix model.

Sampling‐Based Methods for Uncertainty Propagation in Flood Modeling Under Multiple Uncertain Inputs: Finding Out the Most Efficient Choice

AbstractIn probabilistic flood modeling, uncertainty manifests in frequency of occurrence, or histograms, for quantities of interest, including the Flood Extent and hazard rating (HR). Such modeling at the field‐scale requires the identification of a more efficient alternative to the Standard Monte Carlo (SMC) method that can reproduce comparable output probability distributions with a relatively reduced sample size, including detailed histograms of quantities of interest. Latin hypercube sampling (LHS) is the most evaluated alternative for fluvial floods but yields no considerable sample size reduction. Potentially better alternatives include adaptive stratified sampling (ASS), Quasi Monte Carlo (QMC) and Haar‐wavelet expansion (HWE), which are yet unevaluated for probabilistic flood modeling. To fulfill this gap, LHS, ASS, QMC, and HWE are compared to quantify sample size reduction to reproduce output detailed histograms—for Flood Extent, and average and maximum HR—while keeping the difference below 10% to the reference SMC prediction. The comparison is done for two test cases with two (i.e., inflow discharge and Manning's coefficient) and three (i.e., further including the ground elevation) input random variables, and a real case with five input random variables. With two input random variables, all four alternatives yield sample size reductions, with QMC and HWE considerably outperforming the others; with three and more input random variables, HWE becomes inflexible and LHS underperforms. Still, QMC is a better choice than ASS to boost sample size reduction for the real case and shall be preferred in probabilistic flood modeling. Accompanying research codes are openly available online.

DETERMINING THE VALUE OF DOUBLE BARRIER OPTION USING STANDARD MONTE CARLO, ANTITHETIC VARIATE, AND CONTROL VARIATE METHODS

In this paper, we applied the standard Monte Carlo, antithetic variate, and control variates methods to value the double barrier knock-in option price. The underlying asset used in the calculation of double barrier knock-in option is the share of ANTM from April 1, 2019 until March 1, 2022. The value of the double barrier knock-in option is simulated using standard Monte Carlo, antithetic variate, and control variates methods. The results showed that all the methods converge to the exact solution, with the control variate method to be the fastest. Standard Monte Carlo method has the least computational time, followed by control variate and antithetic variate method. Compared to the other methods, control variate is the most effective and efficient in determining the value of double barrier knock-in option, based on the option value, relative error and computational time. Antithetic variate method converges faster to the exact solution compared to standard Monte Carlo. However it has the longest computation time compared to the other methods.

A Monte Carlo method for 3D radiative transfer equations with multifractional singular kernels

We propose in this work a Monte Carlo method for three dimensional scalar radiative transfer equations with non-integrable, space-dependent scattering kernels. Such kernels typically account for long-range statistical features, and arise for instance in the context of wave propagation in turbulent atmosphere, geophysics, and medical imaging in the peaked-forward regime. In contrast to the classical case where the scattering cross section is integrable, which results in a non-zero mean free time, the latter here vanishes. This creates numerical difficulties as standard Monte Carlo methods based on a naive regularization exhibit large jump intensities and an increased computational cost. We propose a method inspired by the finance literature based on a small jumps - large jumps decomposition, allowing us to treat the small jumps efficiently and reduce the computational burden. We demonstrate the performance of the approach with numerical simulations and provide a complete error analysis. The multifractional terminology refers to the fact that the high frequency contribution of the scattering operator is a fractional Laplace-Beltrami operator on the unit sphere with space-dependent index.

Lifestyle, Longevity, and Legacy Risks with Annuities in Retirement Portfolio Decumulation

Investors planning for retirement balance three Ls: (1) lifestyle risk, hoping to maintain a consumption stream that provides a chosen standard of living; (2) longevity risk, hoping to remain solvent throughout their lifetime; and (3) legacy risk, hoping to leave a bequest to their heirs. We solve this multiple objective problem for a wide range of consumption and annuitization scenarios. For each scenario, we apply dynamic programming to optimally evolve the investments in the non-annuitized portion of the portfolio so as to minimize longevity risk. Our dynamic programming approach has the advantages of (1) generating results that are far superior to what standard Monte Carlo methods, static portfolios, and target date fund glide paths can provide and (2) not requiring utility functions, which are hard to specify for individuals. We show that investors who want to minimize their longevity and legacy risk and who are unable to annuitize their full consumption stream are best off avoiding even partial annuitization of their portfolio. For investors who are able to annuitize their full consumption stream, we quantify their longevity versus legacy risk trade-offs, enabling them to select the best annuity for their needs.

