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At least 271 records · Page 15

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NASA NTRS1988

The cryptoendolithic microbial environment in the Ross Desert of Antarctica: mathematical models of the thermal regime

Microbial activity in the Antarctic cryptoendolithic habitat is regulated primarily by temperature. Previous field studies have provided some information on the thermal regime in this habitat, but this type of information is limited by the remoteness of the site and the harsh climatic conditions. Therefore, a mathematical model of the endolithic thermal regime was constructed to augment the field data. This model enabled the parameters affecting the horizontal and altitudinal distribution of the community to be examined. The model predicts that colonization should be possible on surfaces with zenith angle less than 15 degrees. At greater zenith angles, colonization should be restricted to surfaces with azimuth angles less than 135 degrees or greater than 225 degrees. The upper elevational limit of the community should be less than 2,500 m. The thermal regime probably does not influence the zonation of the community within a rock.

Non-NASA Center↗

NASA NTRS2023

A Confidence-Based Approach to Including Survivors in a Probabilistic TID Failure Assessment

A probabilistic total ionizing dose (TID) failure assessment is extended to include survivor data, enabling the bounding of failure probability to a desired confidence level (CL) without failure data. The extension provides an avenue for analyzing microelectronics tested for TID without reaching a failure mode, a scenario often encountered by missions utilizing commercial-off-the-shelf (COTS) technologies. Using the type-I censored likelihood formulation and a realistic upper bound on expected device performance, the failure probability space is bounded by confidence contours within the context of a variable environment. The framework accommodates any type of distribution assumed for the part failure or the environment under consideration. Furthermore, the framework can be utilized pre-emptively to plan future device TID tests, minimizing costs while meeting survival requirements. Heritage data may also be used as survivors to further minimize testing costs when parts are from the same lot, but the amount of constraint derived from heritage is limited. Altogether, the framework enables a formal, mathematically rigorous analysis of radiation tolerant devices tested to a maximum dose, as well as flight heritage, in a hardness assurance methodology.

confidence↗

NASA NTRS2006

Stochastic Representation of Chaos Using Terminal Attractors

A nonlinear version of the Liouville equation based on terminal attractors is part of a mathematical formalism for describing postinstability motions of dynamical systems characterized by exponential divergences of trajectories leading to chaos (including turbulence as a form of chaos). The formalism can be applied to both conservative systems (e.g., multibody systems in celestial mechanics) and dissipative systems (e.g., viscous fluids). The development of the present formalism was undertaken in an effort to remove positive Lyapunov exponents. The means chosen to accomplish this is coupling of the governing dynamical equations with the corresponding Liouville equation that describes the evolution of the flow of error probability. The underlying idea is to suppress the divergences of different trajectories that correspond to different initial conditions, without affecting a target trajectory, which is one that starts with prescribed initial conditions.

Zak, Michail↗

NASA NTRS2003

Extension of Liouville Formalism to Postinstability Dynamics

A mathematical formalism has been developed for predicting the postinstability motions of a dynamic system governed by a system of nonlinear equations and subject to initial conditions. Previously, there was no general method for prediction and mathematical modeling of postinstability behaviors (e.g., chaos and turbulence) in such a system. The formalism of nonlinear dynamics does not afford means to discriminate between stable and unstable motions: an additional stability analysis is necessary for such discrimination. However, an additional stability analysis does not suggest any modifications of a mathematical model that would enable the model to describe postinstability motions efficiently. The most important type of instability that necessitates a postinstability description is associated with positive Lyapunov exponents. Such an instability leads to exponential growth of small errors in initial conditions or, equivalently, exponential divergence of neighboring trajectories. The development of the present formalism was undertaken in an effort to remove positive Lyapunov exponents. The means chosen to accomplish this is coupling of the governing dynamical equations with the corresponding Liouville equation that describes the evolution of the flow of error probability. The underlying idea is to suppress the divergences of different trajectories that correspond to different initial conditions, without affecting a target trajectory, which is one that starts with prescribed initial conditions.

