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

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NASA NTRS1979

Monte Carlo simulation of single accident airport risk profile

A computer simulation model was developed for estimating the potential economic impacts of a carbon fiber release upon facilities within an 80 kilometer radius of a major airport. The model simulated the possible range of release conditions and the resulting dispersion of the carbon fibers. Each iteration of the model generated a specific release scenario, which would cause a specific amount of dollar loss to the surrounding community. By repeated iterations, a risk profile was generated, showing the probability distribution of losses from one accident. Using accident probability estimates, the risks profile for annual losses was derived. The mechanics are described of the simulation model, the required input data, and the risk profiles generated for the 26 large hub airports.

Source record↗
NASA NTRS2014

Service Life Extension of the Propulsion System of Long-Term Manned Orbital Stations

One of the critical non-replaceable systems of a long-term manned orbital station is the propulsion system. Since the propulsion system operates beginning with the launch of station elements into orbit, its service life determines the service life of the station overall. Weighing almost a million pounds, the International Space Station (ISS) is about four times as large as the Russian space station Mir and about five times as large as the U.S. Skylab. Constructed over a span of more than a decade with the help of over 100 space flights, elements and modules of the ISS provide more research space than any spacecraft ever built. Originally envisaged for a service life of fifteen years, this Earth orbiting laboratory has been in orbit since 1998. Some elements that have been launched later in the assembly sequence were not yet built when the first elements were placed in orbit. Hence, some of the early modules that were launched at the inception of the program were already nearing the end of their design life when the ISS was finally ready and operational. To maximize the return on global investments on ISS, it is essential for the valuable research on ISS to continue as long as the station can be sustained safely in orbit. This paper describes the work performed to extend the service life of the ISS propulsion system. A system comprises of many components with varying failure rates. Reliability of a system is the probability that it will perform its intended function under encountered operating conditions, for a specified period of time. As we are interested in finding out how reliable a system would be in the future, reliability expressed as a function of time provides valuable insight. In a hypothetical bathtub shaped failure rate curve, the failure rate, defined as the number of failures per unit time that a currently healthy component will suffer in a given future time interval, decreases during infant-mortality period, stays nearly constant during the service life and increases at the end when the design service life ends and wear-out phase begins. However, the component failure rates do not remain constant over the entire cycle life. The failure rate depends on various factors such as design complexity, current age of the component, operating conditions, severity of environmental stress factors, etc. Development, qualification and acceptance test processes provide rigorous screening of components to weed out imperfections that might otherwise cause infant mortality failures. If sufficient samples are tested to failure, the failure time versus failure quantity can be analyzed statistically to develop a failure probability distribution function (PDF), a statistical model of the probability of failure versus time. Driven by cost and schedule constraints however, spacecraft components are generally not tested in large numbers. Uncertainties in failure rate and remaining life estimates increase when fewer units are tested. To account for this, spacecraft operators prefer to limit useful operations to a period shorter than the maximum demonstrated service life of the weakest component. Running each component to its failure to determine the maximum possible service life of a system can become overly expensive and impractical. Spacecraft operators therefore, specify the required service life and an acceptable factor of safety (FOS). The designers use these requirements to limit the life test duration. Midway through the design life, when benefits justify additional investments, supplementary life test may be performed to demonstrate the capability to safely extend the service life of the system. An innovative approach is required to evaluate the entire system, without having to go through an elaborate test program of propulsion system elements. Evaluating every component through a brute force test program would be a cost prohibitive and time consuming endeavor. ISS propulsion system components were designed and built decades ago. There are no representative ground test articles for some of the components. A 'test everything' approach would require manufacturing new test articles. The paper outlines some of the techniques used for selective testing, by way of cherry picking candidate components based on failure mode effects analysis, system level impacts, hazard analysis, etc. The type of testing required for extending the service life depends on the design and criticality of the component, failure modes and failure mechanisms, life cycle margin provided by the original certification, operational and environmental stresses encountered, etc. When specific failure mechanism being considered and the underlying relationship of that mode to the stresses provided in the test can be correlated by supporting analysis, time and effort required for conducting life extension testing can be significantly reduced. Exposure to corrosive propellants over long periods of time, for instance, lead to specific failure mechanisms in several components used in the propulsion system. Using Arrhenius model, which is tied to chemically dependent failure mechanisms such as corrosion or chemical reactions, it is possible to subject carefully selected test articles to accelerated life test. Arrhenius model reflects the proportional relationship between time to failure of a component and the exponential of the inverse of absolute temperature acting on the component. The acceleration factor is used to perform tests at higher stresses that allow direct correlation between the times to failure at a high test temperature to the temperatures to be expected in actual use. As long as the temperatures are such that new failure mechanisms are not introduced, this becomes a very useful method for testing to failure a relatively small sample of items for a much shorter amount of time. In this article, based on the example of the propulsion system of the first ISS module Zarya, theoretical approaches and practical activities of extending the service life of the propulsion system are reviewed with the goal of determining the maximum duration of its safe operation.

