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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

A unified and constructive framework for the universality of neural networks

Abstract One of the reasons why many neural networks are capable of replicating complicated tasks or functions is their universal approximation property. Though the past few decades have seen tremendous advances in theories of neural networks, a single constructive and elementary framework for neural network universality remains unavailable. This paper is an effort to provide a unified and constructive framework for the universality of a large class of activation functions including most of the existing ones. At the heart of the framework is the concept of neural network approximate identity (nAI). The main result is as follows: any nAI activation function is universal in the space of continuous functions on compacta. It turns out that most of the existing activation functions are nAI, and thus universal. The framework induces several advantages over the contemporary counterparts. First, it is constructive with elementary means from functional analysis, probability theory, and numerical analysis. Second, it is one of the first unified and constructive attempts that is valid for most of the existing activation functions. Third, it provides new proofs for most activation functions. Fourth, for a given activation and error tolerance, the framework provides precisely the architecture of the corresponding one-hidden neural network with a predetermined number of neurons and the values of weights/biases. Fifth, the framework allows us to abstractly present the first universal approximation with a favorable non-asymptotic rate. Sixth, our framework also provides insights into the developments, and hence providing constructive derivations, of some of the existing approaches.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

An integrated data analysis tool for improving measurements on the MST RFP

Many plasma diagnostics contain complementary information. For example, the double-foil soft x-ray system (SXR) and the Thomson Scattering diagnostic (TS) on the Madison Symmetric Torus both measure electron temperature. The complementary information from these diagnostics can be combined using a systematic method based on integrated data analysis techniques, leading to more accurate and sensitive results. An integrated data analysis tool based on Bayesian probability theory was able to estimate electron temperatures that are consistent with both the SXR and TS diagnostics and more precise than either. A Markov Chain Monte Carlo analysis to increase the flexibility of the tool was implemented and benchmarked against a grid search method.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRS1982

A comprehensive model to determine the effects of temperature and species fluctuations on reaction rates in turbulent reacting flows

The use of probability theory to determine the effects of turbulent fluctuations on reaction rates in turbulent combustion systems is briefly reviewed. Results are presented for the effect of species fluctuations in particular. It is found that turbulent fluctuations of species act to reduce the reaction rates, in contrast with the temperature fluctuations previously determined to increase Arrhenius reaction rate constants. For the temperature fluctuations, a criterion is set forth for determining if, in a given region of a turbulent flow field, the temperature can be expected to exhibit ramp like fluctuations. Using the above results, along with results previously obtained, a model is described for testing the effects of turbulent fluctuations of temperature and species on reaction rates in computer programs dealing with turbulent reacting flows. An alternative model which employs three variable probability density functions (temperature and two species) and is currently being formulated is discussed as well.

Chinitz, W.↗
DOE OSTI2018

Multivariate Analogs of Classical Univariate Discrete Distributions and Their Properties

Some discrete distributions such as Bernoulli, binomial, geometric, negative binomial, Poisson, Polya–Aeppli, and others play an important role in applied problems of probability theory and mathematical statistics. We propose a variant of a multivariate distribution whose components have a given univariate discrete distribution. In fact we consider some very general variant of the so-called reduction method. We find the explicit form of the mass function and generating function of such distribution and study their properties. We prove that our construction is unique in natural exponential families of distributions. Our results are the generalization and unification of many results of other authors.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A consistent, Bayesian, approach to the cross section probability distribution in the unresolved resonance region

The cross sections of neutron-induced reactions can be divided into three energy ranges: the resolved resonance region (RRR), the unresolved resonance region (URR), and the fast region. In general, the cross sections in the URR show significant fluctuations that cannot be predicted and cannot be experimentally resolved, thus, it is commonly assumed that the cross section at a specific energy is given by a probability distribution function (PDF) over a range of values that can span several orders of magnitude. The current methodology used to describe such behavior is to construct the PDF by stochastically generating resonance ladders and numerically measuring the PDF. The resonance ladders are sampled using known resonance statistical properties and average resonance widths and spacings extrapolated from the RRR. Although this is a standard and widely used technique, it is computationally very expensive, therefore, an alternative, analytical, approach would be preferable due to the considerable speed up of the computational time in real life applications. Moreover, the current methodology does not take into account existing experimental data, such for total and capture cross sections, that are available for many nuclei. Finally, this approach was developed to be used in reactor-scale applications and it is not suited for use in single-event applications. In this work we will rethink the entire approach to the PDF construction using a Bayesian mindset. This will allow us to provide a different definition of the PDF that allows a much faster calculation of the higher-temperature PDFs and a proper combination of theoretical and experimental PDFs following the probability theory. We will also show that our definition is well suited for single-event applications and we will make an explicit connection between our method and the standard approach. We do this by showing that the central limit theorem applies and our method leads to the same PDF obtained with the standard methodology, for a large number of events per history.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Diagnostics: Chapter 8 of the special issue: on the path to tokamak burning plasma operation