Zak, Michail↗

DOE OSTI2024

Stochastic Trust-Region Algorithm in Random Subspaces with Convergence and Expected Complexity Analyses

Here, this work proposes a framework for large-scale stochastic derivative-free optimization (DFO) by introducing STARS, a trust-region method based on iterative minimization in random subspaces. This framework is both an algorithmic and theoretical extension of a random subspace derivative-free optimization (RSDFO) framework, and an algorithm for stochastic optimization with random models (STORM). Moreover, like RSDFO, STARS achieves scalability by minimizing interpolation models that approximate the objective in low-dimensional affine subspaces, thus significantly reducing per-iteration costs in terms of function evaluations and yielding strong performance on largescale stochastic DFO problems. The user-determined dimension of these subspaces, when the latter are defined, for example, by the columns of so-called Johnson-Lindenstrauss transforms, turns out to be independent of the dimension of the problem. For convergence purposes, inspired by the analyses of RSDFO and STORM, both a particular quality of the subspace and the accuracies of random function estimates and models are required to hold with sufficiently high, but fixed, probabilities. Using martingale theory under the latter assumptions, an almost sure global convergence of STARS to a first-order stationary point is shown, and the expected number of iterations required to reach a desired first-order accuracy is proved to be similar to that of STORM and other stochastic DFO algorithms, up to constants.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Bayesian reduced-order deep learning surrogate model for dynamic systems described by partial differential equations

We propose a reduced-order deep-learning surrogate model for dynamic systems described by time-dependent partial differential equations. This method employs space–time Karhunen–Loève expansions (KLEs) of the state variables and space-dependent KLEs of space-varying parameters to identify the reduced (latent) dimensions. Subsequently, a deep neural network (DNN) is used to map the parameter latent space to the state variable latent space. An approximate Bayesian method is developed for uncertainty quantification (UQ) in the proposed KL-DNN surrogate model. The KL-DNN method is tested for the linear advection–diffusion and nonlinear diffusion equations, and the Bayesian approach for UQ is compared with the deep ensembling (DE) approach, commonly used for quantifying uncertainty in DNN models. It was found that the approximate Bayesian method provides a more informative distribution of the PDE solutions in terms of the coverage of the reference PDE solutions (the percentage of nodes where the reference solution is within the confidence interval predicted by the UQ methods) and log predictive probability. The DE method is found to underestimate uncertainty and introduce bias. For the nonlinear diffusion equation, we compare the KL-DNN method with the Fourier Neural Operator (FNO) method and find that KL-DNN is 10% more accurate and needs less training time than the FNO method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Understanding Pore Filling Processes and Adsorption/Desorption Hysteresis in Nanoporous Metal–Organic Frameworks: Insights from Grand Canonical Monte Carlo Simulations and Free Energy Calculations

Grand canonical Monte Carlo (GCMC) simulations were used to investigate pore filling and hysteresis in nanoporous metal-organic frameworks (MOFs). Adsorption and desorption isotherms were calculated for argon at 87 K in 1866 MOFs from the CoRE MOF database and for short n-alkanes in selected MOFs, keeping the adsorbent structure rigid. Analysis of the molecular configurations showed two different mechanisms and origins of hysteresis: one involving a transition of the adsorbate arrangement in the pores similar to a gas-to-liquid transition associated with a large change in the loading and one more similar to a liquid-to-solid transition associated with a relatively small change in the loading. Our GCMC simulations in MOFs with diverse pore topologies indicate exceptions to an empirical relationship for the minimum diameter of a cylindical pore required for hysteresis as a function of the adsorbate diameter and reduced temperature. The simulations reveal some structures where isotherms exhibit two steps in the adsorption branch and only one step in the desorption branch. Hysteresis loops with a different number of adsorption and desorption steps are not common. Here, to better understand why hysteresis is observed in the GCMC simulations, the concept of the transition probability for observing a step in the adsorption isotherm at a given pressure in a GCMC simulation is introduced. We used two different methods to calculate the transition probabilities and find that these yield comparable results. Furthermore, the transition probability provides a measure for the length of GCMC simulations to yield reliable results.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Accelerating multilevel Markov Chain Monte Carlo using machine learning models

