Fast Monte Carlo Method Research Articles

Fast Resampling and Monte Carlo Methods in Python

Fast Resampling and Monte Carlo Methods in Python

DERGMs: Degeneracy-restricted exponential family random graph models

AbstractExponential random graph models, or ERGMs, are a flexible and general class of models for modeling dependent data. While the early literature has shown them to be powerful in capturing many network features of interest, recent work highlights difficulties related to the models’ ill behavior, such as most of the probability mass being concentrated on a very small subset of the parameter space. This behavior limits both the applicability of an ERGM as a model for real data and inference and parameter estimation via the usual Markov chain Monte Carlo algorithms. To address this problem, we propose a new exponential family of models for random graphs that build on the standard ERGM framework. Specifically, we solve the problem of computational intractability and “degenerate” model behavior by an interpretable support restriction. We introduce a new parameter based on the graph-theoretic notion of degeneracy, a measure of sparsity whose value is commonly low in real-world networks. The new model family is supported on the sample space of graphs with bounded degeneracy and is called degeneracy-restricted ERGMs, or DERGMs for short. Since DERGMs generalize ERGMs—the latter is obtained from the former by setting the degeneracy parameter to be maximal—they inherit good theoretical properties, while at the same time place their mass more uniformly over realistic graphs. The support restriction allows the use of new (and fast) Monte Carlo methods for inference, thus making the models scalable and computationally tractable. We study various theoretical properties of DERGMs and illustrate how the support restriction improves the model behavior. We also present a fast Monte Carlo algorithm for parameter estimation that avoids many issues faced by Markov Chain Monte Carlo algorithms used for inference in ERGMs.

Bayesian seismic inversion: a fast sampling Langevin dynamics Markov chain Monte Carlo method

SUMMARY In this study, we aim to solve the seismic inversion in the Bayesian framework by generating samples from the posterior distribution. This distribution incorporates the uncertainties in the seismic data, forward model, and prior information about the subsurface model parameters; thus, we obtain more information through sampling than through a point estimate (e.g. maximum a posteriori method). Based on the numerical cost of solving the forward problem and the dimensions of the subsurface model parameters and observed data, sampling with Markov chain Monte Carlo (MCMC) algorithms can be prohibitively expensive. Herein, we consider the promising Langevin dynamics MCMC algorithm. However, this algorithm has two central challenges: (1) the step size requires prior tuning to achieve optimal performance and (2) the Metropolis–Hastings acceptance step is computationally demanding. We approach these challenges by proposing an adaptive step-size rule and considering the suppression of the Metropolis–Hastings acceptance step. We highlight the proposed method’s potential through several numerical examples and rigorously validate it via qualitative and quantitative evaluation of the sample quality based on the kernelized Stein discrepancy (KSD) and other MCMC diagnostics such as trace and autocorrelation function plots. We conclude that, by suppressing the Metropolis–Hastings step, the proposed method provides fast sampling at efficient computational costs for large-scale seismic Bayesian inference; however, this inflates the second statistical moment (variance) due to asymptotic bias. Nevertheless, the proposed method reliably recovers important aspects of the posterior, including means, variances, skewness and 1-D and 2-D marginals. With larger computational budget, exact MCMC methods (i.e. with a Metropolis–Hastings step) should be favoured. The results thus obtained can be considered a feasibility study for promoting the approximate Langevin dynamics MCMC method for Bayesian seismic inversion on limited computational resources.

On intersection volumes of confidence hyper-ellipsoids and two geometric Monte Carlo methods

Abstract The intersection or the overlap region of two n-dimensional ellipsoids plays an important role in statistical decision making in a number of applications. For instance, the intersection volume of two n-dimensional ellipsoids has been employed to define dissimilarity measures in time series clustering (see [M. Bakoben, T. Bellotti and N. M. Adams, Improving clustering performance by incorporating uncertainty, Pattern Recognit. Lett. 77 2016, 28–34]). Formulas for the intersection volumes of two n-dimensional ellipsoids are not known. In this article, we first derive exact formulas to determine the intersection volume of two hyper-ellipsoids satisfying a certain condition. Then we adapt and extend two geometric type Monte Carlo methods that in principle allow us to compute the intersection volume of any two generalized convex hyper-ellipsoids. Using the exact formulas, we evaluate the performance of the two Monte Carlo methods. Our numerical experiments show that sufficiently accurate estimates can be obtained for a reasonably wide range of n, and that the sample-mean method is more efficient. Finally, we develop an elementary fast Monte Carlo method to determine, with high probability, if two n-ellipsoids are separated or overlap.

