Variance Reduction Methods Research Articles

Strata Design for Variance Reduction in Stochastic Simulation

Stratified sampling is one of the powerful variance reduction methods for analyzing system performance, such as reliability, with stochastic simulation. It divides the input space into disjoint subsets, called strata, to draw samples from each stratum. Partitioning the input space properly and allocating greater computational effort to crucial strata can help accurately estimate system performance with a limited computational budget. How to create strata, however, has yet to be thoroughly examined. Strata design faces the curse of dimensionality and data scarcity as the input dimension increases. We analytically derive the optimal stratification structure that minimizes the estimation variance for univariate problems. Further, reconciling the optimal stratification into decision trees, we devise a robust algorithm for multi-dimensional problems. Numerical experiments and a wind turbine case study demonstrate the superiority of the proposed method in terms of variance reduction, leading to computational efficiency and scalability.

Doubly-robust and heteroscedasticity-aware sample trimming for causal inference

SUMMARY A popular method for variance reduction in causal inference is propensity-based trimming, the practice of removing units with extreme propensities from the sample. This practice has theoretical grounding when the data are homoscedastic and the propensity model is parametric (Yang & Ding, 2018; Crump et al., 2009), but in modern settings where heteroscedastic data are analyzed with non-parametric models, existing theory fails to support current practice. In this work, we address this challenge by developing new methods and theory for sample trimming. Our contributions are three-fold: first, we describe novel procedures for selecting which units to trim. Our procedures differ from previous works in that we trim not only units with small propensities, but also units with extreme conditional variances. Second, we give new theoretical guarantees for inference after trimming. In particular, we show how to perform inference on the trimmed sub-population without requiring that our regressions converge at parametric rates. Instead, we make only fourth-root rate assumptions like those in the double machine learning literature. This result applies to conventional propensity-based trimming as well and thus may be of independent interest. Finally, we propose a bootstrap-based method for constructing simultaneously valid confidence intervals for multiple trimmed sub-populations, which are valuable for navigating the trade-off between sample size and variance reduction inherent in trimming. We validate our methods in simulation, on the 2007-2008 National Health and Nutrition Examination Survey, and on a semi-synthetic Medicare dataset and find promising results in all settings.

A Monte Carlo Simulation Dataset of Radiation Field Parameters for a Simple Nuclear Facility Scenario

Radioactive safety in nuclear facilities is of utmost importance. Prior to workers entering these areas, a 3D radiation field is needed for accurately estimating their exposure. Due to the complex relationship between radiation measurements and radiation fields, implementing neural networks is a promising approach for reconstruction. However, research on direct 3D radiation field reconstruction using neural networks is limited, and there is no standardized open-source dataset for training and evaluation. To address these issues, we created a simplified model of a nuclear facility and utilized the Monte Carlo program MCShield to simulate 3D radiation parameters. MCShield, which is mainly used for shielding calculations, has been verified for accuracy through benchmark tests. In addition, this paper proves the correctness of the MCShield program and the effectiveness of the AIS variance reduction method through calculations on the WinFrith Iron benchmark experiment and the NUREG/CR-6115 benchmark. The results show that the MCShield program as well as the AIS method can be used for dataset calculations.

Numerical methods for distributed stochastic compositional optimization problems with aggregative structure

ABSTRACT The paper studies the distributed stochastic compositional optimization problems over networks, where all the agents' inner-level function is the sum of each agent's private expectation function. Focusing on the aggregative structure of the inner-level function, we employ the hybrid variance reduction method to obtain the information on each agent's private expectation function, and apply the dynamic consensus mechanism to track the information on each agent's inner-level function. Then by combining with the standard distributed stochastic gradient descent method, we propose a distributed aggregative stochastic compositional gradient descent method. When the objective function is smooth, the proposed method achieves the convergence rate O ( K − 1 / 2 ) . We further combine the proposed method with the communication compression and propose the communication compressed variant distributed aggregative stochastic compositional gradient descent method. The compressed variant of the proposed method maintains the convergence rate O ( K − 1 / 2 ) . Simulated experiments on decentralized reinforcement learning verify the effectiveness of the proposed methods.

