Variance Reduction Techniques Research Articles

A correlated sampling-based Monte Carlo simulation for fast CBCT iterative scatter correction.

In recent years, cone-beam computed tomography (CBCT) has played an important role in medical imaging. However, the applications of CBCT are limited due to the severe scatter contamination. Conventional Monte Carlo (MC) simulation can provide accurate scatter estimation for scatter correction, but the expensive computational cost has always been the bottleneck of MC method in clinical application. In this work, an MC simulation method combined with a variance reduction technique called correlated sampling is proposed for fast iterative scatter correction. Correlated sampling exploits correlation between similar simulation systems to reduce the variance of interest quantities. Specifically, conventional MC simulation is first performed on the scatter-contaminated CBCT to generate the initial scatter signal. In the subsequent correction iterations, scatter estimation is then updated by applying correlated MC sampling to the latest corrected CBCT images by reusing the random number sequences of the task-related photons in conventional MC. Afterward, the corrected projections obtained by subtracting the scatter estimation from raw projections are utilized for FDK reconstruction. These steps are repeated until an adequate scatter correction is obtained. The performance of the proposed framework is evaluated by the accuracy of the scatter estimation, the quality of corrected CBCT images and efficiency. Overall, the difference in mean absolute percentage error between scatter estimation with and without correlated sampling is 0.25% for full-fan case and 0.34% for half-fan case, respectively. In simulation studies, scatter artifacts are substantially eliminated, where the mean absolute error value is reduced from 15 to 2HU in full-fan case and from 53 to 13HU in half-fan case. Scatter-to-primary ratio is reduced to 0.02 for full-fan and 0.04 for half-fan, respectively. In phantom study, the contrast-to-noise ratio (CNR) is increased by a factor of 1.63, and the contrast is increased by a factor of 1.77. As for clinical studies, the CNR is improved by 11% and 14% for half-fan and full-fan, respectively. The contrast after correction is increased by 19% for half-fan and 44% for full-fan. Furthermore, root mean square error is also effectively reduced, especially from 78 to 4HU for full-fan. Experimental results demonstrate that the figure of merit is improved between 23 and 43 folds when using correlated sampling. The proposed method takes less than 25s for the whole iterative scatter correction process. The proposed correlated sampling-based MC simulation method can achieve fast and accurate scatter correction for CBCT, making it suitable for real-time clinical use.

Adaptive stratified sampling for structural reliability analysis

An adaptive framework of stratified sampling is established for estimating the probability of failure in structural reliability analysis in combination with variance reduction employed by strata. Among a variety of variance reduction techniques, stratification of Latin hypercube type makes it possible to apply importance sampling and control variates by strata in an adaptive manner without the need to track differences in computing cost across strata. By batching the resulting paralleled tasks, the proposed algorithm adjusts the allocation of computing budget only occasionally yet effectively taking full account of decreasing stratum variances. A variety of structural reliability problems with sparse structures lie within the scope of the proposed framework in the sense that stratification with gradual adjustment of computing budget avoids wasting the resource on possibly many empty strata, while importance sampling and control variates improve Monte Carlo methods on non-empty strata. We provide numerical results to illustrate the wide applicability and effectiveness of the proposed framework.

Scarce Sample-Based Reliability Estimation and Optimization Using Importance Sampling

Importance sampling is a variance reduction technique that is used to improve the efficiency of Monte Carlo estimation. Importance sampling uses the trick of sampling from a distribution, which is located around the zone of interest of the primary distribution thereby reducing the number of realizations required for an estimate. In the context of reliability-based structural design, the limit state is usually separable and is of the form Capacity (C)–Response (R). The zone of interest for importance sampling is observed to be the region where these distributions overlap each other. However, often the distribution information of C and R themselves are not known, and one has only scarce realizations of them. In this work, we propose approximating the probability density function and the cumulative distribution function using kernel functions and employ these approximations to find the parameters of the importance sampling density (ISD) to eventually estimate the reliability. In the proposed approach, in addition to ISD parameters, the approximations also played a critical role in affecting the accuracy of the probability estimates. We assume an ISD which follows a normal distribution whose mean is defined by the most probable point (MPP) of failure, and the standard deviation is empirically chosen such that most of the importance sample realizations lie within the means of R and C. Since the probability estimate depends on the approximation, which in turn depends on the underlying samples, we use bootstrap to quantify the variation associated with the low failure probability estimate. The method is investigated with different tailed distributions of R and C. Based on the observations, a modified Hill estimator is utilized to address scenarios with heavy-tailed distributions where the distribution approximations perform poorly. The proposed approach is tested on benchmark reliability examples and along with surrogate modeling techniques is implemented on four reliability-based design optimization examples of which one is a multi-objective optimization problem.

