Stratification as a General Variance Reduction Method for Markov Chain Monte Carlo
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Mathematics↗
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Results for “Variance reduction”
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Variance Reduction within Implicit Monte Carlo Thermal Radiation Transport using the Local Importance Function Transform [Slides]
System of equations derived by Fleck and Cummings to model time-dependent thermal radiative transfer (TRT) problems: Solution of radiation specific intensity (I), non-linearly coupled to material internal energy (U); Absorption and re-emission approximated by effective scattering events. Useful for high-energy density physics simulations: i.e., astrophysics, inertial confinement fusion (ICF). Whereas deterministic methods are fully discretized and free of statistical noise, Monte Carlo methods allow for dynamic sampling of the phase space at the price of statistical noise; Monte Carlo methods are also characterized by a slow $\frac{1}{√Ν}$ convergence rate.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Weight window variance reduction on GPUs in the Shift Monte Carlo Code
Explore the source record for details and available documents.
Royston, Katherine↗
Moment Tensor Inversion Toolkit
The MTINV toolkit (2002-present) is a collection of computer codes and applications written to invert for the moment tensor of a seismic source given the three components of ground motion recorded at regional seismic stations (e.g., Ichinose et al., 2003). The computer codes and workflow are organized to generate moment tensor solutions for a range of source depths and origin times because of the trade-off between these two quantities. The metric used is the variance reduction and variance reduction modulated by the percent double-couple to determine the best-fit moment-tensor solution. We can solve for a deviatoric moment tensor with a constraint added for no isotropic component although this constraint can be lifted for estimating the full moment tensor like mining collapses or explosion sources.
Ichinose, GeneA↗
Ensemble approximate control variate estimators: Applications to multi-fidelity 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.
97 MATHEMATICS AND COMPUTING↗
A generalized approximate control variate framework for multifidelity uncertainty quantification
We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelerates the estimation of statistics of computationally expensive simulation models using an ensemble of models with lower cost. These lower cost models — which are typically lower fidelity with unknown statistics — are used to reduce the variance in statistical estimators relative to a MC estimator with equivalent cost. We derive the conditions under which our proposed approximate control variate framework recovers existing multifidelity variance reduction schemes as special cases. We demonstrate that existing recursive/nested strategies are suboptimal because they use the additional low-fidelity models only to efficiently estimate the unknown mean of the first low-fidelity model. As a result, they cannot achieve variance reduction beyond that of a control variate estimator that uses a single low-fidelity model with known mean. However, there often exists about an order-of-magnitude gap between the maximum achievable variance reduction using all low-fidelity models and that achieved by a single low-fidelity model with known mean. We show that our proposed approach can exploit this gap to achieve greater variance reduction by using non-recursive sampling schemes. The proposed strategy reduces the total cost of accurately estimating statistics, especially in cases where only low-fidelity simulation models are accessible for additional evaluations. Several analytic examples and an example with a hyperbolic PDE describing elastic wave propagation in heterogeneous media are used to illustrate the main features of the methodology.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Optimizing foreground mitigation for CMB lensing with combined multifrequency and geometric methods
Here, a key challenge for current and upcoming cosmic microwave background lensing measurements is their sensitivity to biases from extragalactic foregrounds, such as Sunyaev-Zel'dovich signals or cosmic infrared background emission. Several methods have been developed to mitigate these lensing foreground biases, dividing broadly into multifrequency cleaning approaches and modifications to the estimator geometry, but how to optimally combine these methods has not yet been explored in detail. In this paper, we examine which combination of lensing foreground mitigation strategies is best able to reduce the impact of foreground contamination for a Simons Observatory-like experiment while preserving maximal signal-to-noise. Although the optimal combination obtained depends on whether bias reduction or variance reduction is prioritized and on whether polarization data is used, generally, we find that combinations involving both geometric (profile hardening, source hardening, or shear) and multifrequency (symmetric cleaning) methods perform best. For lensing power spectrum measurements from temperature (polarization and temperature), our combined estimator methods are able to reduce the bias below σ/4 or 0.3% (0.1%), a factor of 16 (30) lower than the standard quadratic estimator bias, at a modest signal-to-noise cost of only 18% (12%). In contrast, single-method foreground-mitigation approaches struggle to reduce the bias to a negligible level below σ/2 without incurring a large noise penalty. For upcoming and current experiments, our combined methods therefore represent a promising approach for making lensing measurements with negligible foreground bias.