Here, this work presents an efficient approach for accelerating multilevel Markov Chain Monte Carlo (MCMC) sampling for large-scale problems using low-fidelity machine learning models. While conventional techniques for large-scale Bayesian inference often substitute computationally expensive high-fidelity models with machine learning models, thereby introducing approximation errors, our approach offers a computationally efficient alternative by augmenting high-fidelity models with low-fidelity ones within a hierarchical framework. The multilevel approach utilizes the low-fidelity machine learning model (MLM) for inexpensive evaluation of proposed samples thereby improving the acceptance of samples by the high-fidelity model. The hierarchy in our multilevel algorithm is derived from geometric multigrid hierarchy. We utilize an MLM to accelerate the coarse level sampling. Training machine learning model for the coarsest level significantly reduces the computational cost associated with generating training data and training the model. We present an MCMC algorithm to accelerate the coarsest level sampling using MLM and account for the approximation error introduced. We provide theoretical proofs of detailed balance and demonstrate that our multilevel approach constitutes a consistent MCMC algorithm. Additionally, we derive the expression for cost reduction due to machine learning model to facilitate cost analysis of the hierarchical sampling algorithm. Our technique is demonstrated on a standard benchmark inference problem in groundwater flow, where we estimate the probability density of a quantity of interest using a four-level MCMC algorithm. Our proposed algorithm accelerates multilevel sampling by a factor of two while achieving similar accuracy compared to sampling using the standard multilevel algorithm.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2008

Statistical Issues for Uncontrolled Reentry Hazards

A number of statistical tools have been developed over the years for assessing the risk of reentering objects to human populations. These tools make use of the characteristics (e.g., mass, shape, size) of debris that are predicted by aerothermal models to survive reentry. The statistical tools use this information to compute the probability that one or more of the surviving debris might hit a person on the ground and cause one or more casualties. The statistical portion of the analysis relies on a number of assumptions about how the debris footprint and the human population are distributed in latitude and longitude, and how to use that information to arrive at realistic risk numbers. This inevitably involves assumptions that simplify the problem and make it tractable, but it is often difficult to test the accuracy and applicability of these assumptions. This paper looks at a number of these theoretical assumptions, examining the mathematical basis for the hazard calculations, and outlining the conditions under which the simplifying assumptions hold. In addition, this paper will also outline some new tools for assessing ground hazard risk in useful ways. Also, this study is able to make use of a database of known uncontrolled reentry locations measured by the United States Department of Defense. By using data from objects that were in orbit more than 30 days before reentry, sufficient time is allowed for the orbital parameters to be randomized in the way the models are designed to compute. The predicted ground footprint distributions of these objects are based on the theory that their orbits behave basically like simple Kepler orbits. However, there are a number of factors - including the effects of gravitational harmonics, the effects of the Earth's equatorial bulge on the atmosphere, and the rotation of the Earth and atmosphere - that could cause them to diverge from simple Kepler orbit behavior and change the ground footprints. The measured latitude and longitude distributions of these objects provide data that can be directly compared with the predicted distributions, providing a fundamental empirical test of the model assumptions.

Matney, Mark↗

NASA NTRS1981

Reliability-based optimization under random vibration environment

A methodology of formulating the optimum design problem for structural systems with random parameters and subjected to random vibration as a mathematical programming problem is presented. The proposed method is applied to the optimum design of a cantilever beam with a tip mass and a truss structure supporting a water tank. The excitations are assumed to be Gaussian processes and the geometric and material properties are taken to be normally distributed random variables. The probabilistic constraints are specified for individual failure modes since it is easier to specify the reliability level for each failure mode keeping in view the consequences of failure in that particular mode. The time parameter appearing in the random vibration based constraints is eliminated by replacing the probabilities of failure by suitable upper bounds. The numerical results demonstrate the feasibility and effectiveness of applying the reliability-based design concepts to structures with random parameters and operating in random vibration environment.