Test Results of Variance Reduction Techniques Applied to Deep Penetration Problem

Nowadays, there is a problem of a lack of computer power to conduct high-precision reactor analysis. There are several factors that increase the excessive computational load and make it difficult to calculate nuclear reactor full-scale models using Monte Carlo method. Among them is the large neutron-flux attenuation, which is observed in deep penetration problems. Various reduction techniques are used to increase the efficiency of Monte Carlo calculations in such tasks. It allows to reduce the statistical uncertainty of the functional evaluation without increasing the number of neutron histories. This work is devoted to study and testing of variance reduction techniques in the deep penetration problem. To demonstrate the possibility of using non-analogue Monte Carlo modeling a test problem was formulated. To quantify the efficacy of applying the variance reduction methods, FOM characteristic is considered, which is a function of the relative error in a flux estimate and the computational time of the simulation. The article considers non-analogue techniques implemented in the MCU (Monte Carlo Univarsal) and OpenMC codes. As part of the study, a module of the OpenMC code was developed that allows to automatically generate weight windows. It is shown that variance reduction techniques increase the calculation efficiency by several times, particularly, the weight windows method in OpenMC make it possible to achieve a 7-fold increase in the efficiency of neutron flux estimation with the same number of simulated histories. The formulated recommendations can be used in the calculation of innovative nuclear reactors full-scale models.

Assessment of emission data and transmission factors supporting radiation protection in the use of 225Ac

Purpose225Ac is the most promising alpha emitter for radiopharmaceutical therapy. Labeling PSMA, it showed to be effective in the treatment of prostate cancer and research is undergoing in order to improve its production capacity.Currently, there are still few data published concerning operational radiation protection in its use, both in clinics and in radiopharmacy, and even some basic data are not readily available. This papers aims to estimate the emission gamma-ray constant of 225Ac when at equilibrium with its descendants, and the transmission factors for a broad beam of the gamma-rays emitted by 225Ac and its descendants. Materials & methodsMonte Carlo simulations were performed using FLUKA 4.2, considering firstly the source in air, in absence of any shielding, and secondly by adding an increasing thicknesses of Lead, Concrete or Tungsten. In order to obtain statistically meaningful results, high-statistics simulations were performed by sampling up to 1010 primary decay events. As the shielding thickness increased, an appropriate variance reduction technique (importance biasing) was applied. ResultsThe specific gamma ray emission constant for 225Ac at equilibrium with descendants resulted (3.26 ± 0.03) × 10-5 mSv/h per 1 MBq at a distance of 100 cm. The transmission factors are presented in detail and data have been appropriately interpolated and fitting parameters are reported. ConclusionsThe attenuation curves show a clear bi-exponential trend: performing shielding calculations by adopting a simple approach based on a single value of Half Value Layer (HVL) or Tenth Value Layer (TVL) cannot provide adequate results.In conclusion, our results may be useful in the design of shielded hot cells or accessories necessary for operational radiation protection.

Monte Carlo radiative transfer with explicit absorption to simulate absorption, scattering, and stimulated emission

Context. The Monte Carlo method is probably the most widely used approach to solve the radiative transfer problem, especially in a general 3D geometry. The physical processes of emission, absorption, and scattering are easily incorporated in the Monte Carlo framework. Net stimulated emission, or absorption with a negative cross section, does not fit this method, however.Aims. We explore alterations to the standard photon packet life cycle in Monte Carlo radiative transfer that allow the treatment of net stimulated emission without loss of generality or efficiency.Methods. We present the explicit absorption technique that allows net stimulated emission to be handled efficiently. It uses the scattering rather than the extinction optical depth along a photon packet’s path to randomly select the next interaction location, and offers a separate, deterministic treatment of absorption. We implemented the technique in a special-purpose Monte Carlo code for a two-stream 1D radiative transfer problem and in the fully featured 3D code SKIRT, and we studied its overall performance using quantitative statistical tests.Results. Our special-purpose code is capable of recovering the analytical solutions to the two-stream problem in all regimes, including the one of strong net stimulated emission. The implementation in SKIRT is straightforward, as the explicit absorption technique easily combines with the variance reduction and acceleration techniques already incorporated. In general, explicit absorption tends to improve the efficiency of the Monte Carlo routine in the regime of net absorption.Conclusions. Explicit absorption allows the treatment of net stimulated emission in Monte Carlo radiative transfer, it interfaces smoothly with other variance reduction and acceleration techniques, and it tends to improve the efficiency of the simulations in the net absorption regime. We recommend to always include this new technique in Monte Carlo radiative transfer.