Wireless Federated Langevin Monte Carlo: Repurposing Channel Noise for Bayesian Sampling and Privacy

Most works on federated learning (FL) focus on the most common frequentist formulation of learning whereby the goal is minimizing the global empirical loss. Frequentist learning, however, is known to be problematic in the regime of limited data as it fails to quantify epistemic uncertainty in prediction. Bayesian learning provides a principled solution to this problem by shifting the optimization domain to the space of distribution in the model parameters. {\color{black}This paper proposes a novel mechanism for the efficient implementation of Bayesian learning in wireless systems. Specifically, we focus on a standard gradient-based Markov Chain Monte Carlo (MCMC) method, namely Langevin Monte Carlo (LMC), and we introduce a novel protocol, termed Wireless Federated LMC (WFLMC), that is able to repurpose channel noise for the double role of seed randomness for MCMC sampling and of privacy preservation.} To this end, based on the analysis of the Wasserstein distance between sample distribution and global posterior distribution under privacy and power constraints, we introduce a power allocation strategy as the solution of a convex program. The analysis identifies distinct operating regimes in which the performance of the system is power-limited, privacy-limited, or limited by the requirement of MCMC sampling. Both analytical and simulation results demonstrate that, if the channel noise is properly accounted for under suitable conditions, it can be fully repurposed for both MCMC sampling and privacy preservation, obtaining the same performance as in an ideal communication setting that is not subject to privacy constraints.

Bayesian Logistic Regression Model for Sub-Areas

Many population-based surveys have binary responses from a large number of individuals in each household within small areas. One example is the Nepal Living Standards Survey (NLSS II), in which health status binary data (good versus poor) for each individual from sampled households (sub-areas) are available in the sampled wards (small areas). To make an inference for the finite population proportion of individuals in each household, we use the sub-area logistic regression model with reliable auxiliary information. The contribution of this model is twofold. First, we extend an area-level model to a sub-area level model. Second, because there are numerous sub-areas, standard Markov chain Monte Carlo (MCMC) methods to find the joint posterior density are very time-consuming. Therefore, we provide a sampling-based method, the integrated nested normal approximation (INNA), which permits fast computation. Our main goal is to describe this hierarchical Bayesian logistic regression model and to show that the computation is much faster than the exact MCMC method and also reasonably accurate. The performance of our method is studied by using NLSS II data. Our model can borrow strength from both areas and sub-areas to obtain more efficient and precise estimates. The hierarchical structure of our model captures the variation in the binary data reasonably well.

A computational framework for uncertain locally resonant metamaterial structures

Locally resonant metamaterial structures, i.e., structures artificially engineered with periodic arrays of small resonators, are attracting a growing interest among scientists. The main reason lies in their remarkable attenuation properties of elastic waves over finite frequency ranges, named band gaps, resulting from periodicity and local resonance. Yet, very little attention has been paid to investigate whether and to which extent the behavior of these structures may be affected by uncertainties. This is indeed the purpose of our study, which proposes a comprehensive computational framework to calculate the frequency response of uncertain locally resonant structures, assuming an interval model for all the relevant parameters of the resonators: mass, stiffness and damping.The computational framework is conceived for finite-element models of the locally resonant structure and standard mass-spring-dashpot models of the resonators. The key steps are an exact dynamic condensation of the degrees of freedom within the resonators, the derivation of an exact and elegant expression for the transfer matrix of the locally resonant structure via the Sherman-Morrison-Woodbury formula, the calculation of the interval frequency response via either a sensitivity-based method or a global optimization method, the choice being driven by a preliminary monotonicity test. Considering two typical locally resonant structures, a beam and a plate, the computational framework proves easy to implement, accurate and robust as compared with the standard Monte Carlo method.

IMPLEMENTATION OF MONTE CARLO MOMENT MATCHING METHOD FOR PRICING LOOKBACK FLOATING STRIKE OPTION

Monte Carlo method was a numerical method that was popular in finance. This method had disadvantages at convergences, so the moment matching was used to improve the efficiency from Monte Carlo method. The research has discussed about pricing of the lookback floating strike option using the Monte Carlo moment matching method. The monthly stock price of PT TELKOM from 2004 to 2021 that used in this research. The results obtained by adding variance reduction moment matching in Monte Carlo method, which produces a relatively had smaller error when compared to the relative error of the standard Monte Carlo method. The orders of convergence from Monte Carlo method with variance reduction moment matching for call and put option are about 1.1 and 1.4. The conclusion that addition of the moment matching can increase the efficiency of the Monte Carlo method in determining the price of the lookback floating strike option.