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At least 145 records · Page 8

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NASA NTRS2001

UNIPASS for AvSP? A Broader View

UNIPASS is a general-purpose probabilistic computer program consisting of three major modules, including preprocessor, solver and postprocessor. UNIPASS contains a user-friendly Graphical User Interface (GUI), numerous state-of-the-art probabilistic analysis techniques, a large library of statistical distributions and a function module with a large library of support functions that can easily define any complex limit-state function in a scripting FORTRAN-like syntax format. Its inverse probability analysis and sensitivities analysis capabilities make it a powerful design aid in any product cycle. Its precise numerical analysis engine is accurate enough to push the failure probabilities of a design to well below 10 (exp -50). UNIPASS is equipped with advanced artificial intelligence that is designed to handle systems with an essentially unlimited number of random variables with ease and efficiency. Its modular arrangement allows you to tailor an analysis to the desired level of accuracy and efficiency. The depth and comprehensiveness of UNIPASS are built upon the decades of experience and expertise of industry leaders including Boeing Aircraft, NASA and the DoD. Its rich content also makes UNIPASS a valuable instructional tool for random processes and probabilistic mechanics. The topics include: 1) Reliability in AvSP; 2) Role of UNIPASS in AvSP; and 3) Examples. This paper is in viewgraph form.

Wu, N. Eva↗

NASA NTRS1991

Computational methods for structural load and resistance modeling

An automated capability for computing structural reliability considering uncertainties in both load and resistance variables is presented. The computations are carried out using an automated Advanced Mean Value iteration algorithm (AMV +) with performance functions involving load and resistance variables obtained by both explicit and implicit methods. A complete description of the procedures used is given as well as several illustrative examples, verified by Monte Carlo Analysis. In particular, the computational methods described in the paper are shown to be quite accurate and efficient for a material nonlinear structure considering material damage as a function of several primitive random variables. The results show clearly the effectiveness of the algorithms for computing the reliability of large-scale structural systems with a maximum number of resolutions.

Thacker, B. H.↗

DOE OSTI2021

An adaptive Hessian approximated stochastic gradient MCMC method

Bayesian approaches have been successfully integrated into training deep neural networks. One popular family is stochastic gradient Markov chain Monte Carlo methods (SG-MCMC), which have gained increasing interest due to their ability to handle large datasets and the potential to avoid overfitting. Although standard SG-MCMC methods have shown great performance in a variety of problems, they may be inefficient when the random variables in the target posterior densities have scale differences or are highly correlated. Here, we present an adaptive Hessian approximated stochastic gradient MCMC method to incorporate local geometric information while sampling from the posterior. The idea is to apply stochastic approximation (SA) to sequentially update a preconditioning matrix at each iteration. The preconditioner possesses second-order information and can guide the random walk of a sampler efficiently. Instead of computing and saving the full Hessian of the log posterior, we use limited memory of the samples and their stochastic gradients to approximate the inverse Hessian-vector multiplication in the updating formula. Moreover, by smoothly optimizing the preconditioning matrix via SA, our proposed algorithm can asymptotically converge to the target distribution with a controllable bias under mild conditions. To reduce the training and testing computational burden, we adopt a magnitude-based weight pruning method to enforce the sparsity of the network. Our method is user-friendly and demonstrates better learning results compared to standard SG-MCMC updating rules. The approximation of inverse Hessian alleviates storage and computational complexities for large dimensional models. Numerical experiments are performed on several problems, including sampling from 2D correlated distribution, synthetic regression problems, and learning the numerical solutions of heterogeneous elliptic PDE. The numerical results demonstrate great improvement in both the convergence rate and accuracy.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2002

Probabilistic Study of Fluid Structure Interaction

A combustor liner was computationally simulated and probabilistically evaluated in view of the several uncertainties in the aerodynamic, structural, material and thermal variables that govern the combustor liner. The interconnection between the computational fluid dynamics code and the finite element structural analysis codes was necessary to couple the thermal profiles with structural design. The stresses and their variations were evaluated at critical points on the liner. Cumulative distribution functions and sensitivity factors were computed for stress responses due to the aerodynamic, mechanical and thermal random variables. It was observed that the inlet and exit temperatures have a lot of influence on the hoop stress. For prescribed values of inlet and exit temperatures, the Reynolds number of the flow, coefficient of thermal expansion, gas emissivity and absorptivity and thermal conductivity of the material have about the same impact on the hoop stress. These results can be used to quickly identify the most critical design variables in order to optimize the design and make it cost effective.