A fast Monte-Carlo method to predict failure probability of offshore wind turbine caused by stochastic variations in soil

As spatial seabed soil variations are stochastic and considerable, it is important to predict the changes in the dynamic behavior of offshore structures for safety. Particularly, if the natural frequency of a structure is significantly changed by the soil variations, resonance can cause structural failure; hereafter, this probability is referred to as failure probability. Although several models have been proposed, they require a long computational time to predict failure probabilities. To overcome this limitation, this work proposes a fast method for estimating the failure probability. First, since a very large computation memory is required to construct a finite element model for stochastic soil, block-wise matrix calculations and a dynamic memory allocation technique were adopted. Second, conventional models require a large number of soil samples to calculate the failure probability, which results in heavy computations. A newly developed fast Monte-Carlo (FMC) method is 5.8 times faster than the conventional method with high accuracy (99.41%). This method was applied to offshore wind turbines and successfully predicted various structural characteristics. A noteworthy prediction is that slender and long foundation is more robust to stochastic soil variations than thick and short foundation. The FMC method can be used for preliminary design of offshore structures.

A fast particle-based approach for calibrating a 3-D model of the Antarctic ice sheet

We consider the scientifically challenging and policy-relevant task of understanding the past and projecting the future dynamics of the Antarctic ice sheet. The Antarctic ice sheet has shown a highly nonlinear threshold response to past climate forcings. Triggering such a threshold response through anthropogenic greenhouse gas emissions would drive drastic and potentially fast sea level rise with important implications for coastal flood risks. Previous studies have combined information from ice sheet models and observations to calibrate model parameters. These studies have broken important new ground but have either adopted simple ice sheet models or have limited the number of parameters to allow for the use of more complex models. These limitations are largely due to the computational challenges posed by calibration as models become more computationally intensive or when the number of parameters increases. Here, we propose a method to alleviate this problem: a fast sequential Monte Carlo method that takes advantage of the massive parallelization afforded by modern high-performance computing systems. We use simulated examples to demonstrate how our sample-based approach provides accurate approximations to the posterior distributions of the calibrated parameters. The drastic reduction in computational times enables us to provide new insights into important scientific questions, for example, the impact of Pliocene era data and prior parameter information on sea level projections. These studies would be computationally prohibitive with other computational approaches for calibration such as Markov chain Monte Carlo or emulation-based methods. We also find considerable differences in the distributions of sea level projections when we account for a larger number of uncertain parameters. For example, based on the same ice sheet model and data set, the 99th percentile of the Antarctic ice sheet contribution to sea level rise in 2300 increases from 6.5 m to 13.1 m when we increase the number of calibrated parameters from three to 11. With previous calibration methods, it would be challenging to go beyond five parameters. This work provides an important next step toward improving the uncertainty quantification of complex, computationally intensive and decision-relevant models.

Comprehensive analysis of thermoluminescence signals in MgB4O7:Dy,Na dosimeter

Magnesium borate (MgB4O7) is a material of interest for thermoluminescence (TL) dosimetry of ionizing radiation, due to its low photon energy dependence and the possibility of developing neutron dosimeters, and its potential as a temperature sensor in passive temperature sensing applications. Recent experimental and modeling work has shown that TL glow curves and isothermal luminescence signals in MgB4O7:Dy,Na can be analyzed using a radiative tunneling model, based on localized recombination processes. This paper presents a comprehensive analysis of TL signals in MgB4O7:Dy,Na dosimeters, based on a recently published model which incorporates simultaneous irradiation and tunneling processes. Four different types of TL data are presented and analyzed using the proposed model: (a) TL glow curves at different irradiation doses, (b) TL signals after sample irradiation at different temperatures, (c) TL after isothermal heating for 500 s at various temperatures, and (d) TL after room temperature irradiation, followed by heating to a temperature Tstop. We also present an alternative fast and efficient Monte Carlo method of analyzing the data, which is very useful for quick analysis of experimental data. The experimental data and modeling results show that the shape of the TL glow peaks remains essentially the same in all four experimental protocols, while the height and temperature of maximum TL intensity depends on the irradiation and preheating conditions.

