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Results for “Monte Carlo Sampling”

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NASA NTRS1985

Large-cell Monte Carlo renormalization of irreversible growth processes

Monte Carlo sampling is applied to a recently formulated direct-cell renormalization method for irreversible, disorderly growth processes. Large-cell Monte Carlo renormalization is carried out for various nonequilibrium problems based on the formulation dealing with relative probabilities. Specifically, the method is demonstrated by application to the 'true' self-avoiding walk and the Eden model of growing animals for d = 2, 3, and 4 and to the invasion percolation problem for d = 2 and 3. The results are asymptotically in agreement with expectations; however, unexpected complications arise, suggesting the possibility of crossovers, and in any case, demonstrating the danger of using small cells alone, because of the very slow convergence as the cell size b is extrapolated to infinity. The difficulty of applying the present method to the diffusion-limited-aggregation model, is commented on.

Nakanishi, H.↗

DOE OSTI2015

Generalized correlation integral vectors: A distance concept for chaotic dynamical systems

Several concepts of fractal dimension have been developed to characterise properties of attractors of chaotic dynamical systems. Numerical approximations of them must be calculated by finite samples of simulated trajectories. In principle, the quantities should not depend on the choice of the trajectory, as long as it provides properly distributed samples of the underlying attractor. In practice, however, the trajectories are sensitive with respect to varying initial values, small changes of the model parameters, to the choice of a solver, numeric tolerances, etc. The purpose of this paper is to present a statistically sound approach to quantify this variability. We modify the concept of correlation integral to produce a vector that summarises the variability at all selected scales. The distribution of this stochastic vector can be estimated, and it provides a statistical distance concept between trajectories. Here, we demonstrate the use of the distance for the purpose of estimating model parameters of a chaotic dynamic model. The methodology is illustrated using computational examples for the Lorenz 63 and Lorenz 95 systems, together with a framework for Markov chain Monte Carlo sampling to produce posterior distributions of model parameters.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2015

Bayesian Inference in Risk Assessment (P-102)

This report includes the following sections: Section 1: Course Topics •Section 2: Review of Basic Probability Calculations •Section 3: Introduction to Bayesian Inference •Section 4: Introduction to Monte Carlo Sampling •Section 5: Uncertainty Propagation in Risk Assessment

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2014

Comparison of the Results of MISSE 6 Atomic Oxygen Erosion Yields of Layered Kapton H Films with Monte Carlo Computational Predictions

A space experiment flown as part of the Materials International Space Station Experiment 6B (MISSE 6B) was designed to compare the atomic oxygen erosion yield (Ey) of layers of Kapton H polyimide with no spacers between layers with that of layers of Kapton H with spacers between layers. The results were compared to a solid Kapton H (DuPont, Wilmington, DE) sample. Monte Carlo computational modeling was performed to optimize atomic oxygen interaction parameter values to match the results of both the MISSE 6B multilayer experiment and the undercut erosion profile from a crack defect in an aluminized Kapton H sample flown on the Long Duration Exposure Facility (LDEF). The Monte Carlo modeling produced credible agreement with space results of increased Ey for all samples with spacers as well as predicting the space-observed enhancement in erosion near the edges of samples due to scattering from the beveled edges of the sample holders.

Oxygen atoms↗

DOE OSTI2025

Personalized and uncertainty-aware coronary hemodynamics simulations: From Bayesian estimation to improved multi-fidelity uncertainty quantification