Kamath, Ulhas↗
NASA NTRS1994

The intrinsic shape of NGC 3379

Photometric and kinematic data from the literature are combined with new dynamical models to derive the intrinsic shape of the 'standard' elliptical galaxy NGC 3379. The parameters that are best constrained are the dynamical triaxiality T (essentially the triaxiality of the total mass distribution) and the short-to long axis ratio of the light distribution c(sub L). The inferred shape is given by a Bayesian probability distribution in the (T, c(sub L) plane. Assuming a uniform prior, the most probable shape is oblate with a flattening of c(sub L) = 87. The distribution is strongly non-Gaussian, however, and the expectation values, (T) = .31 (c(sub L) = .75, imply a flatter and more triaxial figure. The 68% highest posterior density region allows more triaxial shapes as long as they are fairly round, or flatter shapes as long as they are nearly oblate. These results are essentially unchanged if the galaxy is assumed to rotate about its short axis, or if it is modeled as an S0 with a negligible-mass disk rather than as an elliptical. The suggestion of Capaccioli et al. (ApJ, 371, 535 (1991)) that NGC 3379 is a rather flat, triaxial S0 galaxy is found to be improbable at the 98% level; this conclusion is largely independent of the bulge-to-disk ratio or the relative rotation speeds of the two components.

Statler, Thomas S.↗
DOE OSTI2014

Modeling X-ray emission around galaxies

Extended X-ray emission can be studied by spatial surface brightness measurements or by spectral analysis, but the two methods can disagree at low intensity levels. Here we present an improved method for spatial analysis that can be extended to include spectral information simultaneously. We construct a model for the entire image in a given energy band and generate a likelihood function to compare the model to the data. A critical goal is disentangling vignetted and unvignetted backgrounds through their different spatial distributions. Employing either maximum likelihood or Markov Chain Monte Carlo, we can derive probability distributions for the source and background parameters together, or we can fit and subtract the background, leaving the description of the source non-parametric. We calibrate this method against a variety of simulated images, and apply it to Chandra observations of the hot gaseous halo around the elliptical galaxy NGC 720. We follow the emission below a tenth of the background and infer a hot gas mass within 35 kpc of 4-5 × 10{sup 9} M {sub ☉}, with some indication that the profile continues to at least 50 kpc and that it steepens. We derive stronger constraints on the surface brightness profile than previous studies that employed the spectral method, and we show that the density profiles inferred from these studies are in conflict with the observed surface brightness profile. Contrary to a previous claim, we find that the X-ray halo does not contain the full complement of missing baryons within the virial radius.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Testing the lognormality of the galaxy and weak lensing convergence distributions from Dark Energy Survey maps

