Variance Reduction Techniques Research Articles

Composite reliability assessment of systems with grid‐edge renewable resources via quasi‐sequential Monte Carlo and cross‐entropy techniques

AbstractThe still growing share of intermittent renewable sources in the electric energy matrix is a factor that greatly increases the complexity of modelling electrical power systems. Although its exploitation is conditioned to the geographic location where the natural resource emerges itself, when this location coincides with the grid‐edge of the system, close to the consumer, the composite reliability assessment is exposed to the combination of two problems: (i) representation of the intermittent energy availability of generators; and (ii) failure event rarity. To address the first problem, a fast Monte Carlo tool capable of representing chronological aspects can be used. For the second problem, an efficient variance reduction technique that adapts well to the composite problem of generation and transmission must be established. Therefore, a quasi‐sequential Monte Carlo simulation tool aided by importance sampling via the cross‐entropy method is proposed as an efficient and robust procedure for evaluating the composite reliability of systems with renewable participation at grid‐edge. The method is evaluated through a modified version of the IEEE Reliability Test System ‐ 1996, which presents renewable generation in consumer load buses and rarity in the composite failure.

Relaxation-based importance sampling for structural reliability analysis

This study presents an importance sampling formulation based on adaptively relaxing parameters from the indicator function and/or the probability density function. The formulation embodies the prevalent mathematical concept of relaxing a complex problem into a sequence of progressively easier sub-problems. Due to the flexibility in constructing relaxation parameters, relaxation-based importance sampling provides a unified framework for various existing variance reduction techniques, such as subset simulation, sequential importance sampling, and annealed importance sampling. More crucially, the framework lays the foundation for creating new importance sampling strategies, tailoring to specific applications. To demonstrate this potential, two importance sampling strategies are proposed. The first strategy couples annealed importance sampling with subset simulation, focusing on low-dimensional problems. The second strategy aims to solve high-dimensional problems by leveraging spherical sampling and scaling techniques. Both methods are desirable for fragility analysis in performance-based engineering, as they can produce the entire fragility surface in a single run of the sampling algorithm. Three numerical examples, including a 1000-dimensional stochastic dynamic problem, are studied to demonstrate the proposed methods.

Meta variance reduction for Monte Carlo estimation of energetic particle confinement during stellarator optimization

This work introduces meta estimators that combine multiple multifidelity techniques based on control variates, importance sampling, and information reuse to yield a quasi-multiplicative amount of variance reduction. The proposed meta estimators are particularly efficient within outer-loop applications when the input distribution of the uncertainties changes during the outer loop, which is often the case in reliability-based design and shape optimization. We derive asymptotic bounds of the variance reduction of the meta estimators in the limit of convergence of the outer-loop results. We demonstrate the meta estimators, using data-driven surrogate models and biasing densities, on a design problem under uncertainty motivated by magnetic confinement fusion, namely the optimization of stellarator coil designs to maximize the estimated confinement of energetic particles. The meta estimators outperform all of their constituent variance reduction techniques alone, ultimately yielding two orders of magnitude speedup compared to standard Monte Carlo estimation at the same computational budget.

A Data-Driven Variance Reduction Technique for Efficiently Modelling Astronaut Radiation Doses in Spacecraft in High-Energy Isotropic Radiation Fields.

The ionizing radiation exposure to crew on current and future space missions can significantly increase their health risks for cancers, degenerative diseases, and other acute and late effects. A common approach for estimating risk to crew is by completing stochastic (e.g., Monte Carlo) or deterministic particle transport simulations. Within the simulated environment, a small fraction of the particle histories tracked will interact with the astronaut or detector, particularly for larger spacecraft such as the International Space Station, Tiangong Space Station or Lunar Gateway. These simulations can be computationally intensive as they require a very large number of particle histories to achieve a low statistical uncertainty. Variance reduction techniques are applied to simulations to reduce the computational time of the simulation while maintaining the same (or less) statistical uncertainty. The variance reduction technique developed herein involves applying a directional source bias to an isotropic radiation field, such as galactic cosmic rays, to reduce the quantity of particles that have a low probability of interacting with the astronaut or detector. A custom application has been developed utilizing the Geant4 Toolkit that computes the trajectories and energies of particles in three dimensions in the International Space Station using the Monte Carlo method. The results demonstrate the impact of our variance reduction technique on effective dose equivalence depending on: primary and secondary particle type (proton, neutron, photon, heavy ion, etc.), geometric volumes and spacecraft materials. Our variance reduction technique can be tuned by the user to optimize the simulation time depending on their objectives and enables rapid testing of different shield configurations and materials. This variance reduction technique is implemented easily using several input parameters for boundary conditions. Recommended values are presented for rapid implementation in simulations.