This chapter presents the activity conducted by the ITPA topical group (TG) on Diagnostics over about the last 15 years. Following a general introduction of the ITER Diagnostics led by their measurement roles, the document is organized in several subchapters detailing the design support, research and development activity conducted by each of the specialist working groups (WGs) of the TG. Please note that the magnetic diagnostics were supported at the TG without a specific WG. Their status is included in the general introduction. In the following some highlights of the subchapter’s contents are provided. Recent advances in ITER first wall (FW) diagnostics for the measurements of plasma-metallic wall interaction in support of the ITER research plan are reported. An InfraRed imaging Video Bolometer for ITER has been developed and tested on several tokamaks to measure the radiated power loss. A laser-induced breakdown spectroscopy (LIBS) technique which utilizes a pulsed laser beam to ablate locally by forming a crater, will measure local tritium inventory in the FW material. Real-time Residual Gas Analyzers will measure the neutral gas composition in a divertor port and an equatorial port during plasma operation. Due to the full metallic FW environment, the plasma-wall interaction in ITER will face several challenges such as the compromised radiated power and divertor heat flux measurements by reflection. Ray tracing and analysis codes have been developed to eliminate and correct the effects of reflection in the measurements. The characteristics of the reflecting surfaces depending on the roughness and angle of the incidence have been measured by dedicated experiments, and the results were applied to the reflection elimination. For the measurement of the metallic impurity radiation induced by eroded metallic atoms, a vacuum ultraviolet spectrometer has been developed and tested. An extensive thermonuclear diagnostic suite will be required to support the operation of ITER and the planned experimental program for future burning plasma experiments. Due to the harsh environmental conditions, the implementation of diagnostic systems in ITER is a major challenge. These conditions include high levels of neutron and gamma fluxes, neutron heating, particle bombardment. Therefore, the selection and design of diagnostic systems must take into account a number of phenomena previously unseen in diagnostic design. For this reason, the measurement of neutrons and confined or lost fast ions, with particular emphasis on alpha particles, is critical to ITER. The diagnostics associated with these measurements will be important for future plasma-burning experiments at ITER. The high neutron emission and very large plasma size in ITER make neutron diagnostics the main diagnostic method used to measure plasma parameters such as fusion power, fusion power density, ion temperature, energy of fast ions and their spatial distributions in the plasma core. Active spectroscopy techniques are methods where a neutral particle beam is injected into the plasma and information on plasma parameters is extracted from the measurement of line emission resulting from the beam-plasma interaction, either by plasma ions or by beam atoms. Spatial localization is achieved by crossing the beamline and multiple observation lines. The ITER plasma will be a high temperature, moderately dense, fully ionized collisional plasma. The plasma facing surfaces are principally metallic being fashioned from beryllium or tungsten but many other elements, arising from either structural or from operational needs, may enter this plasma. The energy range of the emitted photons range from meV (infra-red) to multi keV (x-rays) and originate from all areas of the plasma volume. The primary role of passive emission diagnostics is to identify what is in the plasma from spectral signatures. Extracting quantitative information from these measurements such as impurity content, ion temperature, rotation, degree of detachment and radiated power depends on calibrated instruments, a physics model of the atomic and molecular processes and plasma transport and an analysis workflow that takes into account environmental effects such as reflections. The particular needs for ITER have prompted a multi-machine, many-year effort to address all these aspects and this chapter reviews the work on diagnostic design, experiments and new analysis techniques. An overview of the laser diagnostics to be implemented on ITER is also provided in this paper. This includes descriptions of the Thomson scattering in the core, edge and divertor regions, polarimetry and interferometry diagnostics used for measuring plasma density and also measurements of helium density in the divertor using Laser Induced Flourescence. Techniques which can allow improvements on current measurements are also addressed in particular expanding poloidal polarimetry measurements to measure field fluctuations and proposed use of dispersion interferometery which has a number of advantages over existing methods. This paper identifies particular areas where further research and testing on existing tokamaks is useful even at this advanced stage to inform the design of diagnostics for ITER. Outstanding areas of concern for the implementation of laser diagnostics, in particular with a view to reliable operation are identified. An overview of the latest developments of microwave diagnostic systems and techniques is given. The primary focus is the contributions for ITER—the next step burning plasma experiment—which is supplemented by describing recent progress of techniques applicable for fusion experiments beyond ITER. The contributions are intentionally kept concise, and are being supplemented by a rich list of references for further studies. Radiation induced effects are receiving continuous and well-deserved attention of the ITER diagnostic community and they are in many cases one of the primary design drivers of the ITER diagnostic systems. The paper summarizes recent progress in this area focusing primarily on the ITER diagnostics but in some cases provides also outlook