It is well known that the probability distribution function (PDF) of galaxy density contrast is approximately lognormal; whether the PDF of mass fluctuations derived from weak lensing convergence (kappa_WL) is lognormal is less well established. We derive PDFs of the galaxy and projected matter density distributions via the Counts in Cells (CiC) method. We use maps of galaxies and weak lensing convergence produced from the Dark Energy Survey (DES) Science Verification data over 139 deg^2. We test whether the underlying density contrast is well described by a lognormal distribution for the galaxies, the convergence and their joint PDF. We confirm that the galaxy density contrast distribution is well modeled by a lognormal PDF convolved with Poisson noise at angular scales from 10-40 arcmin (corresponding to physical scales of 3-10 Mpc). We note that as kappa_WL is a weighted sum of the mass fluctuations along the line of sight, its PDF is expected to be only approximately lognormal. We find that the kappa_WL distribution is well modeled by a lognormal PDF convolved with Gaussian shape noise at scales between 10 and 20 arcmin, with a best-fit chi^2/DOF of 1.11 compared to 1.84 for a Gaussian model, corresponding to p-values 0.35 and 0.07 respectively, at a scale of 10 arcmin. Above 20 arcmin a simple Gaussian model is sufficient. The joint PDF is also reasonably fitted by a bivariate lognormal. As a consistency check we compare the variances derived from the lognormal modelling with those directly measured via CiC. Our methods are validated against maps from the MICE Grand Challenge N-body simulation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2012

Review of the WECC EDT phase 2 EIM benefits analysis and results report.

A region-wide Energy Imbalance Market (EIM) was recently proposed by the Western Electricity Coordinating Council (WECC). In order for the Western Area Power Administration (Western) to make more informed decisions regarding its involvement in the EIM, Western asked Argonne National Laboratory (Argonne) to review the EIM benefits study (the October 2011 revision) performed by Energy and Environmental Economics, Inc. (E3). Key components of the E3 analysis made use of results from a study conducted by the National Renewable Energy Laboratory (NREL); therefore, we also reviewed the NREL work. This report examines E3 and NREL methods and models used in the EIM study. Estimating EIM benefits is very challenging because of the complex nature of the Western Interconnection (WI), the variability and uncertainty of renewable energy resources, and the complex decisions and potentially strategic bidding of market participants. Furthermore, methodologies used for some of the more challenging aspects of the EIM have not yet matured. This review is complimentary of several components of the EIM study. Analysts and modelers clearly took great care when conducting detailed simulations of the WI using well-established industry tools under stringent time and budget constraints. However, it is our opinion that the following aspects of the study and the interpretation of model results could be improved upon in future analyses. The hurdle rate methodology used to estimate current market inefficiencies does not directly model the underlying causes of sub-optimal dispatch and power flows. It assumes that differences between historical flows and modeled flows can be attributed solely to market inefficiencies. However, flow differences between model results and historical data can be attributed to numerous simplifying assumptions used in the model and in the input data. We suggest that alternative approaches be explored in order to better estimate the benefits of introducing market structures like the EIM. In addition to more efficient energy transactions in the WI, the EIM would reduce the amount of flexibility reserves needed to accommodate forecast errors associated with variable production from wind and solar energy resources. The modeling approach takes full advantage of variable resource diversity over the entire market footprint, but the projected reduction in flexibility reserves may be overly optimistic. While some reduction would undoubtedly occur, the EIM is only an energy market and would therefore not realize the same reduction in reserves as an ancillary services market. In our opinion the methodology does not adequately capture the impact of transmission constraints on the deployment of flexibility reserves. Estimates of flexibility reserves assume that forecast errors follow a normal distribution. Improved estimates could be obtained by using other probability distributions to estimate up and down reserves to capture the underlying uncertainty of these resources under specific operating conditions. Also, the use of a persistence forecast method for solar is questionable, because solar insolation follows a deterministic pattern dictated by the sun's path through the sky. We suggest a more rigorous method for forecasting solar insolation using the sun's relatively predictable daily pattern at specific locations. The EIM study considered only one scenario for hydropower resources. While this scenario is within the normal range over the WI footprint, it represents a severe drought condition in the Colorado River Basin from which Western schedules power. Given hydropower's prominent role in the WI, we recommend simulating a range of hydropower conditions since the relationship between water availability and WI dispatch costs is nonlinear. Also, the representation of specific operational constraints faced by hydropower operators in the WI needs improvements. The model used in the study cannot fully capture all of the EIM impacts and complexities of power system operations. In particular, a primary benefit of the EIM is a shorter dispatch interval; namely, 5 minutes. However, the model simulates the dispatch hourly. Therefore it cannot adequately measure the benefits of a more frequent dispatch. A tool with a finer time resolution would significantly improve simulation accuracy. When the study was conducted, the rules for the EIM were not clearly defined and it was appropriate to estimate societal benefits of the EIM assuming a perfect market without a detailed specification of the market design. However, incorporating a more complete description of market rules will allow for better estimates of EIM benefits. Furthermore, performing analyses using specific market rules can identify potential design flaws that may be difficult and expensive to correct after the market is established. Estimated cost savings from a more efficient dispatch are less than one percent of the total cost of electricity production.

