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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2013

A stochastic diffusion process for Lochner's generalized Dirichlet distribution

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner’s generalized Dirichlet distribution as its asymptotic solution. Individual samples of a discrete ensemble, obtained from the system of stochastic differential equations, equivalent to the Fokker-Planck equation developed here, satisfy a unit-sum constraint at all times and ensure a bounded sample space, similarly to the process developed in for the Dirichlet distribution. Consequently, the generalized Dirichlet diffusion process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Compared to the Dirichlet distribution and process, the additional parameters of the generalized Dirichlet distribution allow a more general class of physical processes to be modeled with a more general covariance matrix.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Controlling radioisotope proportions when randomly sampling from Dirichlet distributions in PyRIID

As machine learning models for radioisotope quantification become more powerful, likewise the need for high-quality synthetic training data grows as well. For problem spaces that involve estimating the relative isotopic proportions of various sources in gamma spectra it is necessary to generate training data that accurately represents the variance of proportions encountered. In this report, we aim to provide guidance on how to target a desired variance of proportions which are randomly when using the PyRIID Seed Mixer, which samples from a Dirichlet distribution. We provide a method for properly parameterizing the Dirichlet distribution in order to maintain a constant variance across an arbitrary number of dimensions, where each dimension represents a distinct source template being mixed. We demonstrate that our method successfully parameterizes the Dirichlet distribution to target a specific variance of proportions, provided that several conditions are met. This allows us to follow a principled technique for controlling how random mixture proportions are generated which are then used downstream in the synthesis process to produce the final, noisy gamma spectra.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2013

A Stochastic Diffusion Process for the Dirichlet Distribution

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

ATOMIC Simulations and Experimental Data for Basalt-like Compounds

This data consists of simulations and experimental measurements of laser-induced breakdown spectroscopy (LIBS). The simulations are produced by ATOMIC, a general purpose plasma modeling and kinetics code that has been designed to compute emission (or absorption) spectra from plasmas [2]. The makeup of the plasma was considered to be divided into some proportion water, some proportion Martian atmosphere (CO2), and some proportion target (from the rock or object impacted by the laser), where these proportions add to 1. Based on expert knowledge, the proportion of water was kept in the range [0.0,0.5] and the proportion of atmosphere was kept in the range [0.02, 0.9]. Our overall suite of simulations contains six sets of simulations that differ in which elements were considered to make up the target. Within each set, we used uniformly drawn temperatures and log mass densities within pre-specified ranges. The temperature range was [0.5,1.5] eV and the log (base 10) mass density range was [-7,-4]. The proportion of water, atmosphere, and target were drawn from a symmetric Dirichlet distribution, but draws in which the propor- tion of water or atmosphere exceeded the pre-specified limits were rejected from the design. Up to eleven constituent elements (Si, Al, Fe, Mg, Ca, O, Ti, Mn, Na, K, P) were considered for the target, as they are the most common elements found in basalt compounds and were used in [1]. For each run, the proportions of the constituent elements making up the target were drawn from a symmetric Dirichlet distribution. We ran 1,350 simulations that included nonzero proportions of all eleven elements. We also ran simulations which excluded some of these elements. In particular, we ran 1,000 simulations that only included nonzero proportions for the six most common elements (Si, Al, Fe, Mg, Ca, O). We also ran five sets, each with 500 simulations, that only included nonzero proportions for five of the six most common elements (but where all sets included O). Thus, we generated a total of 4,850 spectra representing basalt-like compounds in which the target was comprised of oxygen and between four and ten other elements. The ATOMIC code produced spectra over a range of 240nm - 880nm that roughly mimics the range collected by the ChemCam instrument on the Mars rover Curiosity. Each spectra had 32,000 wavelengths split across three spectrometer ranges (to mimic ChemCam). The experimental data, described in [1], measures a prepared basalt sample. All files are kept in directories whose names indicate the set of elements considered for the target with file names numbered to indicate the line in the design files used to produce the simulation. The designs are provided as text files with names indicating their purpose. The experimental data is provided as a CSV file which contains a header with measurement information, followed by a collection of 50 shots across a collection of wavelengths, along with the computed median and mean across shots.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2015