Gorla, Rama S. R.↗

NASA NTRS2004

Fluid Structure Interaction in a Turbine Blade

An unsteady, three dimensional Navier-Stokes solution in rotating frame formulation for turbomachinery applications is presented. Casting the governing equations in a rotating frame enabled the freezing of grid motion and resulted in substantial savings in computer time. The turbine blade was computationally simulated and probabilistically evaluated in view of several uncertainties in the aerodynamic, structural, material and thermal variables that govern the turbine blade. The interconnection between the computational fluid dynamics code and finite element structural analysis code was necessary to couple the thermal profiles with the structural design. The stresses and their variations were evaluated at critical points on the Turbine blade. Cumulative distribution functions and sensitivity factors were computed for stress responses due to aerodynamic, geometric, mechanical and thermal random variables.

Gorla, Rama S. R.↗

NASA NTRS2006

Exploring the Connection Between Sampling Problems in Bayesian Inference and Statistical Mechanics

The Bayesian and statistical mechanical communities often share the same objective in their work - estimating and integrating probability distribution functions (pdfs) describing stochastic systems, models or processes. Frequently, these pdfs are complex functions of random variables exhibiting multiple, well separated local minima. Conventional strategies for sampling such pdfs are inefficient, sometimes leading to an apparent non-ergodic behavior. Several recently developed techniques for handling this problem have been successfully applied in statistical mechanics. In the multicanonical and Wang-Landau Monte Carlo (MC) methods, the correct pdfs are recovered from uniform sampling of the parameter space by iteratively establishing proper weighting factors connecting these distributions. Trivial generalizations allow for sampling from any chosen pdf. The closely related transition matrix method relies on estimating transition probabilities between different states. All these methods proved to generate estimates of pdfs with high statistical accuracy. In another MC technique, parallel tempering, several random walks, each corresponding to a different value of a parameter (e.g. "temperature"), are generated and occasionally exchanged using the Metropolis criterion. This method can be considered as a statistically correct version of simulated annealing. An alternative approach is to represent the set of independent variables as a Hamiltonian system. Considerab!e progress has been made in understanding how to ensure that the system obeys the equipartition theorem or, equivalently, that coupling between the variables is correctly described. Then a host of techniques developed for dynamical systems can be used. Among them, probably the most powerful is the Adaptive Biasing Force method, in which thermodynamic integration and biased sampling are combined to yield very efficient estimates of pdfs. The third class of methods deals with transitions between states described by rate constants. These problems are isomorphic with chemical kinetics problems. Recently, several efficient techniques for this purpose have been developed based on the approach originally proposed by Gillespie. Although the utility of the techniques mentioned above for Bayesian problems has not been determined, further research along these lines is warranted

Pohorille, Andrew↗

NASA NTRS1989

Probabilities and statistics for backscatter estimates obtained by a scatterometer with applications to new scatterometer design data

The values of the Normalized Radar Backscattering Cross Section (NRCS), sigma (o), obtained by a scatterometer are random variables whose variance is a known function of the expected value. The probability density function can be obtained from the normal distribution. Models for the expected value obtain it as a function of the properties of the waves on the ocean and the winds that generated the waves. Point estimates of the expected value were found from various statistics given the parameters that define the probability density function for each value. Random intervals were derived with a preassigned probability of containing that value. A statistical test to determine whether or not successive values of sigma (o) are truly independent was derived. The maximum likelihood estimates for wind speed and direction were found, given a model for backscatter as a function of the properties of the waves on the ocean. These estimates are biased as a result of the terms in the equation that involve natural logarithms, and calculations of the point estimates of the maximum likelihood values are used to show that the contributions of the logarithmic terms are negligible and that the terms can be omitted.