for the possible solutions for even more demanding radiation environment of fusion reactors beyond ITER. Despite advancements in the area of modeling and simulation of various radiation induced effects, experimental testing in a nuclear environment as close as possible to the target one is still seen as unavoidable for proper qualification of particular diagnostic functional elements. Recent advancement within three diagnostic areas: optical diagnostics, magnetics and bolometers is covered. Encouraging results on qualification of silica glass vacuum window assemblies are presented. In the area of magnetic sensors, progress of irradiation tests performed on ITER in-vessel LTCC inductive sensors is presented with outlook for novel technological approaches to inductive sensors utilizing thick printing and photolithography technologies being highlighted. Summary of advancements in the area of steady state magnetic field sensors based on Hall effect is given. New results of neutron irradiation test of the ITER borosilicate glass inserts for vacuum electrical feedthroughs are summarized finding negligible swelling at target level of neutron fluence. Off-line irradiation tests of fiber optic current sensors for plasma current measurement demonstrated that both for gamma doses up to 5 MGy and a total neutron fluence up to 10 15 cm −2 , radiation induced changes are still compatible with required measurement accuracy on ITER. The ITER bolometers are given as an example how considering radiation effects may influence the diagnostic design. Finally, outlook for future main R&D directions is outlined. All optical and laser-based diagnostics in ITER will be using mirrors to guide plasma radiation toward detectors, cameras and sensors. In the hostile plasma, radiation and particle environment the optical characteristics of diagnostic mirrors will degrade directly affecting the entire performance of involved diagnostic systems. An assessment of factors affecting mirror performance is provided. Among the prime adverse factors are deposition of plasma impurities, sputtering of mirror surface and steam ingress in the vicinity of mirrors. Within the International Tokamak Physics Activity with active support by ITER central team and domestic agencies, the structured research and development (R&D) program on mitigation of risks for diagnostic mirrors is underway. Within this program the mirror material development, the passive mitigation of mirror degradation by using diagnostic ducts and shutters along with an active mirror recovery program comprising the in-situ mirror cleaning and calibration is underway. Recent developments in diagnostic mirror R&D are described in this Chapter along with an example of their implementation of R&D solutions in ITER Infrared Thermography diagnostic. An assessment of still open engineering and physics questions, considerations on mirror risks during an early phase of ITER operation are given along with an overview of diagnostic mirror evolution in the late ITER operation stage toward the demonstration fusion power plant. Several crucial areas of diagnostic R&D outlined in ITER Research Plan are addressed. The basic control groups in a fusion reactor can be broken-down in five categories: (1) plasma position, magnetic configuration, and plasma current control, (2) profile control and confinement optimization, (3) MHD control and suppression, (4) edge dissipation control, radiation and plasma exhaust control and (5) break-down optimization. These categories are coupled via the physics (a control action in one domain will affect the other domains) and via shared actuators (e.g. ECRH for impurity accumulation avoidance, current density distribution control and MHD suppression). Consequently, a supervisory control system should determine the priority of the various control tasks, their couplings, and the interfaces with the safety and interlock system. For the systematic development of the various controllers taking the complexity of the plasma and the control system into account, a model-based approach is required. A short historical overview is given of the developments in systems and control theory and control engineering with special emphasis on those developments that are most relevant for Nuclear Fusion research and operation. An overview is given of the state of the field of fusion plasma control for the control categories. It will be shown how synthetic diagnostics are being developed in ITER and how they are used in diagnostic design and design validation and how they can be in model-based controller synthesis using relatively simple models. In modern control methods, multiple diagnostics are used to constrain relatively simple models. The constrained models provide an estimate for the state. This opens the route to state controllers, such as model predictive control. A major challenge in nuclear fusion research is the coherent combination of data from heterogeneous diagnostics and modeling codes for machine control and safety as well as physics studies. Measured data from different diagnostics often provide information about the same subset of physical parameters. Additionally, information provided by some diagnostics might be needed for the analysis of other diagnostics. A joint analysis of complementary and redundant data allows, e.g. to improve the reliability of parameter estimation, to increase the spatial and temporal resolution of profiles, to obtain synergistic effects, to consider diagnostics interdependencies and to find and resolve data inconsistencies. Physics-based modeling and parameter relationships provide additional information improving the treatment of ill-posed inversion problems. A coherent combination of all kind of available information within a probabilistic framework allows for improved data analysis results. The concept of integrated data analysis (IDA) in the framework of Bayesian probability theory is outlined and contrasted with conventional data analysis. Components of the probabilistic approach are summarized and specific ingredients beneficial for data analysis at fusion devices are discussed.