Non-invasive simulations of coronary hemodynamics have improved clinical risk stratification and treatment outcomes for coronary artery disease, compared to relying on anatomical imaging alone. However, simulations typically use empirical approaches to distribute total coronary flow amongst the arteries in the coronary tree, which ignores patient variability, the presence of disease, and other clinical factors. Further, uncertainty in the clinical data often remains unaccounted for in the modeling pipeline. We present an end-to-end uncertainty-aware pipeline to (1) personalize coronary flow simulations by incorporating vessel-specific coronary flows as well as cardiac function; and (2) predict clinical and biomechanical quantities of interest with improved precision, while accounting for uncertainty in the clinical data. We assimilate patient-specific measurements of myocardial blood flow from clinical CT myocardial perfusion imaging to estimate branch-specific coronary artery flows. Simulated noise in the clinical data is used to estimate the joint posterior distributions of the model parameters using adaptive Markov Chain Monte Carlo sampling. Additionally, the posterior predictive distribution for the relevant quantities of interest is determined using a new approach combining multi-fidelity Monte Carlo estimation with non-linear, data-driven dimensionality reduction. This leads to improved correlations between high- and low-fidelity model outputs. Our framework accurately recapitulates clinically measured cardiac function as well as branch-specific coronary flows under measurement noise uncertainty. We observe substantial reductions in confidence intervals for estimated quantities of interest compared to single-fidelity Monte Carlo estimation and state-of-the-art multi-fidelity Monte Carlo methods. This holds especially true for quantities of interest that showed limited correlation between the low- and high-fidelity model predictions. In addition, the proposed multi-fidelity Monte Carlo estimators are significantly cheaper to compute than traditional estimators, under a specified confidence level or variance. The proposed pipeline for personalized and uncertainty-aware predictions of coronary hemodynamics is based on routine clinical measurements and recently developed techniques for CT myocardial perfusion imaging. The proposed pipeline offers significant improvements in precision and reduction in computational cost.

Bayesian parameter estimation↗

DOE OSTI2019

A Monte Carlo Rectangle Packing Algorithm for Identifying Likely Spatial Distributions of Final Closure Cap Subsidence in the E-Area Low-Level Waste Facility

A Monte Carlo rectangle packing algorithm, summarized in Figure ES-1, has been developed to provide insights into the likely spatial distribution of subsided areas or holes in the final closure cap for the E-Area Low-Level Waste Facility (ELLWF) resulting from the placement of non-crushable waste containers within slit and engineered trench disposal units. Using historical records provided by Solid Waste Management as an indicator of future waste streams, a probability distribution of unique waste containers was constructed, and Monte Carlo sampling was used to sequentially select waste containers for placement in a rectangular region bounded by the disposal unit footprint. Within this Monte Carlo approach, the probability of any container type being selected is weighted by the fractional abundance relative to all other container types. For each Monte Carlo realization, the frequency of non-crushable waste containers appearing at each location on a spatially discretized mesh is updated. After many realizations, the location(s) with the highest frequencies of non-crushable waste containers are regarded as the most likely locations to experience subsidence (i.e., for a hole to form). Utilizing this approach has enabled the down-selection of a computationally intractable set of subsidence hole combinations to be explored in three-dimensional finite element groundwater flow and contaminant transport models. Such an approach can be applied to a number of different packaging problems.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2015

Theoretically informed Monte Carlo simulation of liquid crystals by sampling of alignment-tensor fields

A theoretically informed coarse-grained Monte Carlo method is proposed for studying liquid crystals. The free energy functional of the system is described in the framework of the Landau-de Gennes formalism. The alignment field and its gradients are approximated by finite differences, and the free energy is minimized through a stochastic sampling technique. The validity of the proposed method is established by comparing the results of the proposed approach to those of traditional free energy minimization techniques. Its usefulness is illustrated in the context of three systems, namely, a nematic liquid crystal confined in a slit channel, a nematic liquid crystal droplet, and a chiral liquid crystal in the bulk. It is found that for systems that exhibit multiple metastable morphologies, the proposed Monte Carlo method is generally able to identify lower free energy states that are often missed by traditional approaches. Importantly, the Monte Carlo method identifies such states from random initial configurations, thereby obviating the need for educated initial guesses that can be difficult to formulate.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2013

RAVEN as a tool for dynamic probabilistic risk assessment: Software overview

RAVEN is a software tool under development at the Idaho National Laboratory (INL) that acts as the control logic driver and post-processing tool for the newly developed Thermal-Hydraulic code RELAP-7. The scope of this paper is to show the software structure of RAVEN and its utilization in connection with RELAP-7. A short overview of the mathematical framework behind the code is presented along with its main capabilities such as on-line controlling/ monitoring and Monte-Carlo sampling. A demo of a Station Black Out PRA analysis of a simplified Pressurized Water Reactor (PWR) model is shown in order to demonstrate the Monte-Carlo and clustering capabilities. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