Pierson, Willard J., Jr.↗

NASA NTRS2018

Stochastic Reduced Order Models with Python (SROMPy)

Stochastic Reduced Order Models with Python (SROMPy) is a software package developed to enable user-friendly utilization of the stochastic reduced order model (SROM) approach for uncertainty quantification. A SROM is a low dimensional, discrete approximation to a random quantity that enables efficient and non-intrusive stochastic computations. With SROMPy, a user can easily generate a SROM to approximate a random variable or vector described by several different types of probability distributions using the Python programming language. Once a SROM is constructed, the software can be used to propagate uncertainty through a user-defined computational model to estimate statistics of a given quantity of interest. This report is meant to introduce the SROMPy module and brie y demonstrate its capabilities. A simple example of a spring-mass system with a random input is included to illustrate the practicality of the SROM approach to uncertainty quantification and relative ease of applying it with SROMPy. The example includes a comparison with a solution obtained using classical Monte Carlo simulation, demonstrating the similarities and advantages of using the SROM approach.

Warner, James E.↗

DOE OSTI2024

Distribution of centrality measures on undirected random networks via the cavity method

The Katz centrality of a node in a complex network is a measure of the node’s importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its “global” character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-Cavity algorithm to solve linear systems on tree-like structures, we show that i) the Katz centrality of a single instance can be computed recursively in a very fast way, and ii) the probability P ( K ) that a random node in the ensemble of undirected random graphs has centrality K satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and Scale Free networks in the locally tree-like regime, with excellent agreement. The analytical distribution of centrality for the configuration model conditioned on the degree of each node can be employed as a benchmark to identify nodes of empirical networks with over- and underexpressed centrality relative to a null baseline. We also provide an approximate formula based on a rank- 1 projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1971

Comments on the continuity of distribution functions obtained by superposition.

Assuming that Y is a nonnegative random variable independent of the differential process X(t), attention is given to the question of whether or not the superposition X(Y) can have a continuous probability distribution. If the process has continuous distributions, then the superposition is continuous if and only if P/Y = 0/ = 0. If the process has discontinuous distributions and no trend, then no superposition can have continuous distribution. If the process has discontinuous distributions and nonzero trend, then the superposition onto a random epoch has continuous distribution if and only if Y has continuous distribution.

Huff, B. W.↗

DOE OSTI2022

Behavior and mechanisms of Doppler wind lidar error in varying stability regimes

Abstract. Wind lidars are widespread and important tools in atmospheric observations. An intrinsic part of lidar measurement error is due to atmospheric variability in the remote-sensing scan volume. This study describes and quantifies the distribution of measurement error due to turbulence in varying atmospheric stability. While the lidar error model is general, we demonstrate the approach using large ensembles of virtual WindCube V2 lidar performing a profiling Doppler-beam-swinging scan in quasi-stationary large-eddy simulations (LESs) of convective and stable boundary layers. Error trends vary with the stability regime, time averaging of results, and observation height. A systematic analysis of the observation error explains dominant mechanisms and supports the findings of the empirical results. Treating the error under a random variable framework allows for informed predictions about the effect of different configurations or conditions on lidar performance. Convective conditions are most prone to large errors (up to 1.5 m s−1 in 1 Hz wind speed in strong convection), driven by the large vertical velocity variances in convective conditions and the high elevation angle of the scanning beams (62∘). Range-gate weighting induces a negative bias into the horizontal wind speeds near the surface shear layer (−0.2 m s−1 in the stable test case). Errors in the horizontal wind speed and direction computed from the wind components are sensitive to the background wind speed but have negligible dependence on the relative orientation of the instrument. Especially during low winds and in the presence of large errors in the horizontal velocity estimates, the reported wind speed is subject to a systematic positive bias (up to 0.4 m s−1 in 1 Hz measurements in strong convection). Vector time-averaged measurements can improve the behavior of the error distributions (reducing the 10 min wind speed error standard deviation to <0.3 m s−1 and the bias to <0.1 m s−1 in strong convection) with a predictable effectiveness related to the number of decorrelated samples in the time window. Hybrid schemes weighting the 10 min scalar- and vector-averaged lidar measurements are shown to be effective at reducing the wind speed biases compared to cup measurements in most of the simulated conditions, with time averages longer than 10 min recommended for best use in some unstable conditions. The approach in decomposing the error mechanisms with the help of the LES flow field could be extended to more complex measurement scenarios and scans.