79 ASTRONOMY AND ASTROPHYSICS↗
Quasi-Stochastic Approximation and Off-Policy Reinforcement Learning: Preprint
The Robbins-Monro stochastic approximation algorithm is a foundation of many algorithmic frameworks for reinforcement learning and often an efficient approach to solving (or approximating the solution to) complex optimal control problems. In many cases, however, practitioners are unable to apply these techniques because of an inherent high variance. This paper aims to provide a general foundation for 'quasi-stochastic approximation,' in which all of the processes under consideration are deterministic, much like quasi-Monte Carlo for variance reduction in simulation. The variance reduction can be substantial, subject to tuning of pertinent parameters in the algorithm. This paper introduces a new coupling argument to establish the optimal rate of convergence provided the gain is sufficiently large. These results are established for linear models and tested also in nonideal settings. A major application of these general results is a new class of reinforcement learning algorithms for deterministic state space models. In this setting, the main contribution is a class of algorithms for approximating the value function for a given policy, using a different policy designed to introduce exploration.
24 POWER TRANSMISSION AND DISTRIBUTION↗
A Proof of the Asymptotic Variance of Path Length Estimators for Single-Collision Monte Carlo Source Iteration in the Thick Diffusion Limit
Here, we prove a theorem relating the variance of path length estimators for single-collision Monte Carlo source iteration to a parameter that becomes infinitesimally small in an important physical regime arising in radiative transfer. In our usage, “single-collision Monte Carlo source iteration” refers to Monte Carlo Boltzmann transport methods in which each Monte Carlo particle history includes no more than a single collision, and the physics of multiple scattering is modeled by lagging the scattering source term and iterating until this term converges. Our theorem can be used to construct variance reduction techniques which improve the order of the estimator variance. This enables calculations that would otherwise require impractically large sample sizes to achieve practical estimator uncertainties. We believe this is the first postulation of a theorem relating estimator variance to a limiting case parameter for single-collision Monte Carlo source iteration, and the first proof of such a theorem. We illustrate the theorem’s value with an example in which the authors of a transport method used the theorem to design a variance reduction technique that improved the uncertainty of their solution by a factor of about 500 for a proxy problem from radiative transfer that contains both optically-thick and optically-thin material.
Mathematics and Computing↗
HFIRCON Version 1.0.5 User Guide
The High Flux Isotope Reactor (HFIR) Controller (HFIRCON) code is a collection of python routines and C plugins that automate the workflow for fuel and single- or multicycle target depletion analyses for HFIR at Oak Ridge National Laboratory (ORNL). This code calls the LAVAMINT (LAVA Model Interrogator) parallel (MCNP) Monte Carlo N-Particle model interrogator to stochastically calculate cell volumes and bounding boxes, the ADVANTG (Automatic Variance Reduction Generation) code package for all variance reduction and source biasing calculations, the ORNL-Transformative Neutronics/MCNP5 transport solver for all transport solutions, and the MSX_DEPLETE module to perform all depletion calculations via the ORIGEN (Oak Ridge Isotope Generation) application programming interface. It also performs a robust set of postprocessing functions to automatically provide summaries of several key metrics that are common to a wide variety of typical HFIR design and safety-basis analyses.
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Radiation Dose Modeling for Niowave’s Accelerator Driven Uranium Target Assembly 3
Molybdenum-99 is a high-value radionuclide commonly used for medical purposes within the United States. The National Nuclear Security Administration (NNSA) seeks to reliably produce the radioisotope 99 Mo without the use of highly enriched uranium. NNSA’s Office of Material Management and Minimization (M3) provides funding and government laboratory expertise to private companies to expedite the production process domestically and currently funds designs that use low-enriched uranium or other 99 Mo production pathways. Several production designs are being explored across the industry, including uranium fission and photonuclear conversion of 100 Mo targets. Niowave Inc. seeks to produce 99 Mo via a high-energy electron accelerator that strikes a lead-bismuth eutectic target that ultimately produces a consistent neutron flux. The neutron flux then interacts in a subcritical reactor core configuration to produce fission in low-enriched or natural uranium targets. These fissionable targets are then processed to extract 99 Mo. The purpose of this work is to estimate the neutron and photon dose response across Niowave’s proposed facility for worker safety during operation. Owing to the size of the proposed Niowave facility and necessary shielding, unbiased Monte Carlo radiation transport is impractical, and variance reduction methods are required. This work focuses on the weight window variance reduction method to produce high confidence dose response results within a Monte Carlo radiation transport code. Specifically, an adjoint-informed weight window methodology was created to improve the dose response estimates for accelerator-driven subcritical reactor designs. This adjoint-informed methodology was implemented for Niowave’s proposed design and improved dose results at far-field locations across the facility. Acceptable dose rate contours for the proposed facility were generated across the facility and are presented in this work.