ITER↗
DOE OSTI2019

Approximating Two-Stage Chance-Constrained Programs with Classical Probability Bounds

We consider a joint-chance constraint (JCC) as a union of sets, and approximate this union using bounds from classical probability theory. When these bounds are used in an optimization model constrained by the JCC, we obtain corresponding upper and lower bounds on the optimal objective function value. We compare the strength of these bounds against each other under two different sampling schemes, and observe that a larger correlation between the uncertainties tends to result in more computationally challenging optimization models. We also observe the same set of inequalities to provide the tightest upper and lower bounds in our computational experiments.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Communication: Towards first principles theory of relaxation in supercooled liquids formulated in terms of cooperative motion

A general theory of the long time, low temperature dynamics of glass-forming fluids remains elusive despite the almost 20 years since the famous pronouncement by the Nobel Laureate P. W. Anderson, “The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition” [Science 267, 1615 (1995)]. While recent work indicates that Adam-Gibbs theory (AGT) provides a framework for computing the structural relaxation time of supercooled fluids and for analyzing the properties of the cooperatively rearranging dynamical strings observed in low temperature molecular dynamics simulations, the heuristic nature of AGT has impeded general acceptance due to the lack of a first principles derivation [G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965)]. This deficiency is rectified here by a statistical mechanical derivation of AGT that uses transition state theory and the assumption that the transition state is composed of elementary excitations of a string-like form. The strings are assumed to form in equilibrium with the mobile particles in the fluid. Hence, transition state theory requires the strings to be in mutual equilibrium and thus to have the size distribution of a self-assembling system, in accord with the simulations and analyses of Douglas and co-workers. The average relaxation rate is computed as a grand canonical ensemble average over all string sizes, and use of the previously determined relation between configurational entropy and the average cluster size in several model equilibrium self-associating systems produces the AGT expression in a manner enabling further extensions and more fundamental tests of the assumptions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Local Hamiltonians for quantitative Green's function embedding methods

Embedding calculations that find approximate solutions to the Schrödinger equation for large molecules and realistic solids are performed commonly in a three step procedure involving (i) construction of a model system with effective interactions approximating the low energy physics of the initial realistic system, (ii) mapping the model system onto an impurity Hamiltonian, and (iii) solving the impurity problem. We have developed a novel procedure for parametrizing the impurity Hamiltonian that avoids the mathematically uncontrolled step of constructing the low energy model system. Instead, the impurity Hamiltonian is immediately parametrized to recover the self-energy of the realistic system in the limit of high frequencies or short time. The effective interactions parametrizing the fictitious impurity Hamiltonian are local to the embedded regions, and include all the non-local interactions present in the original realistic Hamiltonian in an implicit way. We show that this impurity Hamiltonian can lead to excellent total energies and self-energies that approximate the quantities of the initial realistic system very well. Moreover, we show that as long as the effective impurity Hamiltonian parametrization is designed to recover the self-energy of the initial realistic system for high frequencies, we can expect a good total energy and self-energy. Lastly, we propose two practical ways of evaluating effective integrals for parametrizing impurity models.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

On the reduction criterion for random quantum states

In this paper, we study the reduction criterion for detecting entanglement of large dimensional bipartite quantum systems. We first obtain an explicit formula for the moments of a random quantum state to which the reduction criterion has been applied. We show that the empirical eigenvalue distribution of this random matrix converges strongly to a limit that we compute, in three different asymptotic regimes. We then employ tools from free probability theory to study the asymptotic positivity of the reduction operators. Finally, we compare the reduction criterion with other entanglement criteria, via thresholds.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Convergence acceleration of Monte Carlo many-body perturbation methods by direct sampling