A bi-criteria algorithm for online non-monotone maximization problems: DR-submodular+concave

In this paper, we study a class of online non-monotone maximization problems under general constraint set. In each round, the function fed back by the environment is of composite structure: the sum of a DR-submodular function and a concave function. This setting covers a wide range of applications. We propose a Frank-Wolfe type online algorithm for solving the considered problem. In our algorithm, there is no need to have the ability to obtain exact gradients of the revealed functions. Adopting the Lyapunov function approach and variance reduction technique, the algorithm is shown to have the bi-criteria competitive ratio (1/4,3/8) with sub-linear regret under selecting suitable parameters.

Calculation of radiation fields in the VVER-1000 concrete biological shield using Monte Carlo code Serpent

To calculate radiation fields in the concrete biological shield (CBS) of the VVER-1000 reactor in this work, we have developed and applied the Monte Carlo code Serpent simulation framework based on the variance reduction technique. We have quantified the radial, axial, and azimuthal variation of neutron and gamma-ray fluxes and the absorbed dose rate in the CBS. Using the calculation results, we estimate maximum neutron fluence and maximum absorbed dose in the VVER-1000 CBS over the period of 60 and 80 years of the reactor operation and localize the domains of highest radiation exposure. The obtained results are in good agreement with the available data on the VVER-1000 and other pressurized water reactors. We show that the fluence of neutrons with energy above 0.1 MeV decreases by half at a depth of 4 cm of concrete, and the gamma-ray absorbed dose decreases by 40 % at a depth of 13.5 cm. The outcomes from this research will help to assess the effects of prolonged irradiation of the VVER-1000 CBS, which is required for reliable risk assessment for extended operation of nuclear power plants.

Multigrid multilevel Monte Carlo for a lattice with a displacement

AbstractThe Hutchinson method can be used to estimate the trace of a function of a matrix A, that is, , stochastically. With the variance of the Hutchinson estimator decreasing as , techniques to reduce the variance of this estimator become necessary due to the high accuracy required by some calculations in scientific computing. In lattice QCD, where A is the Dirac operator D stemming from a discretization of the Dirac equation, the accuracy of some calculations involving renders them practically unfeasible if one uses the Hutchinson estimator to approximate the trace. One such variance reduction technique, known as multigrid multilevel Monte Carlo, has been introduced recently to deal with for the Laplace 2D, Gauge Laplacian, and Schwinger problems. We use it here in the context of lattice QCD, and moreover, we analyze its impact when dealing with , where is the matrix that emerges when a displacement in one of the four spacetime dimensions of the lattice is introduced. This latter trace is of importance when computing generalized parton distributions in lattice QCD.

Optimizing shift selection in multilevel Monte Carlo for disconnected diagrams in lattice QCD

The calculation of disconnected diagram contributions to physical signals is a computationally expensive task in Lattice QCD. To extract the physical signal, the trace of the inverse Lattice Dirac operator, a large sparse matrix, must be stochastically estimated. Because the variance of the stochastic estimator is typically large, variance reduction techniques must be employed. Multilevel Monte Carlo (MLMC) methods reduce the variance of the trace estimator by utilizing a telescoping sequence of estimators. Frequency Splitting is one such method that uses a sequence of inverses of shifted operators to estimate the trace of the inverse lattice Dirac operator, however there is no a priori way to select the shifts that minimize the cost of the multilevel trace estimation. In this article, we present a sampling and interpolation scheme that is able to predict the variances associated with Frequency Splitting under displacements of the underlying space time lattice. The interpolation scheme is able to predict the variances to high accuracy and therefore chooses shifts that correspond to an approximate minimum of the cost for the trace estimation. We show that Frequency Splitting with the chosen shifts displays significant speedups over multigrid deflation, and that these shifts can be used for multiple configurations within the same ensemble with no penalty to performance.

Distributed Stochastic Gradient Tracking Algorithm With Variance Reduction for Non-Convex Optimization.