NASA NTRS2014

Engineering Risk Assessment of Space Thruster Challenge Problem

The Engineering Risk Assessment (ERA) team at NASA Ames Research Center utilizes dynamic models with linked physics-of-failure analyses to produce quantitative risk assessments of space exploration missions. This paper applies the ERA approach to the baseline and extended versions of the PSAM Space Thruster Challenge Problem, which investigates mission risk for a deep space ion propulsion system with time-varying thruster requirements and operations schedules. The dynamic mission is modeled using a combination of discrete and continuous-time reliability elements within the commercially available GoldSim software. Loss-of-mission (LOM) probability results are generated via Monte Carlo sampling performed by the integrated model. Model convergence studies are presented to illustrate the sensitivity of integrated LOM results to the number of Monte Carlo trials. A deterministic risk model was also built for the three baseline and extended missions using the Ames Reliability Tool (ART), and results are compared to the simulation results to evaluate the relative importance of mission dynamics. The ART model did a reasonable job of matching the simulation models for the baseline case, while a hybrid approach using offline dynamic models was required for the extended missions. This study highlighted that state-of-the-art techniques can adequately adapt to a range of dynamic problems.

Assessment↗

DOE OSTI2014

Likelihood-informed dimension reduction for nonlinear inverse problems

The intrinsic dimensionality of an inverse problem is affected by prior information, the accuracy and number of observations, and the smoothing properties of the forward operator. From a Bayesian perspective, changes from the prior to the posterior may, in many problems, be confined to a relatively low-dimensional subspace of the parameter space. We present a dimension reduction approach that defines and identifies such a subspace, called the 'likelihood-informed subspace' (LIS), by characterizing the relative influences of the prior and the likelihood over the support of the posterior distribution. This identification enables new and more efficient computational methods for Bayesian inference with nonlinear forward models and Gaussian priors. In particular, we approximate the posterior distribution as the product of a lower-dimensional posterior defined on the LIS and the prior distribution marginalized onto the complementary subspace. Markov chain Monte Carlo sampling can then proceed in lower dimensions, with significant gains in computational efficiency. We also introduce a Rao–Blackwellization strategy that de-randomizes Monte Carlo estimates of posterior expectations for additional variance reduction. We demonstrate the efficiency of our methods using two numerical examples: inference of permeability in a groundwater system governed by an elliptic PDE, and an atmospheric remote sensing problem based on Global Ozone Monitoring System (GOMOS) observations.

42 ENGINEERING↗

DOE OSTI2025

Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker–Planck Equations

The Fokker-Planck (FP) equation is a foundational partial differential equation (PDE) in stochastic processes involving Brownian motions. However, the curse of dimensionality (CoD) poses a formidable challenge when dealing with high-dimensional FP equations. Although Monte Carlo simulation and (vanilla) Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit significant numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimensionality increases, surpassing the precision of numerical simulations and resulting in substantial errors. In addition, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a notoriously difficult Hamilton-Jacobi-Bellman (HJB) equation, which is impractical for PINN learning, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in stochastic differential equations (SDEs). The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling, offering an effective means to overcome the CoD. Three fitting methods, Score Matching (SM), Sliced Score Matching (SSM), and Score-PINN, are introduced, each contributing unique advantages in computational complexity, accuracy, and generality. The proposed score-based SDE solver operates in two stages: first, employing score matching or Score-PINN to acquire the score function; and second, solving the LL via an ordinary differential equation (ODE) using the obtained score function. Comparative evaluations across these methods showcase varying trade-offs. The proposed methodology is evaluated across diverse SDEs, including anisotropic Ornstein-Uhlenbeck processes, geometric Brownian motion, and Brownian motion with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy distributions. The numerical results demonstrate the score-based SDE solver’s stability, speed, and performance across different experimental settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

A D-term Modeling Code (DMC) for Simultaneous Calibration and Full-Stokes Imaging of Very Long Baseline Interferometric Data

In this paper we present DMC, a model and associated tool for polarimetric imaging of very long baseline interferometry data sets that simultaneously reconstructs the full-Stokes emission structure along with the station-based gain and leakage calibration terms. DMC formulates the imaging problem in terms of posterior exploration, which is achieved using Hamiltonian Monte Carlo sampling. The resulting posterior distribution provides a natural quantification of uncertainty in both the image structure and the data calibration. We run DMC on both synthetic and real data sets, the results of which demonstrate its ability to accurately recover both the image structure and calibration quantities, as well as to assess their corresponding uncertainties. The framework underpinning DMC is flexible, and its specific implementation is under continued development.