An Efficient Direct Search Method for Simulation Optimization With Conditional-Expectation- Based Objectives

In order to generalize the applicability of Conditional Value at Risk, one of the most widely used measurements used in financial risk management, we develop a solution methodology for the conditional expectation (CE)-based simulation optimization problems. To optimize CE-based objective functions in a highly generalized context, we propose a gradient-free, direct search optimization method, called SNM-CE, which inherits the search framework of Stochastic Nelder-Mead (SNM) Simplex Method but further incorporates effective mechanisms designed for handling problems with CE-based objective functions. As we assume the underlying problem is complicated enough that no closed-form expression can represent the objective function, stochastic simulation is applied to estimate CE. We apply Importance Sampling (IS) as a variance reduction technique, which, combined with a newly-developed methodology, called SOCBA-mn, ensures that simulation resources are used with great efficiency. We show that SNM-CE can converge to the true global optimum with probability one (w.p.1) like SNM. An extensive numerical study and a communication system-based empirical study are both conducted to demonstrate the effectiveness, efficiency and viability of this research in both theoretical and practical settings. Note to Practitioners—This paper develops a direct search algorithm, called SNM-CE, used for solving conditional expectation-based simulation optimization problems. By tackling conditional expectation-based problems, SNM-CE fills a gap in the stochastic optimization literature which has traditionally focused on expectation- or quantile-based objective functions. SNM-CE possesses a high degree of flexibility and generalizability which users can benefit from in solving real-world applications. For example, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> value in the CE-based simulation optimization formulation can be freely adjusted. In other words, the quantile value above which the CE is estimated and optimized can be set according to the needs of the practitioner and/or characteristics of the problem to be solved. Also, SNM-CE is designed such that instead of setting the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> value, the user may choose to utilize a particular numerical value of the objective function as the threshold for the CE estimation and optimization. SNM-CE is also simple in terms of implementation and, being a direct search method, does not impose many assumptions about the structure of the underlying objective function and does not necessitate the use of gradient information in the search process. As one application example, SNM-CE can be utilized to select the operational parameters which minimize the average delay in the manufacturing of semiconductors given that the delay exceeds the 90th percentile of simulated delay times.

Ensemble Approximate Control Variate Estimators: Applications to MultiFidelity Importance Sampling

The recent growth in multifidelity uncertainty quantification has given rise to a large set of variance reduction techniques that leverage information from model ensembles to provide variance reduction for estimates of the statistics of a high-fidelity model. In this paper we provide two contributions: (1) we utilize an ensemble estimator to account for uncertainties in the optimal weights of approximate control variate (ACV) approaches and derive lower bounds on the number of samples required to guarantee variance reduction; and (2) we extend an existing multifidelity importance sampling (MFIS) scheme to leverage control variates. Our approach directly addresses a limitation of many multifidelity sampling strategies that require the usage of pilot samples to estimate covariances. As such we make significant progress towards both increasing the practicality of approximate control variates—for instance, by accounting for the effect of pilot samples—and using multifidelity approaches more effectively for estimating low-probability events. The numerical results indicate our hybrid MFIS-ACV estimator achieves up to 50% improvement in variance reduction over the existing state-of-the-art MFIS estimator, which had already shown an outstanding convergence rate compared to the Monte Carlo method, on several problems of computational mechanics.