In the Monte Carlo many-body perturbation (MC-MP) method, the conventional correlation-correction formula, which is a long sum of products of low-dimensional integrals, is first recast into a short sum of high-dimensional integrals over electron-pair and imaginary-time coordinates. These high-dimensional integrals are then evaluated by the Monte Carlo method with random coordinates generated by the Metropolis–Hasting algorithm according to a suitable distribution. The latter algorithm, while advantageous in its ability to sample nearly any distribution, introduces autocorrelation in sampled coordinates, which in turn increases the statistical uncertainty of the integrals and thus the computational cost. It also involves wasteful rejected moves and an initial “burn-in” step as well as displays hysteresis. Here, an algorithm is proposed that directly produces a random sequence of electron-pair coordinates for the same distribution used in the MC-MP method, which is free from autocorrelation, rejected moves, a burn-in step, or hysteresis. Furthermore, this direct-sampling algorithm is shown to accelerate second- (MC-MP2) and third-order Monte Carlo many-body perturbation (MC-MP3) calculations by up to 222% and 38%, respectively.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Performance evaluation of cosmic ray muon trajectory estimation algorithms

Muons, being elementary particles with minimal interaction with nuclear materials and abundant at sea level, have sparked interest in utilizing them for imaging various applications, such as mining [Borselli et al., Sci. Rep. 12, 22329 (2022)], volcano imaging [Nagamine et al., Nucl. Instrum. Meth. A, 356, 585(1995)], and underground tunnel detection [Guardincerri et al., Pure Appl. Geophys. 174, 2133 (2017)]. Recently, their use in nuclear nonproliferation and safeguard verification has gained attention, particularly in cargo screening for nuclear waste smuggling [Baesso et al., J. Instrum. 9, C10041 (2014)], source localization [L. J. Schultz et al., Nucl. Instrum. Meth. A 519, 687 (2004)], and locating nuclear fuel debris in reactors [Borozdin et al., Phys. Rev. Let. 109, 152501 (2012)]. However, the resolution of muon image reconstruction techniques is limited due to multiple Coulomb scattering (MCS) within the target object. To achieve robust muon tomography, it is crucial to develop efficient and flexible physics-based algorithms that can model the MCS process accurately and estimate the most probable trajectory of muons as they pass through the target object. To address this limitation, in this study, a novel algorithmic approach utilizing the Bayesian probability theory and Gaussian approximation of MCS is chosen. Different energy levels, materials, and target sizes were considered in the evaluations. The results demonstrate that the Generalized Muon Trajectory Estimation (GMTE) algorithm offers significant improvements over currently used algorithms. Across all test scenarios, the GMTE algorithm demonstrated ~50% and 38% increase in precision compared to Straight Line Path (SLP) and Point of Closest Approach (PoCA) algorithms, respectively. Furthermore, it exhibited 10%–35% and 10%–15% increases in muon flux utilization for high and medium Z materials, respectively, compared to the PoCA algorithm. In conclusion, the extensive simulations confirm the enhanced performance and efficiency of the GMTE algorithm, offering improved resolution and reduced measurement time for cosmic ray muon imaging compared to the current SLP and PoCA algorithms.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

Where Does Water Go During Hydraulic Fracturing?

Typically, during hydraulic fracturing millions of gallons of water are injected at high pressure into deep shale formations. This water can be housed in fractures, within the shale matrix, and can potentially migrate beyond the shale formation via fractures and/or faults raising environmental concerns. Here, we describe a generic framework for producing estimates of the volume available in fractures and undamaged shale matrix where water injected into a representative shale site could reside during hydraulic fracturing, and apply it to a representative site that incorporates available field data. Moreover, the amount of water that can be stored in the fractures is estimated by calculating the volume of all the fractures associated with a discrete fracture network (DFN) based on real data and using probability theory to estimate the volume of smaller fractures that are below the lower cutoff for the fracture radius in the DFN. The amount of water stored in the matrix is estimated utilizing two distinct methods—one using a two-phase model at the pore-scale and the other using a single-phase model at the continuum scale. Based on these calculations, it appears that most of the water resides in the matrix with a lesser amount in the fractures.