Improving compressed matrix multiplication using control variate method

The seminal work by Pagh [1] proposed a matrix multiplication algorithm for real-valued squared matrices called Compressed Matrix Multiplication (CMM) having a sparse matrix output product. The algorithm is based on a popular sketching technique called Count-Sketch [2] and Fast Fourier Transform (FFT). For input square matrices A and B of order n and the product matrix AB with Frobenius norm ||AB||F, the algorithm offers an unbiased estimate for each entry, i.e., (AB)i,j of the product matrix AB with a variance bounded by ||AB||F2/b, where b is the compressed bucket size. Thus, the variance will eventually become high for a small bucket size. In this work, we address the high variance problem of CMM with the help of a simple and practical technique based on classical variance reduction methods in statistics. Our techniques rely on the Control Variate (CV) method. We suggest rigorous theoretical analysis for variance reduction and complement it via supporting empirical evidence.

Prompt-gamma track-length estimator with time tagging from proton tracking

The design of prompt-gamma detectors necessitates numerous Monte Carlo simulations to precisely develop and optimize the detection stages in proton therapy. Alongside the advancement of MC simulations, various variance reduction methods have been explored to speed-up calculations. Among these techniques, track-length estimators are interesting scoring methods for achieving both speed and accuracy in Monte Carlo simulations of rare events. This paper introduces an extension of the GATE vpgTLE module that incorporates the prompt-gamma emission time, which is tagged from the proton tracking, enhancing its utility for studies focused on detector design and optimization that rely on time measurements. The results obtained from a clinical radiotherapy plan are presented. We demonstrate that the new vpgTLE tally with time tagging is accurate, except for certain prompt-gamma lines corresponding to long mean-life nuclei.

Research on a Monte Carlo global variance reduction method based on an automatic importance sampling method

Research on a Monte Carlo global variance reduction method based on an automatic importance sampling method

Neural network-based source biasing to speed-up challenging MCNP simulations

Nuclear analysis of fusion facilities, especially in the context of ITER, is complex due to the need for precise modeling of complex geometry and radiation sources. Monte Carlo (MC) codes, such as MCNP, are used in this context due to their high precision and capability to deal with these cases. To speed up calculations, variance reduction (VR) techniques are crucial for carrying out the simulations in reasonable times. Particularly, the cases where only a small source phase space contributes significantly to the tally must be optimized by an exhaustive sampling of this phase space. This paper introduces a novel VR method for optimizing the calculation. This method obtains the histories that contribute most to the tally and uses this information for a neural network (NN) to sample the source´s phase space. Here we present a preliminary implementation of the method to study its viability in complex cases. The method's efficacy is demonstrated in a representative source-geometry configuration, significantly reducing computational time for the tally convergence. The promising results suggest applicability to more intricate ITER-like scenarios, prompting further development.

Faster Stochastic Variance Reduction Methods for Compositional MiniMax Optimization

This paper delves into the realm of stochastic optimization for compositional minimax optimization—a pivotal challenge across various machine learning domains, including deep AUC and reinforcement learning policy evaluation. Despite its significance, the problem of compositional minimax optimization is still under-explored. Adding to the complexity, current methods of compositional minimax optimization are plagued by sub-optimal complexities or heavy reliance on sizable batch sizes. To respond to these constraints, this paper introduces a novel method, called Nested STOchastic Recursive Momentum (NSTORM), which can achieve the optimal sample complexity and obtain the nearly accuracy solution, matching the existing minimax methods. We also demonstrate that NSTORM can achieve the same sample complexity under the Polyak-Lojasiewicz (PL)-condition—an insightful extension of its capabilities. Yet, NSTORM encounters an issue with its requirement for low learning rates, potentially constraining its real-world applicability in machine learning. To overcome this hurdle, we present ADAptive NSTORM (ADA-NSTORM) with adaptive learning rates. We demonstrate that ADA-NSTORM can achieve the same sample complexity but the experimental results show its more effectiveness. All the proposed complexities indicate that our proposed methods can match lower bounds to existing minimax optimizations, without requiring a large batch size in each iteration. Extensive experiments support the efficiency of our proposed methods.