07 ISOTOPE AND RADIATION SOURCES↗
SCALE 6.2.4 Validation: Radiation Shielding
For safe and reliable use of computer codes by the community, accuracy must be clearly evaluated. In particular, the nuclear reactor engineering and licensing field needs accurate tools for radiation shielding modeling. Monaco with Automated Variance Reduction using Importance Calculations (MAVRIC) is one such tool, with built-in variance reduction methods distributed within the SCALE code, and its validity is demonstrated in this report for the released version 6.2.4. Representative benchmarks corresponding to shielding analysis are selected for the validation study. Typical experimental results analyzed from those benchmarks include neutron fluxes, detector count rates, detector energy response functions, neutron and gamma doses, foil neutron activation rates and activities, neutron leakage fluxes, and skyshine dose rates. Thousands of points of comparison between experiment and calculation are presented in this work. Other than rare outliers typically explained by either a lack of information or large uncertainties in the experiment conditions, material, or dimensions, MAVRIC agrees well with the experiment results. MAVRIC is also compared to Monte Carlo N-Particle (MCNP) calculations when available, and both codes generally produce good agreements within estimated uncertainties. The selected benchmarks are obtained from reliable sources such as the International Criticality Safety Benchmark Evaluation Project Handbook (ICSBEP Handbook), the Shielding Integral Benchmark Archive & Database (SINBAD), and other shielding validation work found in the literature. Additional datapoints and benchmarks will be added to future versions of this report to incrementally expand the shielding validation suite incrementally.
61 RADIATION PROTECTION AND DOSIMETRY↗
SCALE 6.3 Validation: Radiation Shielding
Safe and reliable use of scientific and engineering computer codes requires validation for the types of applications in which they will be used. An example in the nuclear reactor engineering and licensing field is radiation transport employed in shielding analyses. The validity of computer codes for shielding applications is demonstrated in this report for SCALE version 6.3.0. Representative benchmarks corresponding to shielding analyses are selected for the validation study. Typical measurement results analyzed from these benchmarks include neutron fluxes, detector count rates, detector energy response functions, neutron and gamma dose rates, neutron activation rates and activities, neutron leakage fluxes, and skyshine dose rates. Thousands of points of comparison between measurement and calculation are presented in this work. Other than rare outliers typically explained by either a lack of information or large uncertainties in the experiment conditions, material, or dimensions, the Monaco with Automated Variance Reduction using Importance Calculations (MAVRIC) radiation transport computer code with built-in variance reduction methods distributed with the SCALE computer code system agrees well with the measurement results. In selected benchmarks, MAVRIC is also compared to Monte Carlo N- Particle® (MCNP® ) 1 calculations. Both computer codes generally agree well within the estimated uncertainties. With the release of SCALE 6.3.0, Shift was integrated as an alternative transport solver in MAVRIC, denoted MAVRIC-Shift. Although the traditional MAVRIC using Monaco was used primarily in this validation study, many results have also been generated using MAVRIC-Shift. Agreement between MAVRIC-Monaco and MAVRIC-Shift is generally very good. The benchmarks presented in this report were obtained from reliable sources such as the International Criticality Safety Benchmark Evaluation Project Handbook, the Shielding Integral Benchmark Archive & Database, and other shielding validation work found in the literature. Additional datapoints and benchmarks will be added to future versions of this report to expand the shielding validation suite.