This article proposes a distributed stochastic algorithm with variance reduction for general smooth non-convex finite-sum optimization, which has wide applications in signal processing and machine learning communities. In distributed setting, a large number of samples are allocated to multiple agents in the network. Each agent computes local stochastic gradient and communicates with its neighbors to seek for the global optimum. In this article, we develop a modified variance reduction technique to deal with the variance introduced by stochastic gradients. Combining gradient tracking and variance reduction techniques, this article proposes a distributed stochastic algorithm, gradient tracking algorithm with variance reduction (GT-VR), to solve large-scale non-convex finite-sum optimization over multiagent networks. A complete and rigorous proof shows that the GT-VR algorithm converges to the first-order stationary points with O(1/k) convergence rate. In addition, we provide the complexity analysis of the proposed algorithm. Compared with some existing first-order methods, the proposed algorithm has a lower O(PMε⁻¹) gradient complexity under some mild condition. By comparing state-of-the-art algorithms and GT-VR in numerical simulations, we verify the efficiency of the proposed algorithm.

Investigation of the anisotropic distribution of microdosimetric quantities in the vicinity of X-ray-irradiated gold nanoparticles

High-Z nanoparticles incorporated in tumors have great potential to enhance radiotherapy treatments by increasing the absorbed radiation dose in the vicinity of the nanoparticle. However, current Monte Carlo simulations estimate dose enhancement metrics in 1D, neglecting the anisotropy of the energy deposition around the nanoparticle. This could have radiobiological consequences and affect treatment accuracy. Accurate knowledge of the spatial distribution of the dose or other microdosimetric quantities is necessary for predicting radiobiological outcomes using biophysical models. In this study, we utilized a new combination of two Monte Carlo Variance Reduction Techniques (VRTs) to estimate microdosimetric quantities at the nanoscale, taking into account the anisotropy of the microdosimetric distributions. Our results show significant anisotropy in multi-events frequency-mean specific energy, as well as in microdosimetric distributions. These findings highlight the importance of considering anisotropy in microdosimetric distributions when predicting radiobiological outcomes and suggest that current 1D simulations may not accurately represent the real microdosimetric effects. Our work contributes to the development of multiscale dosimetry approaches for radiotherapy treatments with high-Z nanoparticles.

Stochastic search for a parametric cost function approximation: Energy storage with rolling forecasts

Rolling forecasts have been almost overlooked in the renewable energy storage literature. In this paper, we provide a new approach for handling uncertainty not just in the accuracy of a forecast, but in the evolution of forecasts over time. Our approach shifts the focus from modeling the uncertainty in a lookahead model to accurate simulations in a stochastic base model. We develop a robust policy for making energy storage decisions by creating a parametrically modified lookahead model, where the parameters are tuned in the stochastic base model. Since computing unbiased stochastic gradients with respect to the parameters require restrictive assumptions, we propose a simulation-based stochastic approximation algorithm based on numerical derivatives to optimize these parameters. While numerical derivatives, calculated based on the noisy function evaluations, provide biased gradient estimates, an online variance reduction technique built in the framework of our proposed algorithm, will enable us to control the accumulated bias errors and establish the finite-time rate of convergence of the algorithm. Our numerical experiments show the performance of this algorithm in finding policies outperforming the deterministic benchmark policy.

Development of Monte Carlo hybrid parallel algorithm and application in high resolution spatial-energy flux distribution calculation

Monte Carlo transport calculations for fusion reactors are very challenging due to factors such as large radiation decay gradients, complex geometric structures, and strong neutron anisotropy. For such deep penetration shielding calculation problems, in order to obtain accurate global flux distribution, variance reduction methods and parallel computing are necessary means. When using the global weight window variance reduction technique, the huge deviation of the particle arrival weight and the weight window can lead to an extreme number of splits, which is the long history problem. If a process encounters the long history problem in parallel computing and takes a lot of time to complete, other processes will stop computing because the histories allocated in the computing cycle have been completed, seriously degrading computing efficiency. In addition, the calculation of the shutdown dose rate, a key issue in fusion neutronics calculations, requires a high “space-energy” resolution, which brings huge challenges to computational memory, especially in parallel computing. In order to overcome the degradation of parallel efficiency caused by the long history problem and the memory limitation in large-scale shutdown dose rate calculation, a hybrid parallel algorithm based on shared memory was researched. Finally, the hybrid parallel architecture was built in the Monte Carlo code cosRMC to realize dynamic load balancing between threads. Through the memory sharing among all threads of a single node, the pressure of memory consumption can be effectively relieved. In order to verify the effectiveness of the hybrid parallel algorithm developed in this paper, it was applied to the calculation of CFETR. The results showed that the hybrid parallel algorithm can effectively alleviate the impact of the long history problem on the parallel efficiency. In terms of saving memory, the hybrid parallel algorithm also showed a significant effect, which provides a guarantee for large-scale shutdown dose rate calculation.

