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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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Sampling Functions from Gaussian Processes and Structured Covariance Gaussian Networks

When learning aerodynamic models from data, it is critical to incorporate estimates of model uncertainty. This motivates the design of probabilistic aerodynamic databases which can be sampled to generate physically and statistically plausible aerodynamic models. In this talk we discuss how to sample deterministic functions from two different kinds of probabilistic models and demonstrate their use. First, Gaussian Process Regressors (GPRs) are a widely used probabilistic kernel-based model which can be thought of as Gaussian distributions over functions. GPRs are generally trained by maximizing the marginal likelihood of seeing the training data over the kernel parameter space. Sample functions are easily generated by drawing points from the Gaussian distribution at desired input points. However, when the points are not known ahead of time, the classical sampling approach is not possible since successive function samples will generate different function realizations. We present an approach for sampling consistent function evaluations from a GPR over multiple samples. Second, we describe a neural network architecture which learns a conditional Gaussian distribution by maximizing the marginal likelihood at each point in the input space. We then discuss and compare several options for generating sample functions which match this distribution. Finally, we demonstrate the use of these probabilistic aerodynamic models in an atmospheric reentry simulation.

Gaussian process regression↗
DOE OSTI2021

A Nonstationary and Non-Gaussian Moving Average Model for Solar Irradiance

Historically, power has flowed from large power plants to customers. Increasing penetration of distributed energy resources such as solar power from rooftop photovoltaic has made the distribution network a two-way-street with power being generated at the customer level. The incorporation of renewables introduces additional uncertainty and variability into the power grid. Distribution network operation studies are being adapted to include renewables; however, such studies require high quality solar irradiance data that adequately reflect realistic meteorological variability. Data from satellite-based products are spatially complete, but temporally coarse, whereas solar irradiances exhibit high frequency variation at very fine timescales. We propose a new stochastic method for temporally downscaling global horizontal irradiance (GHI) to 1 min resolution, but we do not consider the spatial aspect due to limited availability of the in situ irradiance measurements. Solar irradiance's first and second-order structures vary diurnally and seasonally, and our model adapts to such nonstationarity. Empirical irradiance data exhibits highly non-Gaussian behavior; we develop a nonstationary and non-Gaussian moving average model that is shown to capture realistic solar variability at multiple timescales. We also propose a new estimation scheme based on Cholesky factors of empirical autocovariance matrices, bypassing difficult and inaccessible likelihood-based approaches. The model is demonstrated for a case study of three locations that are located in diverse climates through the United States. The model is compared against competitors from the literature and is shown to provide better uncertainty and variability quantification on testing data.

Cholesky factor↗
DOE OSTI2020

Kepler Data Analysis: Non-Gaussian Noise and Fourier Gaussian Process Analysis of Stellar Variability

We develop a statistical analysis model of Kepler stellar flux data in the presence of planet transits, non-Gaussian noise, and stellar variability. We first develop a model for the Kepler noise probability distribution in the presence of outliers, which make the noise probability distribution non-Gaussian. We develop a signal likelihood analysis based on this probability distribution, in which we model the signal as a sum of the star variability and planetary transits. We argue that these components need to be modeled together if optimal signal is to be extracted from the data. For the stellar variability model we develop an optimal Gaussian process analysis using a Fourier-based Wiener filter approach, where the power spectrum is non-parametric and learned from the data. We develop high dimensional optimization of the objective function, where we jointly optimize all the model parameters, including thousands of star variability modes, and planet transit parameters. We apply the method to Kepler-90 data and show that it gives a better match to the stellar variability than the existing methods, and robustly handles noise outliers. As a consequence, the planet radii have a higher value than what the existing methods give, including splines and celerite.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Absorptive Weak Plume Detection on Gaussian and Non-Gaussian Background Clutter

For additive signals on Gaussian clutter, the optimal detector is a linear matched filter that is adapted to the known signal and the covariance of the background. This adaptive matched filter is widely used for gas-phase plume detection, even though the effect of the plume on the background is not strictly additive. Here, a derivation of the matched filter for a strictly absorptive plume produces, in the weak plume limit, a quadratic filter. This quadratic matched filter is extended in two ways: an elliptically-contoured multivariate t distribution is used to generalize the Gaussian background clutter, and a generalized likelihood ratio test detector is derived to extend applicability to stronger plumes. In addition to detectors whose purpose is to identify presence versus absence of a plume, expressions are also derived for estimating plume strength. The performance of these various detectors is evaluated by implanting simulated plume into background images that are either real hyperspectral images or simulated images based on different (Gaussian, multivariate t, and lognormal) clutter distributions.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Gaussian FLOWERS: Wind-rose-based analytical integration of Gaussian wake model for extremely fast AEP estimation