Lattice Monte Carlo for quantum Hall states on a torus

Monte Carlo is one of the most useful methods to study the quantum Hall problems. In this paper, we introduce a fast lattice Monte Carlo method based on a mathematically exact reformulation of the torus quantum Hall problems from continuum to lattice. We first apply this new technique to study the Berry phase of transporting composite fermions along different closed paths enclosing or not enclosing the Fermi surface center in the half filled Landau level problem. The Monte Carlo result agrees with the phase structure we found on small systems and confirms it on much larger sizes. Several other quantities including the Coulomb energy in different Landau levels, structure factor, particle-hole symmetry are computed and discussed for various model states. In the end, based on certain knowledge of structure factor, we introduce a algorithm by which the lattice Monte Carlo efficiency is further boosted by several orders.

A Fast Two-Phase Monte Carlo Method for Constructing Polar Codes With Arbitrary Binary Kernel

We propose a two-phase Monte Carlo (TPMC) method to accelerate the original Monte Carlo (MC) method for constructing polar codes with high-dimensional kernels. In the TPMC method, some of the most reliable and unreliable bits are obtained by Gaussian approximate-density evolution (GA-DE) method in the first phase; in the second phase, these most reliable and unreliable bits are viewed as frozen bits. Then, the MC method are used to evaluate the remaining bits and select some best bits from the remaining bits. Finally, these best bits and the most reliable bits are combined as the information bits of the constructed polar code. By our investigation, most bits can be fixed as frozen bits in the second phase without error performance loss of the constructed polar codes. Because computation of frozen bits can be saved and computation of the GA-DE method can be ignored in contrast to that of the MC method, the TPMC method substantially reduces the complexity of the MC method. Simulation results show that 1) For a G ⊗3 15 polar code with block length 3375 and code rate 1/2, the TPMC method can fix 3200 bits as frozen bits in the second phase without error performance loss, which reduces the computation cost by approximately 92.6% over the MC method; 2) Polar codes with high-dimensional kernels constructed by the TPMC method outperforms polar codes with the G 2 kernel constructed by the Tal-Vardy method in terms of error performance; 3) With the same computational cost, the TPMC method can construct better polar codes than the MC method.

Density overwrites of internal tumor volumes in intensity modulated proton therapy plans for mobile lung tumors

The purpose of this study was to investigate internal tumor volume density overwrite strategies to minimize intensity modulated proton therapy (IMPT) plan degradation of mobile lung tumors.Four planning paradigms were compared for nine lung cancer patients. Internal gross tumor volume (IGTV) and internal clinical target volume (ICTV) structures were defined encompassing their respective volumes in every 4DCT phase. The paradigms use different planning CT (pCT) created from the average intensity projection (AIP) of the 4DCT, overwriting the density within the IGTV to account for movement. The density overwrites were: (a) constant filling with 100 HU (C100) or (b) 50 HU (C50), (c) maximum intensity projection (MIP) across phases, and (d) water equivalent path length (WEPL) consideration from beam’s-eye-view. Plans were created optimizing dose-influence matrices calculated with fast GPU Monte Carlo (MC) simulations in each pCT. Plans were evaluated with MC on the 4DCTs using a model of the beam delivery time structure. Dose accumulation was performed using deformable image registration. Interplay effect was addressed applying 10 times rescanning.Significantly less DVH metrics degradation occurred when using MIP and WEPL approaches. Target coverage ( Gy(RBE)) was fulfilled in most cases with MIP and WEPL ( Gy (RBE)), keeping dose heterogeneity low ( Gy(RBE)). The mean lung dose was kept lowest by the WEPL strategy, as well as the maximum dose to organs at risk (OARs). The impact on dose levels in the heart, spinal cord and esophagus were patient specific.Overall, the WEPL strategy gives the best performance and should be preferred when using a 3D static geometry for lung cancer IMPT treatment planning. Newly available fast MC methods make it possible to handle long simulations based on 4D data sets to perform studies with high accuracy and efficiency, even prior to individual treatment planning.