14 SOLAR ENERGY↗
NASA NTRS1994

AutoClass: A Bayesian Approach to Classification

We describe a Bayesian approach to the untutored discovery of classes in a set of cases, sometimes called finite mixture separation or clustering. The main difference between clustering and our approach is that we search for the "best" set of class descriptions rather than grouping the cases themselves. We describe our classes in terms of a probability distribution or density function, and the locally maximal posterior probability valued function parameters. We rate our classifications with an approximate joint probability of the data and functional form, marginalizing over the parameters. Approximation is necessitated by the computational complexity of the joint probability. Thus, we marginalize w.r.t. local maxima in the parameter space. We discuss the rationale behind our approach to classification. We give the mathematical development for the basic mixture model and describe the approximations needed for computational tractability. We instantiate the basic model with the discrete Dirichlet distribution and multivariant Gaussian density likelihoods. Then we show some results for both constructed and actual data.

Stutz, John↗
NASA NTRS1974

The beta distribution - A statistical model for world cloud cover

A statistical distribution representing worldwide cloud cover is obtained as a basis for an empirical global cloud cover model incorporating some 29 regions of the globe which are of interest to NASA's Skylab Program. A total of 160 representative empirical cloud cover distributions were investigated with the conclusion that the beta probability distribution is an adequate model for world cloud cover.

Falls, L. W.↗
NASA NTRS1980

Uncertainty in estimates of the number of extraterrestrial civilizations

Estimation of the number N of communicative civilizations by means of Drake's formula involves the combination of several quantities, each of which is to some extent uncertain. The uncertainty in any quantity may be represented by a probability distribution function, even if that quantity is itself a probability. The uncertainty of current estimates of N is derived principally from uncertainty in estimates of the lifetime of advanced civilizations. It is argued that this is due primarily to uncertainty concerning the existence of a "Galactic Federation" which is in turn contingent upon uncertainty about whether the limitations of present-day physics are absolute or (in the event that there exists a yet-undiscovered "hyperphysics") transient . It is further argued that it is advantageous to consider explicitly these underlying assumptions in order to compare the probable numbers of civilizations operating radio beacons, permitting radio leakage, dispatching probes for radio surveillance or dispatching vehicles for manned surveillance.

Peter A Sturrock↗
DOE OSTI2012

Calibration of a distributed flood forecasting model with input uncertainty using a Bayesian framework

In the process of calibrating distributed hydrological models, accounting for input uncertainty is important, yet challenging. In this study, we develop a Bayesian model to estimate parameters associated with a geomorphology‐based hydrological model (GBHM). The GBHM model uses geomorphic characteristics to simplify model structure and physically based methods to represent hydrological processes. We divide the observed discharge into low‐ and high‐flow data, and use the first‐order autoregressive model to describe their temporal dependence. We consider relative errors in rainfall as spatially distributed variables and estimate them jointly with the GBHM parameters. The joint posterior probability distribution is explored using Markov chain Monte Carlo methods, which include Metropolis‐Hastings, delay rejection adaptive Metropolis, and Gibbs sampling methods. We evaluate the Bayesian model using both synthetic and field data sets. The synthetic case study demonstrates that the developed method generally is effective in calibrating GBHM parameters and in estimating their associated uncertainty. The calibration ignoring input errors has lower accuracy and lower reliability compared to the calibration that includes estimation of the input errors, especially under model structure uncertainty. The field case study shows that calibration of GBHM parameters under complex field conditions remains a challenge. Although jointly estimating input errors and GBHM parameters improves the continuous ranked probability score and the consistency of the predictive distribution with the observed data, the improvement is incremental. To better calibrate parameters in a distributed model, such as GBHM here, we need to develop a more complex model and incorporate much more information.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2015