A major cost in the study of wind farm layout optimization is the repeated evaluation of the annual energy production (AEP). The current approach to estimating AEP requires a large set of flow simulations to be performed that cover each discrete wind speed and direction combination contained within the wind rose, followed by a probability-weighted sum of the power production resulting from each simulation. Even with inexpensive engineering wake models, this numerical integration scheme can lead to high computational costs. In this paper, we derive an analytical formulation for estimating farm AEP across every wind direction, based on a Gaussian wake velocity model, which reduces the number of wind farm simulations to a single function evaluation. As a result, we find that the Gaussian-FLOWERS approach reduces the time for AEP calculations by more than two orders of magnitude with a small trade-off in accuracy when compared to a conventional approach. This massive reduction in computation cost is useful to reduce overall costs in wind farm layout optimization studies.

17 WIND ENERGY↗
DOE OSTI2024

Optimization on Manifolds via Graph Gaussian Processes

This paper integrates manifold learning techniques within a Gaussian process upper confidence bound algorithm to optimize an objective function on a manifold. Our approach is motivated by applications where a full representation of the manifold is not available and querying the objective is expensive. We rely on a point cloud of manifold samples to define a graph Gaussian process surrogate model for the objective. Query points are sequentially chosen using the posterior distribution of the surrogate model given all previous queries. We establish regret bounds in terms of the number of queries and the size of the point cloud. Several numerical examples complement the theory and illustrate the performance of our method.

Bayesian optimization↗
DOE OSTI2021

Reducing Ground-based Astrometric Errors with Gaia and Gaussian Processes

Stochastic field distortions caused by atmospheric turbulence are a fundamental limitation to the astrometric accuracy of ground-based imaging. This distortion field is measurable at the locations of stars with accurate positions provided by the Gaia DR2 catalog; we develop the use of Gaussian process regression (GPR) to interpolate the distortion field to arbitrary locations in each exposure. We introduce an extension to standard GPR techniques that exploits the knowledge that the 2D distortion field is curl-free. Applied to several hundred 90 s exposures from the Dark Energy Survey as a test bed, we find that the GPR correction reduces the variance of the turbulent astrometric distortions ≈12× , on average, with better performance in denser regions of the Gaia catalog. The rms per-coordinate distortion in the riz bands is typically ≈7 mas before any correction and ≈2 mas after application of the GPR model. The GPR astrometric corrections are validated by the observation that their use reduces, from 10 to 5 mas rms, the residuals to an orbit fit to riz-band observations over 5 yr of the r = 18.5 trans-Neptunian object Eris. We also propose a GPR method, not yet implemented, for simultaneously estimating the turbulence fields and the 5D stellar solutions in a stack of overlapping exposures, which should yield further turbulence reductions in future deep surveys.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ-approximate inverse Cholesky factors of the kernel matrices with complexity O(N log d (N/ϵ)) in space and O(N log 2d (N/ϵ)) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1983

A comparison between backscattering coefficients using Gaussian and non-Gaussian surface statistics

Backscattering coefficients computed using Gaussian and non-Gaussian surface statistics are compared. The computations are performed for a class of surface-height distributions and surface correlation functions. Results indicate that the coherent component of the scattering coefficient is strongly dependent upon the surface-height distribution. The noncoherent component also depends on the surface-height distribution and the surface root-mean-square (rms) slope when the rms surface height is large relative to the incident wavelength. On the other hand, when the surface rms height is moderate or small, the noncoherent component is sensitive mainly to the surface-height correlation function.