Sink strength simulations using the Monte Carlo method: Applied to spherical traps

The sink strength is an important parameter for the mean-field rate equations to simulate temporal changes in the micro-structure of materials. However, there are noteworthy discrepancies between sink strengths obtained by the Monte Carlo and analytical methods. In this study, we show the reasons for these differences. We present the equations to estimate the statistical error for sink strength calculations and show the way to determine the sink strengths for multiple traps.We develop a novel, very fast Monte Carlo method to obtain sink strengths. The results show that, in addition to the well-known sink strength dependence of the trap concentration, trap radius and the total sink strength, the sink strength also depends on the defect diffusion jump length and the total trap volume fraction. Taking these factors into account, allows us to obtain a very accurate analytic expression for the sink strength of spherical traps.

Efficient SRAM yield optimization with mixture surrogate modeling

Largely repeated cells such as SRAM cells usually require extremely low failure-rate to ensure a moderate chi yield. Though fast Monte Carlo methods such as importance sampling and its variants can be used for yield estimation, they are still very expensive if one needs to perform optimization based on such estimations. Typically the process of yield calculation requires a lot of SPICE simulation. The circuit SPICE simulation analysis accounted for the largest proportion of time in the process yield calculation. In the paper, a new method is proposed to address this issue. The key idea is to establish an efficient mixture surrogate model. The surrogate model is based on the design variables and process variables. This model construction method is based on the SPICE simulation to get a certain amount of sample points, these points are trained for mixture surrogate model by the lasso algorithm. Experimental results show that the proposed model is able to calculate accurate yield successfully and it brings significant speed ups to the calculation of failure rate. Based on the model, we made a further accelerated algorithm to further enhance the speed of the yield calculation. It is suitable for high-dimensional process variables and multi-performance applications.

Predictions on dynamic evolution of compositional mixing degree in two-component aggregation

The compositional distribution in two-component aggregative mixing of initially bidisperse particle populations can be described by a Guassian-type function, which is determined by the mixing degree χ (assessed quantitatively by the mass-normalized power density of excess component A), and the overall mass fraction ϕ (a known value from the initial feeding condition) of component A. It is known that χ will reach a steady-state value χ∞ over time (factually, after attaining the self-preserving size distribution), and χ∞ is only relevant to ϕ, namely the feeding condition. However, the dynamic evolution of χ before the attainment of a steady-state value is not exactly known. In this paper, the fast differentially-weighted Monte Carlo method for population balance modeling was used to predict the dependence of time-varied χ on initial feeding conditions through hundreds of systematically varied simulations. It is found that χ is subject to an exponential decay, largely depending on the ratio of steady-state mixing degree and its initial value (χ∞/χ0). With the explored exponential formulas for the dynamic mixing degree, it is possible to attain an optimum control on the compositional distributions during two-component aggregation processes through selecting the initial feeding parameters, and the time needed for reaching a steady-state is investigated.

Variational Approximations for Selecting Hierarchical Models of Circular Data in a Small Area Estimation Application

Abstract We consider hierarchical regression models for circular data using the projected normal distribution, applied in the development of weights for the Access Point Angler Intercept Survey, a recreational angling survey conducted by the US National Marine Fisheries Service. Weighted estimates of recreational fish catch are used in stock assessments and fisheries regulation. The construction of the survey weights requires the distribution of daily departure times of anglers from fishing sites, within spatio-temporal domains subdivided by the mode of fishing. Because many of these domains have small sample sizes, small area estimation methods are developed. Bayesian inference for the circular distributions on the 24-hour clock is conducted, based on a large set of observed daily departure times from another National Marine Fisheries Service study, the Coastal Household Telephone Survey. A novel variational/Laplace approximation to the posterior distribution allows fast comparison of a large number of models in this context, by dramatically speeding up computations relative to the fast Markov Chain Monte Carlo method while giving virtually identical results.

Variational approximations for selecting hierarchical models of circular data in a small area estimation application

We consider hierarchical regression models for circular data using the projected normal distribution, applied in the development of weights for the Access Point Angler Intercept Survey, a recreational angling survey conducted by the US National Marine Fisheries Service. Weighted estimates of recreational fish catch are used in stock assessments and fisheries regulation. The construction of the survey weights requires the distribution of daily departure times of anglers from fishing sites, within spatio-temporal domains subdivided by the mode of fishing. Because many of these domains have small sample sizes, small area estimation methods are developed. Bayesian inference for the circular distributions on the 24-hour clock is conducted, based on a large set of observed daily departure times from another National Marine Fisheries Service study, the Coastal Household Telephone Survey. A novel variational/Laplace approximation to the posterior distribution allows fast comparison of a large number of models in this context, by dramatically speeding up computations relative to the fast Markov Chain Monte Carlo method while giving virtually identical results.