Stochastic distributed learning with gradient quantization and double-variance reduction

ABSTRACT We consider distributed optimization over several devices, each sending incremental model updates to a central server. This setting is considered, for instance, in federated learning. Various schemes have been designed to compress the model updates in order to reduce the overall communication cost. However, existing methods suffer from a significant slowdown due to additional variance coming from the compression operator and as a result, only converge sublinearly. What is needed is a variance reduction technique for taming the variance introduced by compression. We propose the first methods that achieve linear convergence for arbitrary compression operators. For strongly convex functions with condition number κ, distributed among n machines with a finite-sum structure, each worker having less than m components, we also (i) give analysis for the weakly convex and the non-convex cases and (ii) verify in experiments that our novel variance reduced schemes are more efficient than the baselines. Moreover, we show theoretically that as the number of devices increases, higher compression levels are possible without this affecting the overall number of communications in comparison with methods that do not perform any compression. This leads to a significant reduction in communication cost. Our general analysis allows to pick the most suitable compression for each problem, finding the right balance between additional variance and communication savings. Finally, we also (iii) give analysis for arbitrary quantized updates.

Reliability analysis of ground movement in tunnelling in spatially variable soil using equivalent parameters

The evaluation of the ground movement induced by tunnelling is of vital importance for the tunnel construction and the safety of the adjacent facilities. As the spatial variability of soil was widely acknowledged, more and more researches take insight into the uncertainty analysis for tunnelling. However, reliability analysis considering the spatial variability of soil is still a tough task since the accurate failure probability generally needs time-consuming calculation of random finite element/difference. Instead of conducting the direct Monte-Carlo simulation, this study utilizes a simplified framework to analyze the reliability of the ground movement of tunnelling. The main concept of this framework is adopting random variable model with equivalent parameters by producing the comparable failure probability as the results of random field model. Coupling with the variance reduction technique, the reliability analysis of tunnel ground movement considering more than one spatial variable can be addressed. Two tunnel cases are studied to explain the adaptability and accuracy of this simplified framework in tunnelling. The characteristic of the spatially variable soil is thoroughly comprehended through various parametric study, based on the robustness of the simplified framework. Results show that the simplified framework precisely predicts the tunnel reliability considering spatially variability of soil in a relatively efficient manner.

Reliability assessment methods to address fast transient of reactor core

In order to enhance the safety of new advanced reactors, reliability based approach to the design of thermal hydraulic system becomes necessary. In this work, “load exceeds capacity” based approach of structural reliability analysis is employed and probability of failure of the system was then assessed in terms of a limit state function while probabilistic measure of limit state function violation is performed through different methods of reliability assessment. Here, we have focused on TRIGA core subjected to reactivity initiated fast transient. Initially, response surface design method has been used for approximating true failure surface, and then FORM-SORM analysis has been carried out. But, due to non linearity involved with failure surface, there have been noticed instability in FORM-SORM implementation. Later, directional simulation approach of Monte Carlo variance reduction techniques has been employed to illustrate such fast transient. In the investigation, there have been several aspects considered and in each case directional simulation method has shown its ability to give valid results. Hence, the method could be recommended as a viable and efficient scheme to solve even fast transient problem in design and analysis of any reactor.

Evaluation of sampling techniques for probabilistic structural integrity assessment of steam generator tubing

Steam generator (SG) tubes for nuclear rectors serve as both pressure and containment boundaries. Rupture of a SG tube is considered a significant event from the perspective of safety and potential for release of radiation. Therefore, SG tubes are designed and operated at high reliability to maintain the pressure boundary's structural integrity. Whereas probabilistic approaches become more widely used in the operational assessment to adequately account for uncertainties from various sources, the high reliability level exerts a high demand for computational efforts, which sometimes may be beyond present computational capabilities. Using the crude Monte Carlo simulation as the baseline approach, this paper investigates several variance reduction techniques that reduce the computational effort for the desired small failure probability of a SG tube rupture. It is shown that Latin hypercubic sampling, directional simulation and subset simulation can improve the accuracy of the failure probability estimate, and directional simulation can improve both accuracy and efficiency dramatically. The probabilistic operational assessment of a specific example shows the calculated failure frequency satisfies the probabilistic structural integrity performance criteria (SIPC) under the normal operating condition, which is consistent with the results when deterministic SIPC is applied.