Variational Dirichlet Blur Kernel Estimation

Blind image deconvolution involves two key objectives: 1) latent image and 2) blur estimation. For latent image estimation, we propose a fast deconvolution algorithm, which uses an image prior of nondimensional Gaussianity measure to enforce sparsity and an undetermined boundary condition methodology to reduce boundary artifacts. For blur estimation, a linear inverse problem with normalization and nonnegative constraints must be solved. However, the normalization constraint is ignored in many blind image deblurring methods, mainly because it makes the problem less tractable. In this paper, we show that the normalization constraint can be very naturally incorporated into the estimation process by using a Dirichlet distribution to approximate the posterior distribution of the blur. Making use of variational Dirichlet approximation, we provide a blur posterior approximation that considers the uncertainty of the estimate and removes noise in the estimated kernel. Furthermore, experiments with synthetic and real data demonstrate that the proposed method is very competitive to the state-of-the-art blind image restoration methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Basin-Size Mapping: Prediction of Metastable Polymorph Synthesizability Across TaC–TaN Alloys

The sizes of the basins of attraction on the potential energy surface are helpful indicators in determining the experimental synthesizability of metastable phases. In principle, these basins can be controlled with changes in thermodynamic conditions such as composition, pressure, and surface energy. Herein, we use random structure sampling to computationally study how alloying smoothly perturbs basin of attraction sizes. The TaC 1-x N x pseudobinary is an ideal test system given the structural and polymorphic contrast of its parent compounds and their technological relevance as epitaxial substrates for Al 1-x Ga x N. While we find limited thermodynamic stability across all computationally observed phases, random structure sampling shows a significant composition region where the rocksalt basin dominates. As such, we predict the potential for the nonequilibrium synthesis of metastable rocksalt TaC 1-x N x alloys as substrates for Al 1-x Ga x N. At higher nitrogen concentrations, other low-energy metastable polymorphs emerge that continue to retain the hexagonal close packing suitable for III-N growth. Confidence in these trends was established through uncertainty quantification of the basin sizes and energy distributions; such analysis utilized the Beta and Dirichlet distributions. In conclusion, we also find (a) polymorph basin sizes can be rationalized in terms of energetic preferences for different coordination environments; and (b) basin sizes universally shrink with increasing nitrogen content, making the system more prone to amorphous growth.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Direct numerical simulations of three-component Rayleigh–Taylor mixing and an improved model for multicomponent reacting mixtures

We present direct numerical simulations of a three-layer Rayleigh–Taylor instability (RTI) problem with a configuration based on the experiments of Suchandra & Ranjan ( J. Fluid Mech. , vol. 974, 2023, A35) and Jacobs & Dalziel ( J. Fluid Mech. , vol. 542, 2005, pp. 251–279). The problem consists of a layer of light fluid between two layers of heavy fluid with an Atwood number of 0.3. These simulations are first validated through comparison with available experimental data. The validated simulations are then utilized to analyse statistics in this three-component flow. First, length scales are examined utilizing spectra and two-point spatial correlations of velocity and species concentration fluctuations. Next, joint probability density functions (p.d.f.s) of species concentration are compared against several model p.d.f.s representing generalizations of the bivariate beta distribution. Notably, the joint p.d.f.s do not appear to be accurately described by a Dirichlet distribution, indicating the marginal distributions do not conform to a beta distribution. Finally, similarity of the present configuration to three-component mixing found in inertial confinement fusion (ICF) applications is exploited to develop and validate an improved model for the impact of multicomponent mixing on thermonuclear (TN) reaction rates. A single time instant from the present simulations is chosen for a TN burn calculation under the hypothetical assumption of ICF materials and temperatures. Total TN output from this second calculation is then compared against the prediction of the improved model. The new model is found to accurately predict TN reaction rates in both premixed and non-premixed configurations.