47 OTHER INSTRUMENTATION↗

DOE OSTI2020

Effects of cluster expansion on the locations of phase transition boundary as a first step to quantify uncertainty in first principles statistical mechanics framework

Predicting phase diagrams from first principle calculations eliminates the need of tedious experimental trials and errors. Fully automating first principle phase diagram calculations without any sort of human intervention has been a long daunting task and troubling scientific communities for decades. This grand problem remains not fully resolved, largely due to the vastly high-dimensional parameter space associated with density functional theory, cluster expansion, lattice Monte Carlo, and the substantial uncertainty propagating through a set of complex simulations. As a first step to tackle this grand problem, we reported a first demonstration of how sensitive phase boundary locations can be to various cluster expansion fittings and input DFT training data. To the best knowledge of the authors, this study reported the first ever attempt to quantify uncertainty in first principles statistical mechanics framework. In addition, a semi-automated phase transition detection algorithm has been devised in this paper to deal with the associated statistical errors and uncertainties from Monte Carlo method and its predecessor DFT calculations and cluster expansions. This algorithm has been applied in a classical cluster expansion Mg-Cd binary alloy system to detect phase transitions at various locations of phase diagram using different cluster expansions to demonstrate its predictive power and quantify uncertainties. The results suggested that using Chebyshev basis function shifted the transition locations toward the dilute solid solution phase and expanded the phase stability of concentrated ordered phases to wider composition range. The addition of perturbed defect configurations into training data set lowered the order-disorder transition temperature to be away from true transition temperature, suggesting the transition nature is indeed configurational disorder dominated rather than defect assisted. Finally, we found that the weighting has negligible effect on the transition locations, except for the case of using Chebyshev basis function to fit all non-weighted configurations that can be susceptible to Monte Carlo sampling hysteresis.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Quantifying electron temperature distributions from time-integrated x-ray emission spectra

K-shell x-ray emission spectroscopy is a standard tool used to diagnose the plasma conditions created in high-energy-density physics experiments. In the simplest approach, the emissivity-weighted average temperature of the plasma can be extracted by fitting an emission spectrum to a single temperature condition. It is known, however, that a range of plasma conditions can contribute to the measured spectra due to a combination of the evolution of the sample and spatial gradients. In this work, we define a parameterized model of the temperature distribution and use Markov Chain Monte Carlo sampling of the input parameters, yielding uncertainties in the fit parameters to assess the uniqueness of the inferred temperature distribution. Here we present the analysis of time-integrated S and Fe x-ray spectroscopic data from the Orion laser facility and demonstrate that while fitting each spectral region to a single temperature yields two different temperatures, both spectra can be fit simultaneously with a single temperature distribution. We find that fitting both spectral regions together requires a maximum temperature of $1310^{+90}_{-70}$ eV with significant contributions from temperatures down to 200 eV.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems

We propose an ensemble score filter (EnSF) for solving high-dimensional nonlinear filtering problems with superior accuracy. A major drawback of existing filtering methods, e.g., particle filters or ensemble Kalman filters, is the low accuracy in handling high-dimensional and highly nonlinear problems. EnSF attacks this challenge by exploiting the score-based diffusion model, defined in a pseudo-temporal domain, to characterizing the evolution of the filtering density. EnSF stores the information of the recursively updated filtering density function in the score function, in stead of storing the information in a set of finite Monte Carlo samples (used in particle filters and ensemble Kalman filters). Unlike existing diffusion models that train neural networks to approximate the score function, we develop a training-free score estimation that uses mini-batch-based Monte Carlo estimator to directly approximate the score function at any pseudo-spatial-temporal location, which provides sufficient accuracy in solving high-dimensional nonlinear problems as well as saves tremendous amount of time spent on training neural networks. Another essential aspect of EnSF is its analytical update step, gradually incorporating data information into the score function, which is crucial in mitigating the degeneracy issue faced when dealing with very high-dimensional nonlinear filtering problems. High-dimensional Lorenz systems are used to demonstrate the performance of our method. EnSF provides surprisingly impressive performance in reliably tracking extremely high-dimensional Lorenz systems (up to 1,000,000 dimension) with highly nonlinear observation processes, which is a well-known challenging problem for existing filtering methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2015