Achieving Efficiency in Black-Box Simulation of Distribution Tails with Self-Structuring Importance Samplers

Scalable and efficient importance sampling for managing tail risks As the models employed in the realm of risk analytics and optimization become increasingly sophisticated, it is crucial that risk management tools, such as variance reduction techniques, that are typically designed for stylized models on a case by case basis evolve to scale well and gain broader applicability. In the paper titled “Achieving efficiency in black-box simulation of distribution tails with self-structuring importance samplers,” the authors take a step toward this goal by introducing a novel importance sampling scheme for estimating tail risks of objectives modeled with a diverse range of tools, including linear programs, integer linear programs, feature maps specified with neural networks, etc. Instead of explicitly tailoring the change of distribution for each specific model, as conventionally done, the paper identifies an elementary transformation of the samples. This transformation, when applied alike across a wide variety of models, yields a near-optimal reduction in variance for estimating/optimizing over tail expectations.

Monte Carlo Simulation for Trading Under a Lévy-Driven Mean-Reverting Framework

We present a Monte Carlo approach to pairs trading on mean-reverting spreads modelled by Lévy-driven Ornstein-Uhlenbeck processes. Specifically, we focus on using a variance gamma driving process, an infinite activity pure jump process to allow for more flexible models of the price spread than is available in the classical model. However, this generalization comes at the cost of not having analytic formulas, so we apply Monte Carlo methods to determine optimal trading levels and develop a variance reduction technique using control variates. Within this framework, we numerically examine how the optimal trading strategies are affected by the parameters of the model. In addition, we extend our method to bivariate spreads modelled using a weak variance alpha-gamma driving process, and explore the effect of correlation on these trades.

Generation of Optimal Weight Values Based on the Recursive Monte Carlo Method for Use in Monte Carlo Deep Penetration Calculations

In nuclear engineering, Monte Carlo (MC) methods are commonly used for reactor analysis and radiation shielding problems. These methods are capable of dealing with both simple and complex system models with accuracy. The application of MC methods experiences challenges when dealing with the deep penetration problems that are typically encountered in radiation shielding cases. It is difficult to produce statistically reliable results due to poor particle sampling in the region of interest. Therefore, such calculations are performed by the Monte Carlo N-Particle Transport (MCNP) code in association with the weight window (WW) variance reduction technique, which increases the particle statistics in the desired tally region. However, for large problems, MCNP’s built-in weight window generator (WWG) produces zero WW parameters for tally regions located far from the source. To address this issue, the recursive Monte Carlo (RMC) method was proposed. This paper focuses on the RMC methodology and its implementation in the Helmholtz-Zentrum Dresden-Rossendorf’s (HZDR’s) in-house code TRAWEI, which is responsible for producing optimal zone weight parameters used for optimizing deep penetration MC calculations. In addition, this paper discusses the verification of the TRAWEI weight generator program to that of an existing MCNP WWG. The performance of TRAWEI-generated weight values is assessed using a handful of test cases involving two shield materials. Globally, the TRAWEI-generated weight values improved not only the statistical variance and computational efficiency of the MC run compared to the analog MCNP simulation but also those of the simulation with WW values generated by the standard MCNP WWG.

Efficient Personalized PageRank Computation: The Power of Variance-Reduced Monte Carlo Approaches

Personalized PageRank (PPR) computation is a fundamental problem in graph analysis. The state-of-the-art algorithms for PPR computation are based on a bidirectional framework which include a deterministic forward push and a Monte Carlo sampling procedure. The Monte Carlo sampling procedure, however, often has a relatively-large variance, thus reducing the performance of the PPR computation algorithms. To overcome this issue, we develop two novel variance-reduced Monte Carlo techniques for PPR computation. Our first technique is to apply power iterations to reduce the variance of the Monte Carlo sampling procedure. We prove that conducting few power iterations can significantly reduce the variance of existing Monte Carlo estimators, only with few additional costs. Moreover, we show that such a simple and novel variance-reduced Monte Carlo technique can achieve comparable estimation accuracy and the same time complexity as the state-of-the-art bidirectional algorithms. Our second technique is a novel progressive sampling method which uses the historical information of former samples to reduce the variance of the Monte Carlo estimator. We develop several novel PPR computation algorithms by integrating both of these variance reduction techniques with two existing Monte Carlo sampling approaches, including random walk sampling and spanning forests sampling. Finally, we conduct extensive experiments on 5 real-life large graphs to evaluate our solutions. The results show that our algorithms can achieve much higher PPR estimation accuracy by using much less time, compared to the state-of-the-art bidirectional algorithms.