42 ENGINEERING↗
DOE OSTI2023

Local Bayesian Dirichlet mixing of imperfect models

Abstract To improve the predictability of complex computational models in the experimentally-unknown domains, we propose a Bayesian statistical machine learning framework utilizing the Dirichlet distribution that combines results of several imperfect models. This framework can be viewed as an extension of Bayesian stacking. To illustrate the method, we study the ability of Bayesian model averaging and mixing techniques to mine nuclear masses. We show that the global and local mixtures of models reach excellent performance on both prediction accuracy and uncertainty quantification and are preferable to classical Bayesian model averaging. Additionally, our statistical analysis indicates that improving model predictions through mixing rather than mixing of corrected models leads to more robust extrapolations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Adaptive mitigation of time-varying quantum noise

Current quantum computers suffer from non-stationary noise channels with high error rates, which undermines their reliability and reproducibility. We propose a Bayesian inference based adaptive algorithm that can learn and mitigate quantum noise in response to changing channel conditions. Our study emphasizes the need for dynamic inference of critical channel parameters to improve program accuracy. We use the Dirichlet distribution to model the stochasticity of the Pauli channel. This allows us to perform Bayesian inference, which can improve the performance of probabilistic error cancellation (PEC) under time-varying noise. Our work demonstrates the importance of characterizing and mitigating temporal variations in quantum noise, which is crucial for developing more accurate and reliable quantum technologies. Our results demonstrate that Bayesian PEC can outperform non-adaptive approaches by a factor of 4.5x when measured using Hellinger distance from the ideal distribution.

Dasgupta, Samudra↗
DOE OSTI2018

A Bayesian Approach for Extracting Fluorescence Lifetimes from Sparse Data Sets and Its Significance for Imaging Experiments

The measurement of fluorescence lifetimes, especially in small sample volumes, presents the dual challenge of probing a small number of fluorophores and fitting the concomitant sparse data set to the appropriate excited-state decay function. A common method of analysis, such as the maximum likelihood (ML) technique, assumes a uniform probability distribution of the parameters describing the fluorescence decay function. An improvement is thus suggested by implementing a suitable nonuniform distribution, as is provided by a Bayesian framework, where the distribution of parameters is obtained from both their prior knowledge and the evidence-based likelihood of an event for a given set of parameters. We have also considered the Dirichlet prior distribution, whose mathematical form enables analytical solutions of the fitting parameters to be rapidly obtained. If Gaussian and exponential prior distributions are judiciously chosen, they reproduce the experimental target lifetime to within 20% with as few as 20 total photon counts for the data set, as does the Dirichlet prior distribution. But because of the analytical solutions afforded by the Dirichlet prior distribution, it is proposed to employ a Dirichlet prior to search parameter space rapidly to provide, if necessary, appropriate parameters for subsequent employment of a Gaussian or exponential prior distribution

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

A robust Bayesian approach to modeling epistemic uncertainty in common-cause failure models

In a standard Bayesian approach to the alpha-factor model for common-cause failure, a precise Dirichlet prior distribution models epistemic uncertainty in the alpha-factors. This Dirichlet prior is then updated with observed data to obtain a posterior distribution, which forms the basis for further inferences. In this paper, we adapt the imprecise Dirichlet model of Walley to represent epistemic uncertainty in the alpha-factors. In this approach, epistemic uncertainty is expressed more cautiously via lower and upper expectations for each alpha-factor, along with a learning parameter which determines how quickly the model learns from observed data. For this application, we focus on elicitation of the learning parameter, and find that values in the range of 1 to 10 seem reasonable. The approach is compared with Kelly and Atwood's minimally informative Dirichlet prior for the alpha-factor model, which incorporated precise mean values for the alpha-factors, but which was otherwise quite diffuse. Next, we explore the use of a set of Gamma priors to model epistemic uncertainty in the marginal failure rate, expressed via a lower and upper expectation for this rate, again along with a learning parameter. As zero counts are generally less of an issue here, we find that the choice of this learning parameter is less crucial. Finally, we demonstrate how both epistemic uncertainty models can be combined to arrive at lower and upper expectations for all common-cause failure rates. Thereby, we effectively provide a full sensitivity analysis of common-cause failure rates, properly reflecting epistemic uncertainty of the analyst on all levels of the common-cause failure model.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2018