The Effect of Branchless Collisions and Population Control on Correlations in Monte Carlo Power Iteration

The investigation of correlations in Monte Carlo power iteration has long been dominated by the question of generational correlations and their effects on the estimation of statistical uncertainties. More recently, there has been a growing interest in spatial correlations, prompted by the discovery of neutron clustering. Despite several attempts, a comprehensive framework concerning how Monte Carlo sampling strategies, population control, and variance reduction methods affect the strength of such correlations is still lacking. In this work, we propose a set of global and local (i.e., space-dependent) tallies that can be used to characterize the impact of correlations. These tallies encompass Shannon entropy, pair distance, normalized variance, and Feynman moment. In order to have a clean yet fully meaningful setting, we carry out our analysis in a few homogeneous and heterogeneous benchmark problems of varying dominance ratio. Several classes of collision sampling strategies, population control, and variance reduction techniques are tested, and their relative advantages and drawbacks are assessed with respect to the proposed tallies. The major finding of our study is that branchless collisions, which suppress the emergence of branches in neutron histories, also considerably reduce the effects of correlations in most of the explored configurations.

The methods of CADIS-NEE and CADIS-DXTRAN in NECP-MCX and their applications

This paper presents two new methods for variance reduction for shielding calculation in Monte Carlo radiation transport. One method is CADIS-NEE, which combines Consistent Adjoint Driven Importance Sampling (CADIS) and next-event estimator (NEE) methods to increase the calculation efficiency of tallies at points. The other is CADIS-deterministic transport (DXTRAN), which combines CADIS and DXTRAN to obtain higher performance than using CADIS and DXTRAN separately. The combination processes are derived and implemented in the hybrid Monte-Carlo-Deterministic particle-transport code NECP-MCX. Various problems are tested to demonstrate the effectiveness of the two methods. According to the results, the two combination methods have higher efficiency than using CADIS, NEE or DXTRAN separately. In a long-distance photon-transport problem, CADIS-NEE converges faster than NEE and the figure of merit (FOM) of CADIS-NEE is 75.6 times of NEE. In a labyrinthine problem, CADIS-DXTRAN's FOM surpasses that of DXTRAN and CADIS by a factor of 45.3 and 17.7, respectively. Therefore, it is advisable to employ these two novel methods selectively in appropriate scenarios to reduce variance.

Effectiveness of Radiation Transport Variance Reduction Methods for Wide-Area Environmental Contamination Assay Applications

This study compares the accuracy, efficiency, and reliability of variance reduction (VR) methods for Monte Carlo radiation transport simulations involving wide-area ground plane (i.e., “surface”) and buried (i.e., “volumetric”) gamma source emissions from environmental soil. The simulation models are idealized external exposure scenarios intended as a basis for deriving site-specific dose-based or carcinogenic risk–based regulatory limits in the radiological site remediation process. These simulations are computationally resource intensive since particle tracks are transported from an extremely large source region to a relatively small detector region. For each simulation, several VR methods are compared with metrics of accuracy, efficiency, and reliability. The MCNP deterministic transport (DXTRAN) VR method was most effective for problems involving sources emitting low-energy gamma rays, and a coupled multicode method was more effective for problems involving sources emitting higher-energy gamma rays that undergo significant attenuation in the soil.