58 GEOSCIENCES↗
DOE OSTI2014

Developing a Comprehensive Risk Assessment Framework for Geological Storage CO 2

The operational risks for CCS projects include: risks of capturing, compressing, transporting and injecting CO₂; risks of well blowouts; risk that CO 2 will leak into shallow aquifers and contaminate potable water; and risk that sequestered CO 2 will leak into the atmosphere. This report examines these risks by using information on the risks associated with analogue activities such as CO 2 based enhanced oil recovery (CO 2 -EOR), natural gas storage and acid gas disposal. We have developed a new analysis of pipeline risk based on Bayesian statistical analysis. Bayesian theory probabilities may describe states of partial knowledge, even perhaps those related to non-repeatable events. The Bayesian approach enables both utilizing existing data and at the same time having the capability to adsorb new information thus to lower uncertainty in our understanding of complex systems. Incident rates for both natural gas and CO 2 pipelines have been widely used in papers and reports on risk of CO 2 pipelines as proxies for the individual risk created by such pipelines. Published risk studies of CO 2 pipelines suggest that the individual risk associated with CO2 pipelines is between 10 -3 and 10 -4 , which reflects risk levels approaching those of mountain climbing, which many would find unacceptably high. This report concludes, based on a careful analysis of natural gas pipeline failures, suggests that the individual risk of CO 2 pipelines is likely in the range of 10-6 to 10-7, a risk range considered in the acceptable to negligible range in most countries. If, as is commonly thought, pipelines represent the highest risk component of CCS outside of the capture plant, then this conclusion suggests that most (if not all) previous quantitative- risk assessments of components of CCS may be orders of magnitude to high. The potential lethality of unexpected CO 2 releases from pipelines or wells are arguably the highest risk aspects of CO 2 enhanced oil recovery (CO2-EOR), carbon capture, and storage (CCS). Assertions in the CCS literature, that CO 2 levels of 10% for ten minutes, or 20 to 30% for a few minutes are lethal to humans, are not supported by the available evidence. The results of published experiments with animals exposed to CO 2 , from mice to monkeys, at both normal and depleted oxygen levels, suggest that lethal levels of CO 2 toxicity are in the range 50 to 60%. These experiments demonstrate that CO 2 does not kill by asphyxia, but rather is toxic at high concentrations. It is concluded that quantitative risk assessments of CCS have overestimated the risk of fatalities by using values of lethality a factor two to six lower than the values estimated in this paper. In many dispersion models of CO 2 releases from pipelines, no fatalities would be predicted if appropriate levels of lethality for CO 2 had been used in the analysis.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Sampling Speeds from a Relativistic Maxwell Distribution

Monte Carlo calculations in astrophysics or other fields of physics involving hot ionized gases may require sarmpling from a relativistic Maxwellian distribution. Winslow 4 described a previously unpublished technique attributed to Von Neumann for such sampling. The technique, based on an approxir.,ation to the relativistic equation, seems adequate 2 for temperatures up to about 1/4 the rest energy (i.e., kT$\sim\atop{<}$ $\frac{mc^2}{4}$) except for inherent inaccuracies in the high energy tail. For much higher temperatures, the approximation is clearly in error. It is the purpose of this paper to give a method for sampling from a relativistic Maxwellian distribution which is mathematically exact and valid for all temperatures. Most (or all) the mathematics involved is scattered through the literature on probability theory. But since many of the arguments are perhaps new to the prospective Monte Carlo user, we have included, for the sake of completeness, a good deal of mathematical rigor in the report.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Probabilistic Error Bounds for Low-Rank Tensor Decompositions Used in Large-Scale Data Analysis Applications (LDRD Final Report)

This report documents a research project on analyzing low-rank tensor models for data analysis that took place at Sandia National Laboratories from October 2023–September 2025. The focus of this work was to extend theoretical frameworks from statistics and probability theory for use with models for scalar, vector, and matrix data to models with tensor, or general multi-dimensional array, data. Through this work, we have provided a new set of tools for bounding errors on low-rank tensor models of both complete and sampled data. The remainder of this report is organized as follows. In Section 1, we describe the proposed work at the start of the project. Section 2 describes the research advances made as part of the project. Other research contributions in the form of conference presentations and software development is provided in Section 3. Workforce development at Sandia and Florida Atlantic University (via a subcontract on this project) is provided in Section 4.

97 MATHEMATICS AND COMPUTING↗