Analysis of Repeatability and Reproducibility Studies With Ordinal Measurements

A Bayesian inferential approach with a noninformative prior is introduced to analyze ordinal repeatability and reproducibility (R&R) data using the De Mast–Van Wieringen model. This approach is extended with a weakly informative prior and random effects to allow for the consideration of a population of raters and prediction of a new rater. This random-effects approach is also shown to result in partial pooling of estimates across raters. In addition, match-probability-based measures to decompose ordinal R&R study data into contributions due to repeatability and due to reproducibility are defined. All extensions involving Bayesian inference (for fixed or random effects) and measures are illustrated on real and simulated ordinal R&R study data and are applicable in business and industry settings. This methodology can be implemented using the supplemental R package ordinalRR available from CRAN. Additional supplementary material for this article is available online.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Bayesian model mixing with multireference energy density functional

Reliably predicting nuclear properties across the entire chart of isotopes is important for applications ranging from nuclear astrophysics to superheavy science to nuclear technology. To this day, however, all the theoretical models that can scale at the level of the chart of isotopes remain semiphenomenological. Because they are fitted locally, their predictive power can vary significantly; different versions of the same theory provide different predictions. Bayesian model mixing takes advantage of such imperfect models to build a local mixture of a set of models to make improved predictions. Earlier attempts to use Bayesian model mixing for mass table calculations relied on models treated at single-reference energy density functional level, which fail to capture some of the correlations caused by configuration mixing or the restoration of broken symmetries. In this study we have applied Bayesian model mixing techniques within a multireference energy density functional (MR-EDF) framework. We considered predictions of two-particle separation energies from particle number projection or angular momentum projection with four different energy density functionals—a total of eight different MR-EDF models. We used a hierarchical Bayesian stacking framework with a Dirichlet prior distribution over weights together with an inverse log-ratio transform to enable positive correlations between different models. We found that Bayesian model mixing provides significantly improved predictions compared to the participating models. Published by the American Physical Society 2025

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2014

A family of random walks with generalized Dirichlet steps

We analyze a class of continuous time random walks in R{sup d},d≥2, with uniformly distributed directions. The steps performed by these processes are distributed according to a generalized Dirichlet law. Given the number of changes of orientation, we provide the analytic form of the probability density function of the position (X{sub {sub d}}(t),t>0) reached, at time t > 0, by the random motion. In particular, we analyze the case of random walks with two steps. In general, it is a hard task to obtain the explicit probability distributions for the process (X{sub {sub d}}(t),t>0). Nevertheless, for suitable values for the basic parameters of the generalized Dirichlet probability distribution, we are able to derive the explicit conditional density functions of (X{sub {sub d}}(t),t>0). Furthermore, in some cases, by exploiting the fractional Poisson process, the unconditional probability distributions of the random walk are obtained. This paper extends in a more general setting, the random walks with Dirichlet displacements introduced in some previous papers.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

Spectral properties of the Dirichlet operator Σ{sub i=1}{sup d}(−∂{sub i}{sup 2}){sup s} on domains in d-dimensional Euclidean space

In this article, we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator Σ{sub i=1}{sup d}(−∂{sub i}{sup 2}){sup s}, s∈(0,1] on an open and bounded subdomain Ω⊂R{sup d} and predict bounds on the sum of the first N eigenvalues, the counting function, the Riesz means, and the trace of the heat kernel. Moreover, utilizing the connection of coherent states to the semi-classical approach of quantum mechanics, we determine the sum for moments of eigenvalues of the associated Schrödinger operator.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

bayesian_tensor_regression

We plan to release the code used to perform the experiment described in our upcoming publication, entitled “Bayesian Tensor Modeling for Distribution-on-Distribution Regression.” This code a Bayesian regression model with a multi-way Dirichlet prior to tensor input distributions. All code to be released implements a new model that is intended for open-source publication.

Murph, Alexander C. [Los Alamos National Lab]↗
DOE OSTI2012

New criterion for conformational polymorphism

An analysis of nonvalent interactions in 29 crystal structures of compounds that have the C{sub a}H{sub b}N{sub c}O{sub d} composition (salicylaldoxime, glycine, and 2,4,6,8,10,12-hexaazaisowurzitane) and form polymorphic modifications is performed using molecular Voronoi-Dirichlet polyhedra. It is found that each conformational polymorph is characterized by a unique combination of the types of intramolecular and intermolecular nonvalent interactions. It is shown that a criterion which takes into account the total number of intramolecular nonvalent contacts and their distribution depending on the nature of neighboring atoms and the rank of faces of molecular Voronoi-Dirichlet polyhedra can be used to reveal conformational polymorphs.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