The azimuthal dependence of outflows and accretion detected using O vi absorption

We report a bimodality in the azimuthal angle (Φ) distribution of gas around galaxies traced by O vi absorption. We present the mean Φ probability distribution function of 29 Hubble Space Telescope-imaged O vi absorbing (EW > 0.1 Å) and 24 non-absorbing (EW < 0.1 Å) isolated galaxies (0.08 < z < 0.67) within ∼200 kpc of background quasars. We show that equivalent width (EW) is anti-correlated with impact parameter and O vi covering fraction decreases from 80% within 50 kpc to 33% at 200 kpc. The presence of O vi absorption is azimuthally dependent and occurs between ±10°–20° of the galaxy projected major axis and within ±30° of the projected minor axis. We find higher EWs along the projected minor axis with weaker EWs along the project major axis. Highly inclined galaxies have the lowest covering fractions due to minimized outflow/inflow cross-section geometry. Absorbing galaxies also have bluer colors while non-absorbers have redder colors, suggesting that star formation is a key driver in the O vi detection rate. O vi surrounding blue galaxies exists primarily along the projected minor axis with wide opening angles while O vi surrounding red galaxies exists primarily along the projected major axis with smaller opening angles, which may explain why absorption around red galaxies is less frequently detected. Our results are consistent with a circumgalactic medium (CGM) originating from major axis-fed inflows/recycled gas and from minor axis-driven outflows. Non-detected O vi occurs between Φ = 20°–60°, suggesting that O vi is not mixed throughout the CGM and remains confined within the outflows and the disk-plane. We find low O vi covering fractions within ±10{sup ∘} of the projected major axis, suggesting that cool dense gas resides in a narrow planer geometry surrounded by diffuse O vi gas.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

Identification of topology changes in power grids using phasor measurements

Phasor measurement units (PMUs) are increasingly important for monitoring the state of an electrical power grid and quickly detecting topology changes caused by events such as lines going down or large loads being dropped. Phasors are complex‐valued measurements of voltage and current at various points of generation and consumption. If a line goes down or a load is removed, power flows change throughout the grid according to known physical laws, and the probability distribution of phasor measurements changes accordingly. This paper develops a method to estimate the current topology of a power grid from phasor measurements and considers the design goal of placing PMUs at strategic points in a distribution system to achieve good sensitivity to single‐line outages. From a vector of phasor measurements, probabilities are computed corresponding to the scenario that all power lines are operational and to alternate scenarios in which each line goes down individually. These probabilities are functions of the joint distributions of phasor measurements under each possible scenario, obtained through Monte Carlo simulations with random load profiles. We use log‐spline densities to estimate marginal distributions of phasor measurements and fold these into a multivariate Gaussian copula to capture important correlations. Sensitivity to outages varies according to which line goes down and where PMUs are placed on the grid. A greedy search algorithm is demonstrated for placing PMUs at locations that provide good sensitivity to single‐line outages. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.

Wiel, Scott Vander↗
DOE OSTI2014

Ignition probability of polymer-bonded explosives accounting for multiple sources of material stochasticity