Efficiency of variance-reduction techniques using TRIPOLI-4®: Application to equivalent dose rate calculations in a spent-fuel cask

This paper presents the comparison of different variance-reduction methods implemented in the Monte Carlo code TRIPOLI-4® in the case of a deep penetration problem. In this complex radiation transport situation, where the shielding is very thick to ensure protection, variance-reduction methods are necessary to obtain results with a good accuracy in a reasonable calculation time. The configuration investigated in this work is a spent-fuel transport cask. The various variance-reduction methods used here consist either of Importance Sampling with the Exponential Transform method, or of sophisticated population-control methods. These methods are detailed, then the results are presented and compared and the efficiency of these methods is investigated. While all methods implemented in TRIPOLI-4® are efficient versus analog simulations, the pre-calculation of importance maps with deterministic methods enables a faster convergence of the Monte Carlo simulations, especially when combined with the Exponential Transform.

A weight window approach for dose rate calculations in Lead-cooled Fast Reactors with OpenMC

This paper presents the application of a variance reduction technique for radiation shielding calculations in Lead-cooled Fast Reactors (LFRs) using the OpenMC Monte Carlo code. The study focuses on assessing the computational efficiency of the Method of Automatic Generation of Importances by Calculation (MAGIC). This technique is employed to generate weight windows for Monte Carlo simulations, aiming at enhancing the efficiency of dose rate estimations for LFRs. Through a detailed study of a LFR mock-up model, the performance differences between a MAGIC-accelerated and a non-accelerated simulation are investigated in terms of both accuracy and computational cost. The results demonstrate the benefits of the variance reduction method, showing its effectiveness in spreading particles throughout the model while improving the accuracy of Monte Carlo estimates far from the core. In addition, this work demonstrates the performances of the MAGIC method in eigenvalue calculations. This work shows an interesting optimisation of Monte Carlo simulations for LFRs and can be considered as a first milestone for newcleo’s radiation shielding studies.

Monte Carlo global variance reduction method combining source bias and weight window

The global tally problem has a wide range of applications in major research fields such as Monte Carlo simulations of pin-by-pin reactor models and time-dependent particle transport problems in multi-physics coupling calculations. Due to the uneven power distribution of the simulated system, the statistical errors of all tallies are unevenly distributed, resulting in some low global efficiency. For this kind of problem with global characteristics, it is necessary to develop global variance reduction techniques to obtain the accurate distribution of target tallies in the entire space. A large number of global variance reduction algorithms have been studied based on the consideration of flattening global tally error distribution, so as to improve global efficiency. This work focuses on the combination of two efficient global variance reduction algorithms, namely, the uniform tally density algorithm and the weight window algorithm, which belong to source bias and transport process bias, respectively. In tally, a method is proposed to adjust the weight window parameters by using the bias factor of the uniform tally density algorithm. Then, the weight window method will be used to reduce the weight fluctuation caused by the uniform tally density method. In this way, an organic combination of these two algorithms can be realized. A series of comparative tests are carried out based on the Hoogenboom-Martin pressurized water reactor benchmark, and it is verified that the hybrid global variance reduction algorithm proposed in this work is better than the single weight window algorithm or the uniform tally density algorithm. In terms of reducing the maximum error, the global efficiency of the hybrid algorithm is 2.6 times and 3 times that of the weight window algorithm and the uniform tally density algorithm, respectively. In addition, through the comparative analysis of computational asymmetry degree and computational efficiency, it is verified that the uniform tally density algorithm has better performance than the classical uniform fission site algorithm, and the performance advantages of the uniform tally density algorithm are quantitatively evaluated based on some new indicators. The results show that the hybrid global variance reduction algorithm proposed in this work can solve the global tally problem efficiently, thereby further promoting research in related fields.