Impact of correlated data in validation procedures - 14483

The increasing demand on accuracy of criticality calculations leads to the question, how to treat correlated data of benchmark experiments correctly. Correlations in the process of validation arise for example if a configuration in a series of experiments shares certain components, e.g. fuel rods. Traditional methods of code validation do not treat correlations explicitly, but give a more conservative estimation of the bias. Including correlations can lead to a more precise estimation. In the work at hand, a method is discussed based on Bayes theorem to include correlations in the experimental data when estimating an application case k{sub eff}. After a short introduction to the applied method and the demonstration of the general capability of it to predict the k{sub eff} correctly, it is shown how the calculated k{sub eff} of an application case depends on correlations in experimental data for a simple Toy Model. The impact of ignoring correlations and taking to account lower and higher correlations is shown and discussed. Further results are shown using a database of correlated experimental data from the International Handbook of Criticality Safety Experiments to estimate the bias of two application cases. The experiments were chosen following the suggestion of the OECD/NEA EGUACSA Benchmark phase IV and consists of 21 experiments from the LEU-COMP-THERM series 007 and 039. With the extracted data results the calculated k{sub eff} values of the application cases are shown and discussed for different examples of correlations. The calculations were done by applying Monte-Carlo sampling methods with the GRS tool SUnCISTT, using ORNL's SCALE 6.1.2. This work discusses a possible way of treating correlations in the recalculation of critical experiments to get a more precise bias estimation of the used code system. It will be shown, that the presented method is capable to treat similarities in experimental setups of data used to calculate the bias correctly, independent of the size of similarities. The presented method is in principle capable of assessing regimes with little experimental data to estimate a bias. (authors)

42 ENGINEERING↗

DOE OSTI2022

Atomic structure of an FeCrMoCBY metallic glass revealed by high energy x-ray diffraction

Amorphous bulk metallic glasses with the composition Fe 48 Cr 15 Mo 14 C 15 B 6 Y 2 have been of interest due to their special mechanical and electronic properties, including corrosion resistance, high yield-strength, large elasticity, catalytic performance, and soft ferromagnetism. Here, we apply a reverse Monte Carlo technique to unravel the atomic structure of these glasses. The pair-distribution functions for various atomic pairs are computed based on the high-energy x-ray diffraction data we have taken from an amorphous sample. Monte Carlo cycles are used to move the atomic positions until the model reproduces the experimental pair-distribution function. The resulting fitted model is consistent with our ab initio simulations of the metallic glass. Our study contributes to the understanding of functional properties of Fe-based bulk metallic glasses driven by disorder effects.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Sign Problem in Tensor-Network Contraction

We investigate how the computational difficulty of contracting tensor networks depends on the sign structure of the tensor entries. Using results from computational complexity, we observe that the approximate contraction of tensor networks with only positive entries has lower computational complexity as compared to tensor networks with general real or complex entries. This raises the question of how this transition in computational complexity manifests itself in the hardness of different tensor-network-contraction schemes. We pursue this question by studying random tensor networks with varying bias toward positive entries. First, we consider contraction via Monte Carlo sampling and find that the transition from hard to easy occurs when the tensor entries become predominantly positive; this can be understood as a tensor-network manifestation of the well-known negative-sign problem in quantum Monte Carlo. Second, we analyze the commonly used contraction based on boundary tensor networks. The performance of this scheme is governed by the number of correlations in contiguous parts of the tensor network (which by analogy can be thought of as entanglement). Remarkably, we find that the transition from hard to easy—i.e., from a volume-law to a boundary-law scaling of entanglement—already occurs for a slight bias of the tensor entries toward a positive mean, scaling inversely with the bond dimension D , and thus the problem becomes easy the earlier the larger D occurs. This is in contrast both to expectations and to the behavior found in Monte Carlo contraction, where the hardness at fixed bias increases with the bond dimension. To provide insight into this early breakdown of computational hardness and the accompanying entanglement transition, we construct an effective classical statistical-mechanical model that predicts a transition at a bias of the tensor entries of 1 / D , confirming our observations. We conclude by investigating the computational difficulty of computing expectation values of tensor-network wave functions (projected entangled-pair states, PEPSs) and find that in this setting, the complexity of entanglement-based contraction always remains low. We explain this by providing a local transformation that maps PEPS expectation values to a positive-valued tensor network. This not only provides insight into the origin of the observed boundary-law entanglement scaling but also suggests new approaches toward PEPS contraction based on positive decompositions. Published by the American Physical Society 2025

Chen, Jielun (ORCID:0000000178411545)↗