17 WIND ENERGY↗

DOE OSTI2022

An efficient method to propagate model uncertainty when inverting seismic data for time domain seismic moment tensors

SUMMARY We present a computationally efficient method to approximately propagate uncertainty when linearly inverting seismic data for point source, time variable moment tensor components. The method is based on the assumption that the data residual, given by the difference between the observed seismic data and the data predicated by a linear inversion, contains the effects of both data and model uncertainty. Our method uses a distribution of data residuals, added directly to the data, in a pseudo-Monte Carlo scheme. Using the assumption that the data residual is a stochastic process, we use the well-known Karhunen–Loève (KL) theorem to construct a distribution of data residuals, where the required basis functions are constructed using Fourier series. The Fourier series are scaled by a product of a random variable and the real-valued spectral amplitudes of the original data residual’s spectrum. Thus, the Fourier series and spectral amplitudes are eigenfunction-eigenvalue pairs used in the KL-based construction of data residual distribution. Using tests with synthetic data, we show that our method compares closely with a Finite Difference Monte Carlo (FDMC) method that we presented previously. More importantly, the method presented here is computationally several orders of magnitude faster than our previous FDMC method, and requires no a priori assumptions of model and/or data uncertainty.

Poppeliers, Christian (ORCID:0000000159526849)↗

DOE OSTI2025

A Class of Sparse Johnson–Lindenstrauss Transforms and Analysis of their Extreme Singular Values

The Johnson–Lindenstrauss (JL) lemma is a powerful tool for dimensionality reduction in modern algorithm design. The lemma states that any set of high-dimensional points in a Euclidean space can be projected into lower dimensions while approximately preserving pairwise Euclidean distances. Random matrices satisfying this lemma are called JL transforms (JLTs). Inspired by existing $s$-hashing JLTs with exactly $s$ nonzero elements on each column, the present work introduces an ensemble of sparse matrices encompassing so-called $s$-hashing-like matrices whose expected number of nonzero elements on each column is $s$. The independence of the sub-Gaussian entries of these matrices and the knowledge of their exact distribution play an important role in their analyses. Using properties of independent sub-Gaussian random variables, these matrices are demonstrated to be JLTs, and their smallest nontrivial singular values and largest singular values are estimated nonasymptotically using a technique from geometric functional analysis. As the dimensions of the matrix grow to infinity, these singular values are proved to converge almost surely to fixed quantities (by using the universal Bai–Yin law) and in distribution to the Gaussian orthogonal ensemble Tracy–Widom law after proper rescalings. Understanding the behaviors of extreme singular values is important in general because they are often used to define a measure of stability of matrix algorithms. For example, JLTs were recently used in derivative-free optimization algorithmic frameworks to select random subspaces in which are constructed random models or poll directions to achieve scalability, and hence estimating their smallest singular value in particular helps determine the dimension of these subspaces.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2014