Accounting for the combined effect of multiple sources of stochasticity in material attributes, we develop an approach that computationally predicts the probability of ignition of polymer-bonded explosives (PBXs) under impact loading. The probabilistic nature of the specific ignition processes is assumed to arise from two sources of stochasticity. The first source involves random variations in material microstructural morphology; the second source involves random fluctuations in grain-binder interfacial bonding strength. The effect of the first source of stochasticity is analyzed with multiple sets of statistically similar microstructures and constant interfacial bonding strength. Subsequently, each of the microstructures in the multiple sets is assigned multiple instantiations of randomly varying grain-binder interfacial strengths to analyze the effect of the second source of stochasticity. Critical hotspot size-temperature states reaching the threshold for ignition are calculated through finite element simulations that explicitly account for microstructure and bulk and interfacial dissipation to quantify the time to criticality (t{sub c}) of individual samples, allowing the probability distribution of the time to criticality that results from each source of stochastic variation for a material to be analyzed. Two probability superposition models are considered to combine the effects of the multiple sources of stochasticity. The first is a parallel and series combination model, and the second is a nested probability function model. Results show that the nested Weibull distribution provides an accurate description of the combined ignition probability. The approach developed here represents a general framework for analyzing the stochasticity in the material behavior that arises out of multiple types of uncertainty associated with the structure, design, synthesis and processing of materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Self-similarity of k -nearest neighbour distributions in scale-free simulations

ABSTRACT We use the k-nearest neighbour probability distribution function (kNN-PDF; Banerjee & Abel 2021a) to assess convergence in a scale-free N-body simulation. Compared to our previous two-point analysis, the kNN-PDF allows us to quantify our results in the language of haloes and numbers of particles, while also incorporating non-Gaussian information. We find good convergence for 32 particles and greater at densities typical of haloes, while 16 particles and fewer appear unconverged. Halving the softening length extends convergence to higher densities, but not to fewer particles. Our analysis is less sensitive to voids, but we analyse a limited range of underdensities and find evidence for convergence at 16 particles and greater even in sparse voids.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

THE DISTRIBUTION OF MASS SURFACE DENSITIES IN A HIGH-MASS PROTOCLUSTER

We study the probability distribution function (PDF) of mass surface densities, Σ, of infrared dark cloud (IRDC) G028.37+00.07 and its surrounding giant molecular cloud. This PDF constrains the physical processes, such as turbulence, magnetic fields, and self-gravity, that are expected to be controlling cloud structure and star formation activity. The chosen IRDC is of particular interest since it has almost 100,000 solar masses within a radius of 8 pc, making it one of the most massive, dense molecular structures known and is thus a potential site for the formation of a “super star cluster.” We study Σ in two ways. First, we use a combination of NIR and MIR extinction maps that are able to probe the bulk of the cloud structure up to Σ ∼ 1 g cm{sup −2}( A {sub V}≃ 200 mag). Second, we study the FIR and submillimeter dust continuum emission from the cloud utilizing Herschel -PACS and SPIRE images and paying careful attention to the effects of foreground and background contamination. We find that the PDFs from both methods, applied over a ∼20′(30 pc)-sized region that contains ≃1.5 × 10{sup 5} M {sub ⊙} and enclosing a minimum closed contour with Σ ≃ 0.013 g cm{sup −2} ( A {sub V} ≃ 3 mag), shows a lognormal shape with the peak measured at Σ ≃ 0.021 g cm{sup −2} ( A {sub V}≃ 4.7 mag). There is tentative evidence for the presence of a high-Σ power-law tail that contains from ∼3% to 8% of the mass of the cloud material. We discuss the implications of these results for the physical processes occurring in this cloud.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

A general maximum entropy framework for thermodynamic variational principles

Minimum free energy principles are familiar in equilibrium thermodynamics, as expressions of the second law. They also appear in statistical mechanics as variational approximation schemes, such as the mean-field and steepest-descent approximations. These well-known minimum free energy principles are here unified and extended to any system analyzable by MaxEnt, including non-equilibrium systems. The MaxEnt Lagrangian associated with a generic MaxEnt distribution p defines a generalized potential Ψ for an arbitrary probability distribution p-hat, such that Ψ is a minimum at (p-hat) = p. Minimization of Ψ with respect to p-hat thus constitutes a generic variational principle, and is equivalent to minimizing the Kullback-Leibler divergence between p-hat and p. Illustrative examples of min–Ψ are given for equilibrium and non-equilibrium systems. An interpretation of changes in Ψ is given in terms of the second law, although min–Ψ itself is an intrinsic variational property of MaxEnt that is distinct from the second law.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