Rao, S. S.↗

NASA NTRS2006

The International Reference Ionosphere - Climatological Standard for the Ionosphere

The International Reference Ionosphere (IRI) a joint project of URSI and COSPAR is the defacto standard for a climatological specification of ionospheric parameters. IRI is based on a wide range of ground and space data and has been steadily improved since its inception in 1969 with the ever-increasing volume of ionospheric data and with better mathematical descriptions of the observed global and temporal variation patterns. The IRI model has been validated with a large amount of data including data from the most recent ionospheric satellites (KOMPSAT, ROCSAT and TIMED) and data from global network of ionosondes. Several IRI teams are working on specific aspects of the IRI modeling effort including an improved representation of the topside ionosphere with a seamless transition to the plasmasphere, a new effort to represent the global variation of F2 peak parameters using the Neural Network (NN) technique, and the inclusion of several additional parameters in IRI, e.g., spread-F probability and ionospheric variability. Annual IRI workshops are the forum for discussions of these efforts and for all science activities related to IRI as well as applications of the IRI model in engineering and education. In this paper I will present a status report about the IRI effort with special emphasis on the presentations and results from the most recent IRI Workshops (Paris, 2004; Tortosa, 2005) and on the most important ongoing IRI activities. I will discuss the latest version of the IRI model, IRI-2006, highlighting the most recent changes and additions. Finally, the talk will review some of the applications of the IRI model with special emphasis on the use for radiowave propagation studies and communication purposes.

Bilitza, Dieter↗

DOE OSTI2025

Generative AI for Wind-Wave Inflows

Extreme wind-wave conditions can create dangerous or damaging environmental conditions for sea-based structures and watercrafts. However, the model of consistent, coupled wind-wave flow conditions is computational expensive, even for simple flow scenarios. This work examines the use of diffusion-based generative artificial intelligence tools to accelerate the generation of these two-phase flows and to characterize probabilities of various states to help with the generation of extreme patterns.

17 WIND ENERGY↗

DOE OSTI2025

The Poisson tensor completion non-parametric differential entropy estimator

We introduce the Poisson tensor completion (PTC) estimator, a non-parametric differential entropy estimator. The PTC estimator leverages inter-sample relationships to compute a low-rank Poisson tensor decomposition of the frequency histogram. Our crucial observation is that the histogram bins are an instance of a space partitioning of counts and thus can be identified with a spatial Poisson process. The Poisson tensor decomposition leads to a completion of the intensity measure over all bins—including those containing few to no samples—and leads to our proposed PTC differential entropy estimator. A Poisson tensor decomposition models the underlying distribution of the count data and guarantees non-negative estimated values and so can be safely used directly in entropy estimation. Our estimator is the first tensor-based estimator that exploits the underlying spatial Poisson process related to the histogram explicitly when estimating the probability density with low-rank tensor decompositions for the purpose of tensor completion. Furthermore, we demonstrate that our PTC estimator is a substantial improvement over standard histogram-based estimators for sub-Gaussian probability distributions because of the concentration of norm phenomenon.

42 ENGINEERING↗

DOE OSTI2026

The Poisson tensor completion parametric estimator

We introduce the Poisson tensor completion (PTC) estimator that exploits inter-sample relationships to compute a low-rank Poisson tensor decomposition of the frequency histogram for samples of a multivariate distribution. Our crucial observation is that the histogram bins are an instance of a space partitioning of counts and thus can be identified with a spatial non-homogeneous Poisson process. The Poisson tensor decomposition leads to a completion of the mean measure over all bins—including those containing few to no samples—and leads to our proposed estimator. A Poisson tensor decomposition models the underlying distribution of the count data and guarantees non-negative estimated values obviating the need for additional constraints to ensure non-negativity. Furthermore, we demonstrate that our PTC estimator is a substantial improvement over standard histogram-based estimators for sub-Gaussian probability distributions because of the concentration of norm phenomenon.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Operationally induced preferred basis in unitary quantum mechanics