61 RADIATION PROTECTION AND DOSIMETRY↗
Toward quantum Monte Carlo forces on heavier ions: Scaling properties
Quantum Monte Carlo (QMC) forces have been studied extensively in recent decades because of their importance with spectroscopic observables and geometry optimization. Here, we benchmark the accuracy and computational cost of QMC forces. The zero-variance zero-bias (ZVZB) force estimator is used in standard variational and diffusion Monte Carlo simulations with mean-field based trial wavefunctions and atomic pseudopotentials. Statistical force uncertainties are obtained with a recently developed regression technique for heavy tailed QMC data [P. Lopez Rios and G. J. Conduit, Phys. Rev. E 99, 063312 (2019)]. By considering selected atoms and dimers with elements ranging from H to Zn (1 ≤ Z eff ≤ 20), we assess the accuracy and the computational cost of ZVZB forces as the effective pseudopotential valence charge, Z eff , increases. We find that the costs of QMC energies and forces approximately follow simple power laws in Zeff. The force uncertainty grows more rapidly, leading to a best case cost scaling relationship of approximately Z e ff 6.5 ( 3 ) for diffusion Monte Carlo. We find that the accessible system size at fixed computational cost scales as Z e ff − 2 , insensitive to model assumptions or the use of the “space warp” variance-reduction technique. Our results predict the practical cost of obtaining forces for a range of materials, such as transition metal oxides where QMC forces have yet to be applied, and underscore the importance of further developing force variance-reduction techniques, particularly for atoms with high Z eff .
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
GANISP: A GAN-Assisted Importance Splitting Probability Estimator
Designing manufacturing processes with high yield and strong reliability relies on effective methods for rare event estimation. Genealogical importance splitting reduces the variance of rare event probability estimators by iteratively selecting and replicating realizations that are headed towards a rare event. The replication step is difficult when applied to deterministic systems where the initial conditions of the offspring realizations need to be modified. Typically, a random perturbation is applied to the offspring to differentiate their trajectory from the parent realization. However, this random perturbation strategy may be effective for some systems while failing for others, preventing variance reduction in the probability estimate. This work seeks to address this limitation using a generative model such as a Generative Adversarial Network (GAN) to generate perturbations that are consistent with the attractor of the dynamical system. The proposed GAN-assisted Importance SPlitting method (GANISP) improves the variance reduction for the system targeted. An implementation of the method is available in a companion repository (https://github.com/NREL/GANISP).
generative adversarial network↗
GANISP: A GAN-Assisted Importance Splitting Probability Estimator: Preprint
To reduce the variance of rare event probability estimator, genealogical importance splitting marches towards a rare event by iteratively selecting and replicating realizations that are headed towards a rare event. The replication step is made difficult when applied to deterministic systems as the initial conditions of the offspring realizations need to be modified. Typically, a random perturbation is applied to the offspring to differentiate their trajectory from the parent realization. It is shown that a random perturbation strategy may be effective for some systems but may also fail for others, thereby preventing variance reduction in the probability estimate. To address this limitation, it is proposed to use a generative model such as a Generative Adversarial Network (GAN) to generate perturbations that are consistent with the attractor of the dynamical system. The GAN-assisted Importance SPlitting method (GANISP) improves the variance reduction for the system targeted. An implementation of the method is available in a companion repository (https://github.com/NREL/GANISP).
generative adversarial network↗
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 60 Co 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. Here 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.
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
FedOSAA: Improving Federated Learning with One-Step Anderson Acceleration
Federated learning (FL) is a distributed machine learning approach that enables multiple local clients and a central server to collaboratively train a model while keeping the data on their own devices. First-order methods, particularly those incorporating variance reduction techniques, are the most widely used FL algorithms due to their simple implementation and stable performance. However, these methods tend to be slow and require a large number of communication rounds to reach the global minimizer. We propose FedOSAA, a novel approach that preserves the simplicity of first-order methods while achieving the rapid convergence typically associated with second-order methods. Our approach applies one Anderson acceleration (AA) step following classical local updates based on first-order methods with variance reduction, such as FedSVRG and SCAFFOLD, during local training. This AA step is able to leverage curvature information from the history points and gives a new update that approximates the Newton-GMRES direction, thereby significantly improving the convergence. We establish a local linear convergence rate to the global minimizer of FedOSAA for smooth and strongly convex loss functions. Numerical comparisons show that FedOSAA substantially improves the communication and computation efficiency of the original first-order methods, achieving performance comparable to second-order methods like GIANT.
Feng, Xue [University of California, Davis]↗