Optimal Monte Carlo particle splitting for neutron transport equation

Global tallying problem is hard to be solved efficiently when using Monte carlo method to simulate neutron transport equation, especially for sys- tem with big scale and/or strong imbalanced characteristics. Among numerous global variance reduction methods, particle splitting is an easy-to-implement method, but the efficiency is hard to be satisfying for practical problems. The essential challenge lies in the fact there are too many parameters to be opti- mized. In this paper, a systematic approach for setting all splitting parameters is proposed based on maximization of some global efficiency indicator. Results of two models show this approach can increase the global efficiency indicator up to two orders of magnitude while keeping the results unbiased.

A variance-reduction strategy for the sensitivity of βeff

The Monte Carlo computation of the GPT-based sensitivity of the effective delayed neutron fraction βeff to nuclear data proves to be quite difficult to converge due to the small amount of delayed neutrons that are sampled in k-eigenvalue calculations. This paper describes a variance-reduction method aimed at efficiently computing the sensitivity coefficients of βeff, reducing the associated Monte Carlo uncertainty and increasing the Figure Of Merit. This variance-reduction technique allows also computing the sensitivities of βj eff for a specific precursor family j. Verification and performance evaluation are achieved using simple configurations admitting analytical solutions and several continuousenergy benchmarks.

Ensemble Variance Reduction Methods for Stochastic Mixed-Integer Programming and their Application to the Stochastic Facility Location Problem

Sample average approximation (SAA), the standard approach to stochastic mixed-integer programming, does not provide guidance for cases with limited computational budgets. In such settings, variance reduction is critical in identifying good decisions. This paper explores two closely related ensemble methods to determine effective decisions with a probabilistic guarantee. (a) The first approach recommends a decision by coordinating aggregation in the space of decisions as well as aggregation of objective values. This combination of aggregation methods generalizes the bagging method and the “compromise decision” of stochastic linear programming. Combining these concepts, we propose a stopping rule that provides an upper bound on the probability of early termination. (b) The second approach applies efficient computational budget allocation for objective function evaluation and contributes to identifying the best solution with a predicted lower bound on the probability of correct selection. It also reduces the variance of the upper bound estimate at optimality. Furthermore, it adaptively selects the evaluation sample size. Both approaches provide approximately optimal solutions even in cases with a huge number of scenarios, especially when scenarios are generated by using oracles/simulators. Finally, we demonstrate the effectiveness of these methods via extensive computational results for “megascale” (extremely large scale) stochastic facility location problems. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by The Office of Naval Research [Grant N00014-20-1-2077] and the Air Force Office of Scientific Research [Grant FA9550-20-1-0006]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0324 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0324 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Code and Data Repository for Ensemble Variance Reduction Methods for Stochastic Mixed-Integer Programming and their Application to the Stochastic Facility Location Problem

Code and Data Repository for Ensemble Variance Reduction Methods for Stochastic Mixed-Integer Programming and their Application to the Stochastic Facility Location Problem

REIN: Reliability Estimation via Importance sampling with Normalizing flows

We introduce a novel framework called REIN: Reliability Estimation by learning an Importance sampling (IS) distribution with Normalizing flows (NFs). The NFs learn probability space maps that transform the probability distribution of the input random variables into a quasi-optimal IS distribution. NFs stack together invertible neural networks to construct differentiable bijections with efficiently computed Jacobian determinants. The NF ‘pushes forward’ a realization from the input probability distribution into a realization from the IS distribution, with importance weights calculated using the change of variables formula. We also propose a loss function to learn a NF map that minimizes the reverse Kullback–Leibler divergence between the ‘pushforward’ distribution and a sequentially updated target distribution obtained by modifying the optimal IS distribution. We demonstrate REIN’s efficacy on a set of benchmark problems that feature very low failure rates, multiple failure modes and high dimensionality, while comparing against other variance reduction methods. We also consider two simple applications, the reliability analyses of a thirty-four story building and a cantilever tube, to demonstrate the applicability of REIN to practical problems of interest. As compared to other methods, REIN is shown to be useful for high-dimensional reliability estimation problems with very small failure probabilities.