Towards efficient Monte Carlo N-Particle simulation of a positron emission tomography (PET) via source volume definition

Monte Carlo N-Particle (MCNP) simulation has been extensively proven in nuclear medicine imaging systems, most notably in designing and optimizing new medical imaging tools. It enables more complicated geometries and the simulation of particles passing through and interacting with materials. However, a relatively long simulation time is a drawback of Monte Carlo simulation, mainly when complex geometry exists. The current study presents an alternative variance reduction technique for a modeled positron emission tomography (PET) camera by reducing the height of the source volume definition while maintaining the geometry of the simulated model. The National Electrical Manufacturers Association (NEMA) of the International Electrotechnical Commission (IEC) PET's phantom was used with a 1 cm diameter and 7 cm height of line source placed in the middle. The first geometry was fully filled the line source with 0.50 mCi radioactivity. In contrast, the second geometry decreased the source definition to 2.4 cm in height, covering 1 cm above and below the sub-block detector level. The source volume definition approach led to a 71% reduction in the total photons to be simulated. Results showed that the proposed variance reduction strategy could produce spatial resolution as precise as fully filled geometry and sped up the simulation time by approximately 65%. Hence, this strategy can be utilized for further PET optimizing simulation studies.

ESTIMATION OF RADIATION DOSES IN THE AREA OF THE MAIN PRODUCTION BLOCK OF THE NUCLEAR POWER PLANT AFTER A REACTOR ACCIDENT.

The aim of this work is to propose methodology for estimations the characteristics of the radiation fields (and derived quantities from the field of dosimetry and radiation protection) in knowledge of a distribution of radionuclides released into the main production unit (or just a containment) area after a nuclear reactor accident. For such task stochastic Monte Carlo method has been chosen. Because of dimensions and thick shielding (concrete) barriers in the facility, application of the variance reduction techniques has been necessary. Monte Carlo code Monaco in sequence MAVRIC (from package SCALE 6.2.3) with variance reduction techniques using CADIS methodology has been employed for designing the proposed methodology. Procedure has been tested on a simulation model of a main production unit described by inspiration from the block of a nuclear power plant with a VVER-1000 reactor (installed in the Czech Republic, Central Europe).

A high fidelity general purpose 3-D Monte Carlo particle transport program JMCT3.0

JMCT is a large-scale, high-fidelity, three-dimensional general neutron–photon–electron–proton transport Monte Carlo software system. It was developed based on the combinatorial geometry parallel infrastructure JCOGIN and the adaptive structured mesh infrastructure JASMIN. JMCT is equipped with CAD modeling and visualizes the image output. It supports the geometry of the body and the structured/unstructured mesh. JMCT has most functions, variance reduction techniques, and tallies of the traditional Monte Carlo particle transport codes. Two energy models, multi-group and continuous, are provided. In recent years, some new functions and algorithms have been developed, such as Doppler broadening on-the-fly (OTF), uniform tally density (UTD), consistent adjoint driven importance sampling (CADIS), fast criticality search of boron concentration (FCSBC) domain decomposition (DD), adaptive control rod moving (ACRM), and random geometry (RG) etc. The JMCT is also coupled with the discrete ordinate SN code JSNT to generate source-biasing factors and weight-window parameters. At present, the number of geometric bodies, materials, tallies, depletion zones, and parallel processors are sufficiently large to simulate extremely complicated device problems. JMCT can be used to simulate reactor physics, criticality safety analysis, radiation shielding, detector response, nuclear well logging, and dosimetry calculations etc. In particular, JMCT can be coupled with depletion and thermal-hydraulics for the simulation of reactor nuclear-hot feedback effects. This paper describes the progress in advanced modeling, high-performance numerical simulation of particle transport, multiphysics coupled calculations, and large-scale parallel computing.

Gramian-based model reduction for unstable stochastic systems

This paper considers large-scale linear stochastic systems representing, e.g., spatially discretized stochastic partial differential equations. Since asymptotic stability can often not be ensured in such a stochastic setting (e.g., due to larger noise), the main focus is on establishing model order reduction (MOR) schemes applicable to unstable systems. MOR is vital to reduce the dimension of the problem in order to lower the enormous computational complexity of for instance sampling methods in high dimensions. In particular, a new type of Gramian-based MOR approach is proposed in this paper that can be used in very general settings. The considered Gramians are constructed to identify dominant subspaces of the stochastic system as pointed out in this work. Moreover, they can be computed via Lyapunov equations. However, covariance information of the underlying systems enters these equations which is not directly available. Therefore, efficient sampling-based methods relying on variance reduction techniques are established to derive the required covariances and hence the Gramians. Alternatively, an ansatz to compute the Gramians by deterministic approximations of covariance functions is investigated. An error bound for the studied MOR methods is proved yielding an a priori criterion for the choice of the reduced system dimension. This bound is new and beneficial even in the deterministic case. The paper is concluded by numerical experiments showing the efficiency of the proposed MOR schemes.