Statistical Mechanics of Turbulent Dynamos

Incompressible magnetohydrodynamic (MHD) turbulence and magnetic dynamos, which occur in magnetofluids with large fluid and magnetic Reynolds numbers, will be discussed. When Reynolds numbers are large and energy decays slowly, the distribution of energy with respect to length scale becomes quasi-stationary and MHD turbulence can be described statistically. In the limit of infinite Reynolds numbers, viscosity and resistivity become zero and if these values are used in the MHD equations ab initio, a model system called ideal MHD turbulence results. This model system is typically confined in simple geometries with some form of homogeneous boundary conditions, allowing for velocity and magnetic field to be represented by orthogonal function expansions. One advantage to this is that the coefficients of the expansions form a set of nonlinearly interacting variables whose behavior can be described by equilibrium statistical mechanics, i.e., by a canonical ensemble theory based on the global invariants (energy, cross helicity and magnetic helicity) of ideal MHD turbulence. Another advantage is that truncated expansions provide a finite dynamical system whose time evolution can be numerically simulated to test the predictions of the associated statistical mechanics. If ensemble predictions are the same as time averages, then the system is said to be ergodic; if not, the system is nonergodic. Although it had been implicitly assumed in the early days of ideal MHD statistical theory development that these finite dynamical systems were ergodic, numerical simulations provided sufficient evidence that they were, in fact, nonergodic. Specifically, while canonical ensemble theory predicted that expansion coefficients would be (i) zero-mean random variables with (ii) energy that decreased with length scale, it was found that although (ii) was correct, (i) was not and the expected ergodicity was broken. The exact cause of this broken ergodicity was explained, after much investigation, by greatly extending the statistical theory of ideal MHD turbulence. The mathematical details of broken ergodicity, in fact, give a quantitative explanation of how coherent structure, dynamic alignment and force-free states appear in turbulent magnetofluids. The relevance of these ideal results to real MHD turbulence occurs because broken ergodicity is most manifest in the ideal case at the largest length scales and it is in these largest scales that a real magnetofluid has the least dissipation, i.e., most closely approaches the behavior of an ideal magnetofluid. Furthermore, the effects grow stronger when cross and magnetic helicities grow large with respect to energy, and this is exactly what occurs with time in a real magnetofluid, where it is called selective decay. The relevance of these results found in ideal MHD turbulence theory to the real world is that they provide at least a qualitative explanation of why confined turbulent magnetofluids, such as the liquid iron that fills the Earth's outer core, produce stationary, large-scale magnetic fields, i.e., the geomagnetic field. These results should also apply to other planets as well as to plasma confinement devices on Earth and in space, and the effects should be manifest if Reynolds numbers are high enough and there is enough time for stationarity to occur, at least approximately. In the presentation, details will be given for both theoretical and numerical results, and references will be provided.

Shebalin, John V.↗

NASA NTRS2014

MHD Turbulence and Magnetic Dynamos

Shebalin, John V↗

NASA NTRS2001

Distribution functions of probabilistic automata

Each probabilistic automaton M over an alphabet A defines a probability measure Prob sub(M) on the set of all finite and infinite words over A. We can identify a k letter alphabet A with the set {0, 1,..., k-1}, and, hence, we can consider every finite or infinite word w over A as a radix k expansion of a real number X(w) in the interval [0, 1]. This makes X(w) a random variable and the distribution function of M is defined as usual: F(x) := Prob sub(M) { w: X(w) < x }. Utilizing the fixed-point semantics (denotational semantics), extended to probabilistic computations, we investigate the distribution functions of probabilistic automata in detail. Automata with continuous distribution functions are characterized. By a new, and much more easier method, it is shown that the distribution function F(x) is an analytic function if it is a polynomial. Finally, answering a question posed by D. Knuth and A. Yao, we show that a polynomial distribution function F(x) on [0, 1] can be generated by a prob abilistic automaton iff all the roots of F'(x) = 0 in this interval, if any, are rational numbers. For this, we define two dynamical systems on the set of polynomial distributions and study attracting fixed points of random composition of these two systems.

Denotational semantics↗

NASA NTRS2003

Structural Reliability Using Probability Density Estimation Methods Within NESSUS