Eom, H. J.↗
DOE OSTI2021

Enhanced Data Efficiency Using Deep Neural Networks and Gaussian Processes for Aerodynamic Design Optimization

Adjoint-based optimization methods are attractive for aerodynamic shape design primarily due to their computational costs being independent of the dimensionality of the input space and their ability to generate high-fidelity gradients that can then be used in a gradient-based optimizer. This makes them very well suited for high-fidelity simulation based aerodynamic shape optimization of highly parametrized geometries such as aircraft wings. However, the development of adjoint-based solvers involve careful mathematical treatment and their implementation require detailed software development. Furthermore, they can become prohibitively expensive when multiple optimization problems are being solved, each requiring multiple restarts to circumvent local optima. In this work, we propose a machine learning enabled, surrogate-based framework that replaces the expensive adjoint solver, without compromising on predicting predictive accuracy. Specifically, we first train a deep neural network (DNN) from training data generated from evaluating the high-fidelity simulation model on a model-agnostic design of experiments on the geometry shape parameters. The optimum shape may then be computed by using a gradient-based optimizer coupled with the trained DNN. Subsequently, we also perform a gradient-free Bayesian optimization, where the trained DNN is used as the prior mean. We observe that the latter framework (DNN-BO) improves upon the DNN-only based optimization strategy for the same computational cost. Overall, this framework predicts the true optimum with very high accuracy, while requiring far fewer high-fidelity function calls compared to the adjoint-based method. Furthermore, we show that multiple optimization problems can be solved with the same machine learning model with high accuracy, to amortize the offline costs associated with constructing our models. Our methodology finds applications in the early stages of aerospace design. (C) 2021 Published by Elsevier Masson SAS.

Renganathan, S. Ashwin↗
NASA NTRS2011

On the Response of a Nonlinear Structure to High Kurtosis Non-Gaussian Random Loadings

This paper is a follow-on to recent work by the authors in which the response and high-cycle fatigue of a nonlinear structure subject to non-Gaussian loadings was found to vary markedly depending on the nature of the loading. There it was found that a non-Gaussian loading having a steady rate of short-duration, high-excursion peaks produced essentially the same response as would have been incurred by a Gaussian loading. In contrast, a non-Gaussian loading having the same kurtosis, but with bursts of high-excursion peaks was found to elicit a much greater response. This work is meant to answer the question of when consideration of a loading probability distribution other than Gaussian is important. The approach entailed nonlinear numerical simulation of a beam structure under Gaussian and non-Gaussian random excitations. Whether the structure responded in a Gaussian or non-Gaussian manner was determined by adherence to, or violations of, the Central Limit Theorem. Over a practical range of damping, it was found that the linear response to a non-Gaussian loading was Gaussian when the period of the system impulse response is much greater than the rate of peaks in the loading. Lower damping reduced the kurtosis, but only when the linear response was non-Gaussian. In the nonlinear regime, the response was found to be non-Gaussian for all loadings. The effect of a spring-hardening type of nonlinearity was found to limit extreme values and thereby lower the kurtosis relative to the linear response regime. In this case, lower damping gave rise to greater nonlinearity, resulting in lower kurtosis than a higher level of damping.

Rizzi, Stephen A.↗
NASA NTRS1999

Gaussian Decomposition of Laser Altimeter Waveforms

We develop a method to decompose a laser altimeter return waveform into its Gaussian components assuming that the position of each Gaussian within the waveform can be used to calculate the mean elevation of a specific reflecting surface within the laser footprint. We estimate the number of Gaussian components from the number of inflection points of a smoothed copy of the laser waveform, and obtain initial estimates of the Gaussian half-widths and positions from the positions of its consecutive inflection points. Initial amplitude estimates are obtained using a non-negative least-squares method. To reduce the likelihood of fitting the background noise within the waveform and to minimize the number of Gaussians needed in the approximation, we rank the "importance" of each Gaussian in the decomposition using its initial half-width and amplitude estimates. The initial parameter estimates of all Gaussians ranked "important" are optimized using the Levenburg-Marquardt method. If the sum of the Gaussians does not approximate the return waveform to a prescribed accuracy, then additional Gaussians are included in the optimization procedure. The Gaussian decomposition method is demonstrated on data collected by the airborne Laser Vegetation Imaging Sensor (LVIS) in October 1997 over the Sequoia National Forest, California.