The preferred-basis problem and the definite-outcome aspect of the measurement problem persist even if the detector is modeled unitarily, because experimental data are necessarily represented in a Boolean event algebra of mutually exclusive records whereas the theoretical description is naturally formulated in a noncommutative operator algebra with continuous unitary symmetry. This change of mathematical type constitutes the core of the 'cut': a structurally necessary interface from group-based kinematics to set-based counting. In the presented view the basis relevant for recorded outcomes is not determined by the system Hamiltonian alone; it is induced by the measurement mapping, i.e., by the detector channel together with the coarse-grained readout that defines an instrument. The probabilistic mapping is anchored in symmetry and measure theory: by Gleason-type uniqueness (Gleason for projections in $d>2$ and Busch's extension for Positive Operator-Valued Measures (POVMs) including $d=2$), the trace rule is the unique probability measure consistent with additivity over exclusive events and basis-independence of the unitary sector. A compact qubit--pointer model yields an induced unsharp POVM $E_\pm=\tfrac12(\id\pm η\,σ_z)$ with $η$ fixed by pointer resolution, displaying explicitly how the detector induces the relevant basis. Finally, nested-observer paradoxes are tightened into a non-composability lemma: joint assignment of outcome propositions is obstructed unless a joint instrument exists. This relocates the origin of randomness to the stochasticity of the transition rules.

Pronskikh, Vitaly [Fermilab] (ORCID:00000002518174↗

NASA NTRS2016

Statistical Issues for Calculating Reentry Hazards

A number of statistical tools have been developed over the years for assessing the risk of reentering object to human populations. These tools make use of the characteristics (e.g., mass, shape, size) of debris that are predicted by aerothermal models to survive reentry. This information, combined with information on the expected ground path of the reentry, is used to compute the probability that one or more of the surviving debris might hit a person on the ground and cause one or more casualties. The statistical portion of this analysis relies on a number of assumptions about how the debris footprint and the human population are distributed in latitude and longitude, and how to use that information to arrive at realistic risk numbers. This inevitably involves assumptions that simplify the problem and make it tractable, but it is often difficult to test the accuracy and applicability of these assumptions. This paper builds on previous IAASS work to re-examine many of these theoretical assumptions, including the mathematical basis for the hazard calculations, and outlining the conditions under which the simplifying assumptions hold. This study also employs empirical and theoretical information to test these assumptions, and makes recommendations how to improve the accuracy of these calculations in the future.

Matney, Mark↗

NASA NTRS2003

Intelligent machines in the twenty-first century: foundations of inference and inquiry

The last century saw the application of Boolean algebra to the construction of computing machines, which work by applying logical transformations to information contained in their memory. The development of information theory and the generalization of Boolean algebra to Bayesian inference have enabled these computing machines, in the last quarter of the twentieth century, to be endowed with the ability to learn by making inferences from data. This revolution is just beginning as new computational techniques continue to make difficult problems more accessible. Recent advances in our understanding of the foundations of probability theory have revealed implications for areas other than logic. Of relevance to intelligent machines, we recently identified the algebra of questions as the free distributive algebra, which will now allow us to work with questions in a way analogous to that which Boolean algebra enables us to work with logical statements. In this paper, we examine the foundations of inference and inquiry. We begin with a history of inferential reasoning, highlighting key concepts that have led to the automation of inference in modern machine-learning systems. We then discuss the foundations of inference in more detail using a modern viewpoint that relies on the mathematics of partially ordered sets and the scaffolding of lattice theory. This new viewpoint allows us to develop the logic of inquiry and introduce a measure describing the relevance of a proposed question to an unresolved issue. Last, we will demonstrate the automation of inference, and discuss how this new logic of inquiry will enable intelligent machines to ask questions. Automation of both inference and inquiry promises to allow robots to perform science in the far reaches of our solar system and in other star systems by enabling them not only to make inferences from data, but also to decide which question to ask, which experiment to perform, or which measurement to take given what they have learned and what they are designed to understand.

Review↗

NASA NTRS1973

Mathematical models for the reflection coefficients of dielectric half-spaces

The reflection coefficients at normal incidence are found for a large class of one-dimensionally inhomogeneous or stratified half-spaces, which contain a homogeneous half-space. The formulation of the problem involves a combination of the classical boundary value technique, and the nonclassical principle of invariant imbedding. Solutions are in closed form and expressible in terms of Bessel functions. All results are given in terms of the ratio of the distance between free space and the homogeneous half-space to the wavelength in vacuo. One special case is that of an arbitrary number of layers lying on a homogeneous half-space where the dielectric constant of each layer has a constant gradient. A number of other special cases, limiting cases, and generalizations are developed including one in which the thickness of the top layer obeys a probability distribution.

Evans, D. D.↗