Built in importance estimation in forward Monte Carlo calculations

Recently, Monte Carlo code development has shown the benefits of using deterministic importance functions in radiation transport calculation; however, using Monte Carlo based importance function is less considered regarding the computational costs in this calculation.This paper offers a Monte Carlo based importance calculation method with the capability of application in forward Monte Carlo simulations. The technique uses the tracked particle’s history in the forward Monte Carlo calculation for the estimation of importance function. Moreover, the particle population, weight, and pathways can be considered for dynamic Monte Carlo calculation. The ability of the purposed method is shown by some examples in simple and complex geometries. The results show that the method can be used in the Monte Carlo code development that can benefit from a dynamic variance reduction technique in radiation transport calculations.

Accelerated radiation transport modeling techniques for pencil beam computed tomography using gamma rays

Monte Carlo radiation transport modeling studies were performed for a compact, and high-resolution gamma-ray computed tomography system designed for imaging irradiated nuclear fuel. The system comprises a 60Co source – chosen for its highly penetrating 1173 keV and 1332 keV gamma rays – a pair of high-aspect-ratio pencil beam collimators, and an inorganic scintillator detector. Two acceleration methods are proposed to rapidly model a transmission type gamma-ray tomography system. The first, a variance reduction technique, is based on performing Monte Carlo simulations with a monodirectionally-biased source, sampled from a characteristic sub-volume of the full source volume. The second acceleration method is based on the deterministic calculations using the Beer–Lambert law and detector response characteristics. Comparison of simulations using acceleration approaches with analog simulations of the fully isotropic, full-volume equivalent, show that the Monte Carlo variance reduction technique gives quantitatively accurate predictions for large collimator aspect ratios while the deterministic calculations are semi-quantitative but converge close to the correct result as the collimator aspect ratio increases. As such, these techniques can be used to reduce the computational cost in generating simulated radiographs and tomographs by several orders of magnitude. Experimental validation efforts are currently underway and will be demonstrated in future work.

Development of Monte Carlo based treatment planning system for BNCT

A treatment planning system (TPS) plays a major role for a successful radiotherapy, directly influencing the calculated dose accuracy, and thus the treatment quality. Multifunctional Arithmetic for Neutron Transportation Analysis (NeuMANTA), a TPS especially designed for complete workflow for BNCT, which aims to provide high precision dose calculation with maximizing the efficacy. The dedicated MC engine COMPASS seamlessly embedded in NeuMANTA can be called for dose distribution calculation in background. Optimized cross-section library, parallel computing, and various variance reduction techniques are applied to speed up the particle transportations. A benchmark was performed between NeuMANTA and MCNP6, and results agreed well with no significant statistical difference, however, NeuMANTA needs only half of the calculation time. The accuracy of estimated doses, and complete treatment workflow are the main features in NeuMANTA.

Convergence analysis of asynchronous stochastic recursive gradient algorithms

Asynchronous stochastic algorithms with variance reduction techniques have been empirically shown to be useful for many large-scale machine learning problems. By making a parallel optimization algorithm asynchronous, one can reduce the synchronization cost and improve the practical efficiency. Recently, the stochastic recursive gradient algorithm has shown superior theoretical performance; however, it is not scalable enough in the current big data era. To make it more practical, we propose a class of asynchronous stochastic recursive gradient methods and analyze them in the shared memory model. The analysis results show that our asynchronous algorithms can linearly converge to the solution in the strongly convex case and complete the iteration faster. In addition, we analyze the “price of asynchrony” and give the sufficient conditions required for linear speedup. To the best of our knowledge, our speedup conditions match the optimal bounds of asynchronous stochastic algorithms known thus far. Finally, we demonstrate our theoretical analyses from a practical implementation on a multicore machine.