Reliability Estimation for Stochastic Flow Networks With Dependent Arcs

The creation and the destruction processes (CPs and DPs) are the basis of many efficient Monte Carlo methods for estimating the unreliability of highly reliable networks on both, the static and the stochastic flow network models, for the case of independent components. Some of these methods are based on the splitting variance reduction techniques. Due to the splitting basic mechanism, they operate over CP. Here, we propose a splitting-based Monte Carlo method, using the Marshall–Olkin copula for the case of nonindependent components. This proposal operates on the DP, because the Marshall–Olkin copula model over networks is quite related—and somehow similar—to DP, which is unusual but here necessary. Before addressing the proposal, the article presents a review of the methods based on CP and DP. At the end, a comparative experimental analysis shows the efficiency of the proposed approach.

Optimized Monte Carlo simulations for voxel-based internal dosimetry

Objective. The scientific community has considered internal dosimetry by the Monte Carlo method the gold standard. However, there is a trade-off between simulation processing time and the statistical quality of the results that makes it a challenge to obtain accurate absorbed dose values in some situations, such as dose estimation in organs affected by cross-irradiation or limited computing power. Variance reduction techniques are used to reduce computational processing time without impairing the statistical quality of the results, such as tracking energy cutoff, secondary particle production threshold, and parallelism of different types of emissions from radionuclides. Approach. In this work, GATE Monte Carlo code and its variance reduction techniques were evaluated to calculate S values of organs from the international commission on radiological protection (ICRP) report 110 male phantom for the lutetium-177, iodine-131, yttrium-90, and radium-223 radionuclides. The results are compared with the data from the OpenDose collaboration. Main results. A cutoff of 5 MeV for local electron deposition and 2.0 mm of secondary particle production range resulted in a computational efficiency increase of 7.9 and 1.05 times, respectively. Simulation of ICRP 107 spectra-based source proved to be about 5 times more efficient when compared to a decay simulation using G4RadioactiveDecay (Geant4-based radioactive decay processes). Track length estimator (TLE) and split exponential track length estimator (seTLE) techniques were used to calculate the absorbed dose of photon emissions, resulting in computational efficiency up to 29.4 and 62.5 times higher when compared to traditional simulations, respectively. In particular, the seTLE technique accelerates the simulation time by up to 1426 times, achieving a statistical uncertainty of 10% in volumes affected by cross-irradiation. Significance. The variance reduction techniques used in this work drastically reduced the simulation time and maintained the statistical quality of the calculated absorbed dose values, proving the feasibility of the use of the Monte Carlo method in internal dosimetry under challenging situations and making it viable for clinical routine or web applications.

Variance Reduction Techniques Based on Binary Option Pricing

Aiming at the binary option pricing model, control variates and antithetic variates methods are focused on comparing the reasons and influencing factors for the difference of variance reduction effect between them. By adjusting the independent variable of the control variates method and the range of binary options indication function, and by sensitivity analysis of four parameters of the pricing model, the study finds a way to improve the variance reduction ability. Through the detailed mathematical derivation of the antithetic variates method to confirm the law of the 4 parameters in the Black-Scholes model and the binary option price found by the former. And construct more accurate theoretical values to provide better direction for setting options prices in the real world.

Generational Variance Reduction in Monte Carlo Criticality Simulations as a Way of Mitigating Unwanted Correlations

Monte Carlo criticality simulations are widely used in nuclear safety demonstrations, as they offer an arbitrarily precise estimation of global and local tallies while making very few assumptions. However, since the inception of such numerical approaches, it is well known that bias might affect both the estimation of errors on these tallies and the tallies themselves. In particular, stochastic modeling approaches developed in the past decade have shed light on the prominent role played by spatial correlations through a phenomenon called neutron clustering. This effect is particularly of great significance when simulating loosely coupled systems (i.e., with a high dominance ratio). In order to tackle this problem, this paper proposes to recast the power iteration technique of Monte Carlo criticality codes into a variance reduction technique called Adaptive Multilevel Splitting. The central idea is that iterating over neutron generations can be seen as pushing a subpopulation of neutrons toward a generational detector (instead of a spatial detector as variance reduction techniques usually do). While both approaches allow for neutron population control, the former blindly removes or splits neutrons. In contrast, the latter optimizes the spatial, generational, and spectral attributes of neutrons when they are removed or split through an adjoint flux estimation, hence tempering both generational and spatial correlations. This is illustrated in the present paper with a simple case of a bare slab reactor in the one-speed theory on which the Adaptive Multilevel Splitting was applied and compared to variations of the Monte Carlo power iteration method used in neutron transport. Besides looking at the resulting efficiency of the methods, this work also aims to highlight the main mechanisms of the Adaptive Multilevel Splitting in criticality calculations.