Seismic fluid identification using a nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo method

Elastic impedance inversion with high efficiency and high stability has become one of the main directions of seismic pre-stack inversion. The nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo (MCMC) method is proposed in this paper, combining conventional MCMC method based on global optimization with a preconditioned conjugate gradient (PCG) algorithm based on local optimization, so this method does not depend strongly on the initial model. It converges to the global optimum quickly and efficiently on the condition that efficiency and stability of inversion are both taken into consideration at the same time. The test data verify the feasibility and robustness of the method, and based on this method, we extract the effective pore-fluid bulk modulus, which is applied to reservoir fluid identification and detection, and consequently, a better result has been achieved.

An Efficient SRAM Yield Analysis and Optimization Method With Adaptive Online Surrogate Modeling

SRAM cells usually require extremely low failure rate or equivalently extremely high production yield, making it impractical to perform yield analysis using Monte Carlo (MC) method as huge amount of samples are needed. Fast MC methods, e.g., importance sampling methods, are still too expensive as the anticipated failure rate is very low. In this paper, a new SRAM yield analysis method is proposed to tackle this issue. The key idea is to improve traditional importance sampling method with an efficient online surrogate model. Experimental results show that the proposed yield analysis method achieves $5\times $ – $22\times $ speedup over existing state-of-the-art techniques without sacrificing estimation accuracy. Sigma distribution and schmoo plot can be quickly generated by the proposed method, which is very useful for realistic applications. Based on the proposed yield analysis method, an efficient yield optimization method has been developed to further automate the SRAM cell design procedure where process variations can be fully considered. Experimental results show that a fully automatic yield optimization for SRAM cells can be done within only a few hours.

MO‐F‐CAMPUS‐I‐03: GPU Accelerated Monte Carlo Technique for Fast Concurrent Image and Dose Simulation

Purpose:To develop an accurate and fast Monte Carlo (MC) method of simulating CT that is capable of correlating dose with image quality using voxelized phantoms.Methods:A realistic voxelized phantom based on patient CT data, XCAT, was used with a GPU accelerated MC code for helical MDCT. Simulations were done with both uniform density organs and with textured organs. The organ doses were validated using previous experimentally validated simulations of the same phantom under the same conditions. Images acquired by tracking photons through the phantom with MC require lengthy computation times due to the large number of photon histories necessary for accurate representation of noise. A substantial speed up of the process was attained by using a low number of photon histories with kernel denoising of the projections from the scattered photons. These FBP reconstructed images were validated against those that were acquired in simulations using many photon histories by ensuring a minimal normalized root mean square error.Results:Organ doses simulated in the XCAT phantom are within 10% of the reference values. Corresponding images attained using projection kernel smoothing were attained with 3 orders of magnitude less computation time compared to a reference simulation using many photon histories.Conclusion:Combining GPU acceleration with kernel denoising of scattered photon projections in MC simulations allows organ dose and corresponding image quality to be attained with reasonable accuracy and substantially reduced computation time than is possible with standard simulation approaches.

Enhanced-Precision LHSMC of Electrical Circuit Considering Low Discrepancy

The Monte-Carlo (MC) technique is very efficient solution for statistical problem. Various MC methods can easily be applied to statistical circuit performance analysis. Recently, as the number of process parameters and their impact, has increasingly affected circuit performance, a sufficient sample size is required in order to consider high dimensionality, profound nonlinearity, and stringent accuracy requirements. Also, it is important to identify the performance of circuit as soon as possible. In this paper, Fast MC method is proposed for efficient analysis of circuit performance. The proposed method analyzes performance using enhanced-precision Latin Hypercube Sampling Monte Carlo (LHSMC). To increase the accuracy of the analysis, we calculate the effective dimension for the low discrepancy value on critical parameters. This will guarantee a robust input vector for the critical parameters. Using a 90㎚ process parameter and OP-AMP, we verified the accuracy and reliability of the proposed method in comparison with the standard MC, LHS and Quasi Monte Carlo (QMC).

Fast Monte Carlo Method in Stochastic Modelling of Charged Particle Multiplication

Fast Monte Carlo Method in Stochastic Modelling of Charged Particle Multiplication