A reliability analysis studies a mathematical model of a physical system taking into account uncertainties of design variables and common results are estimations of a response density, which also implies estimations of its parameters. Some common density parameters include the mean value, the standard deviation, and specific percentile(s) of the response, which are measures of central tendency, variation, and probability regions, respectively. Reliability analyses are important since the results can lead to different designs by calculating the probability of observing safe responses in each of the proposed designs. All of this is done at the expense of added computational time as compared to a single deterministic analysis which will result in one value of the response out of many that make up the density of the response. Sampling methods, such as monte carlo (MC) and latin hypercube sampling (LHS), can be used to perform reliability analyses and can compute nonlinear response density parameters even if the response is dependent on many random variables. Hence, both methods are very robust; however, they are computationally expensive to use in the estimation of the response density parameters. Both methods are 2 of 13 stochastic methods that are contained within the Numerical Evaluation of Stochastic Structures Under Stress (NESSUS) program. NESSUS is a probabilistic finite element analysis (FEA) program that was developed through funding from NASA Glenn Research Center (GRC). It has the additional capability of being linked to other analysis programs; therefore, probabilistic fluid dynamics, fracture mechanics, and heat transfer are only a few of what is possible with this software. The LHS method is the newest addition to the stochastic methods within NESSUS. Part of this work was to enhance NESSUS with the LHS method. The new LHS module is complete, has been successfully integrated with NESSUS, and been used to study four different test cases that have been proposed by the Society of Automotive Engineers (SAE). The test cases compare different probabilistic methods within NESSUS because it is important that a user can have confidence that estimates of stochastic parameters of a response will be within an acceptable error limit. For each response, the mean, standard deviation, and 0.99 percentile, are repeatedly estimated which allows confidence statements to be made for each parameter estimated, and for each method. Thus, the ability of several stochastic methods to efficiently and accurately estimate density parameters is compared using four valid test cases. While all of the reliability methods used performed quite well, for the new LHS module within NESSUS it was found that it had a lower estimation error than MC when they were used to estimate the mean, standard deviation, and 0.99 percentile of the four different stochastic responses. Also, LHS required a smaller amount of calculations to obtain low error answers with a high amount of confidence than MC. It can therefore be stated that NESSUS is an important reliability tool that has a variety of sound probabilistic methods a user can employ and the newest LHS module is a valuable new enhancement of the program.

Chamis, Chrisos C.↗

NASA NTRS2002

A simple way to measure daily lifestyle regularity

A brief diary instrument to quantify daily lifestyle regularity (SRM-5) is developed and compared with a much longer version of the instrument (SRM-17) described and used previously. Three studies are described. In Study 1, SRM-17 scores (2 weeks) were collected from a total of 293 healthy control subjects (both genders) aged between 19 and 92 years. Five items (1) Get out of bed, (2) First contact with another person, (3) Start work, housework or volunteer activities, (4) Have dinner, and (5) Go to bed were then selected from the 17 items and SRM-5 scores calculated as if these five items were the only ones collected. Comparisons were made with SRM-17 scores from the same subject-weeks, looking at correlations between the two SRM measures, and the effects of age and gender on lifestyle regularity as measured by the two instruments. In Study 2 this process was repeated in a group of 27 subjects who were in remission from unipolar depression after treatment with psychotherapy and who completed SRM-17 for at least 20 successive weeks. SRM-5 and SRM-17 scores were then correlated within an individual using time as the random variable, allowing an indication of how successful SRM-5 was in tracking changes in lifestyle regularity (within an individual) over time. In Study 3 an SRM-5 diary instrument was administered to 101 healthy control subjects (both genders, aged 20-59 years) for two successive weeks to obtain normative measures and to test for correlations with age and morningness. Measures of lifestyle regularity from SRM-5 correlated quite well (about 0.8) with those from SRM-17 both between subjects, and within-subjects over time. As a detector of irregularity as defined by SRM-17, the SRM-5 instrument showed acceptable values of kappa (0.69), sensitivity (74%) and specificity (95%). There were, however, differences in mean level, with SRM-5 scores being about 0.9 units [about one standard deviation (SD)] above SRM-17 scores from the same subject-weeks. SRM-5 scores also deviated more from a Gaussian distribution than did SRM-17 ones. In a study with a sample size of 101, the new SRM-5 instrument yielded scores with a mean of 4.11 and an SD of 1.13. Correlations between lifestyle regularity and age, and between lifestyle regularity and morningness appeared similar whether 5-item or 17-item SRM measures were used. When a gender difference in lifestyle regularity appeared, it was detected by both SRM-5 and SRM-17 measures.

Non-NASA Center↗