Hofton, Michelle A.↗
NASA NTRS2015

Rao-Blackwellization for Adaptive Gaussian Sum Nonlinear Model Propagation

When dealing with imperfect data and general models of dynamic systems, the best estimate is always sought in the presence of uncertainty or unknown parameters. In many cases, as the first attempt, the Extended Kalman filter (EKF) provides sufficient solutions to handling issues arising from nonlinear and non-Gaussian estimation problems. But these issues may lead unacceptable performance and even divergence. In order to accurately capture the nonlinearities of most real-world dynamic systems, advanced filtering methods have been created to reduce filter divergence while enhancing performance. Approaches, such as Gaussian sum filtering, grid based Bayesian methods and particle filters are well-known examples of advanced methods used to represent and recursively reproduce an approximation to the state probability density function (pdf). Some of these filtering methods were conceptually developed years before their widespread uses were realized. Advanced nonlinear filtering methods currently benefit from the computing advancements in computational speeds, memory, and parallel processing. Grid based methods, multiple-model approaches and Gaussian sum filtering are numerical solutions that take advantage of different state coordinates or multiple-model methods that reduced the amount of approximations used. Choosing an efficient grid is very difficult for multi-dimensional state spaces, and oftentimes expensive computations must be done at each point. For the original Gaussian sum filter, a weighted sum of Gaussian density functions approximates the pdf but suffers at the update step for the individual component weight selections. In order to improve upon the original Gaussian sum filter, Ref. [2] introduces a weight update approach at the filter propagation stage instead of the measurement update stage. This weight update is performed by minimizing the integral square difference between the true forecast pdf and its Gaussian sum approximation. By adaptively updating each component weight during the nonlinear propagation stage an approximation of the true pdf can be successfully reconstructed. Particle filtering (PF) methods have gained popularity recently for solving nonlinear estimation problems due to their straightforward approach and the processing capabilities mentioned above. The basic concept behind PF is to represent any pdf as a set of random samples. As the number of samples increases, they will theoretically converge to the exact, equivalent representation of the desired pdf. When the estimated qth moment is needed, the samples are used for its construction allowing further analysis of the pdf characteristics. However, filter performance deteriorates as the dimension of the state vector increases. To overcome this problem Ref. [5] applies a marginalization technique for PF methods, decreasing complexity of the system to one linear and another nonlinear state estimation problem. The marginalization theory was originally developed by Rao and Blackwell independently. According to Ref. [6] it improves any given estimator under every convex loss function. The improvement comes from calculating a conditional expected value, often involving integrating out a supportive statistic. In other words, Rao-Blackwellization allows for smaller but separate computations to be carried out while reaching the main objective of the estimator. In the case of improving an estimator's variance, any supporting statistic can be removed and its variance determined. Next, any other information that dependents on the supporting statistic is found along with its respective variance. A new approach is developed here by utilizing the strengths of the adaptive Gaussian sum propagation in Ref. [2] and a marginalization approach used for PF methods found in Ref. [7]. In the following sections a modified filtering approach is presented based on a special state-space model within nonlinear systems to reduce the dimensionality of the optimization problem in Ref. [2]. First, the adaptive Gaussian sum propagation is explained and then the new marginalized adaptive Gaussian sum propagation is derived. Finally, an example simulation is presented.

state estimation↗
DOE OSTI2025

Decoherence dynamics in molecular qubits: Exponential, Gaussian and beyond

In this article, we examine how the structure of system–bath interactions can determine commonly encountered temporal decoherence patterns, such as Gaussian and exponential decay, in molecular and other qubits coupled to a thermal bosonic bath. The analysis, based on a pure dephasing picture that admits analytical treatment, shows that decoherence, in general, is neither purely Gaussian nor exponential but rather the exponential of oscillatory functions, with periods determined by the bath’s frequencies. For initially unentangled qubit-bath states, Gaussian decay is always present at early times. It becomes increasingly dominant with increasing temperature, qubit–bath interaction strength, and bath correlation time. Initial system–bath entanglement that arises due to displacement in the position of the bath states preserves the Gaussian decay. By contrast, strict exponential decay arises only in very specific models that we isolate. However, it becomes dominant for times longer than the bath correlation time or for early times when there is initial entanglement due to momentum displacement of the bath states. For molecular electronic decoherence, the long-time exponential regime plays a limited role as it emerges after most coherence is lost. Thus, the Gaussian decay provides a more suitable (albeit imperfect) model of such decoherence. Furthermore, we discuss the connection between electronic decoherence dynamics and electronic spectroscopic line shape theory, where Gaussian spectral peaks correspond to Gaussian coherence decay and Lorentzian peaks correspond to exponential coherence decay. We find that Gaussian spectral peaks, usually associated with inhomogeneous broadening, can emerge from the entangling unitary system–bath dynamics even when there is no inhomogeneity in the initial conditions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