Wishart variate generators
Subroutine WSHRT for generating matrix of variances and covariances of Wishart distribution
Smith, W. B.↗
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Results for “Covariance Matrix”
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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Combining Multiple Gyroscope Outputs for Increased Accuracy
A proposed method of processing the outputs of multiple gyroscopes to increase the accuracy of rate (that is, angular-velocity) readings has been developed theoretically and demonstrated by computer simulation. Although the method is applicable, in principle, to any gyroscopes, it is intended especially for application to gyroscopes that are parts of microelectromechanical systems (MEMS). The method is based on the concept that the collective performance of multiple, relatively inexpensive, nominally identical devices can be better than that of one of the devices considered by itself. The method would make it possible to synthesize the readings of a single, more accurate gyroscope (a virtual gyroscope) from the outputs of a large number of microscopic gyroscopes fabricated together on a single MEMS chip. The big advantage would be that the combination of the MEMS gyroscope array and the processing circuitry needed to implement the method would be smaller, lighter in weight, and less power-hungry, relative to a conventional gyroscope of equal accuracy. The method (see figure) is one of combining and filtering the digitized outputs of multiple gyroscopes to obtain minimum-variance estimates of rate. In the combining-and-filtering operations, measurement data from the gyroscopes would be weighted and smoothed with respect to each other according to the gain matrix of a minimum- variance filter. According to Kalman-filter theory, the gain matrix of the minimum-variance filter is uniquely specified by the filter covariance, which propagates according to a matrix Riccati equation. The present method incorporates an exact analytical solution of this equation.
Bayard, David S.↗
$\mathcal{R}$ -matrix analysis of η + nat Cl reactions relevant to molten-salt reactor designs [Abstract]
Performed with support from the U.S. Nuclear Criticality Safety Program in an effort to provide improved chlorine cross section and corresponding covariance data, the R-matrix analysis of neutron induced reactions for two stable chlorine isotopes, 35,37 Cl, was performed in the energy range of thermal up to 1.2 MeV. Starting from the repository of the ENDF/B-VIII.0 library and following recent measurement campaigns, this work represents a significant improvement in the evaluation of the ( η , p ) reaction channel. The evaluation methodology uses the $\mathcal{R}$ -matrix code SAMMY to generate a set of Reich-Moore resonance parameters. As predicted by recently measured 35 Cl( η , p ) data, the presented evaluation features a dramatic increase in the magnitude of the ( η , p ) reaction channel over the ENDF/B-VIII.0 and previous nuclear data ENDF/B released libraries. Together with details of the evaluation procedure, the impact of the updates in the outgoing proton emissions on reactivity coefficients for different molten-salt reactor designs is presented and discussed.
charged-particle↗
Hierarchical off-diagonal low-rank approximation of Hessians in inverse problems, with application to ice sheet model initialization
Obtaining lightweight and accurate approximations of discretized objective functional Hessians in inverse problems governed by partial differential equations (PDEs) is essential to make both deterministic and Bayesian statistical large-scale inverse problems computationally tractable. The cubic computational complexity of dense linear algebraic tasks, such as Cholesky factorization, that provide a means to sample Gaussian distributions and determine solutions of Newton linear systems is a computational bottleneck at large-scale. These tasks can be reduced to log-linear complexity by utilizing hierarchical off-diagonal low-rank (HODLR) matrix approximations. In this work, we show that a class of Hessians that arise from inverse problems governed by PDEs are well approximated by the HODLR matrix format. In particular, we study inverse problems governed by PDEs that model the instantaneous viscous flow of ice sheets. In these problems, we seek a spatially distributed basal sliding parameter field such that the flow predicted by the ice sheet model is consistent with ice sheet surface velocity observations. Here, we demonstrate the use of HODLR Hessian approximation to efficiently sample the Laplace approximation of the posterior distribution with covariance further approximated by HODLR matrix compression. Computational studies are performed which illustrate ice sheet problem regimes for which the Gauss–Newton data-misfit Hessian is more efficiently approximated by the HODLR matrix format than the low-rank (LR) format. We then demonstrate that HODLR approximations can be favorable, when compared to global LR approximations, for large-scale problems by studying the data-misfit Hessian associated with inverse problems governed by the first-order Stokes flow model on the Humboldt glacier and Greenland ice sheet.
97 MATHEMATICS AND COMPUTING↗
Resolved resonance region evaluations of n+ 206,207,208 Pb for fast spectrum applications
Resolved resonance region evaluations of the major isotopes of natural lead, 206 Pb, 207 Pb, 208 Pb, have been per formed to support the development of Generation IV reactors. In this study, validation of nuclear data for lead fast reactors was performed with simulations of shielding benchmarks, integral critical benchmarks, and quasi-differential scattering measurements. Sensitivity analyses of these systems showed that elastic scattering reactions above 100 keV were the dominant reactions driving system performance. The resolved resonance regions (RRRs) of the lead isotopes extend past 100 keV, making the RRR an ideal starting point to evaluate lead cross sections. Since the R-matrix requires knowledge of bound, distant, and observed resonances, it was necessary to evaluate from 10 -5 eV up to the respective limit of the RRR. The 208 Pb RRR evaluation was extended to 1.5 MeV in order to obtain resonance parameters used to calculate new elastic scattering angular distributions up to 1.5 MeV. Resonance parameter uncertainties and covariance were generated using the R-matrix code SAMMY. The new RRR parameters show a direct improvement to the scattering kernel below 1.5 MeV which in turn greatly improves fast critical experiments over ENDF/B-VIII.0.
206Pb↗
On the Possibility of Ill-Conditioned Covariance Matrices in the First-Order Two-Step Estimator
The first-order two-step nonlinear estimator, when applied to a problem of orbital navigation, is found to occasionally produce first step covariance matrices with very low eigenvalues at certain trajectory points. This anomaly is the result of the linear approximation to the first step covariance propagation. The study of this anomaly begins with expressing the propagation of the first and second step covariance matrices in terms of a single matrix. This matrix is shown to have a rank equal to the difference between the number of first step states and the number of second step states. Furthermore, under some simplifying assumptions, it is found that the basis of the column space of this matrix remains fixed once the filter has removed the large initial state error. A test matrix containing the basis of this column space and the partial derivative matrix relating first and second step states is derived. This square test matrix, which has dimensions equal to the number of first step states, numerically drops rank at the same locations that the first step covariance does. It is formulated in terms of a set of constant vectors (the basis) and a matrix which can be computed from a reference trajectory (the partial derivative matrix). A simple example problem involving dynamics which are described by two states and a range measurement illustrate the cause of this anomaly and the application of the aforementioned numerical test in more detail.
Garrison, James L.↗
Estimation of proportions in mixed pixels through their region characterization
A region of mixed pixels can be characterized through the probability density function of proportions of classes in the pixels. Using information from the spectral vectors of a given set of pixels from the mixed pixel region, expressions are developed for obtaining the maximum likelihood estimates of the parameters of probability density functions of proportions. The proportions of classes in the mixed pixels can then be estimated. If the mixed pixels contain objects of two classes, the computation can be reduced by transforming the spectral vectors using a transformation matrix that simultaneously diagonalizes the covariance matrices of the two classes. If the proportions of the classes of a set of mixed pixels from the region are given, then expressions are developed for obtaining the estmates of the parameters of the probability density function of the proportions of mixed pixels. Development of these expressions is based on the criterion of the minimum sum of squares of errors. Experimental results from the processing of remotely sensed agricultural multispectral imagery data are presented.
Chittineni, C. B.↗
Autonomous satellite navigation methods using the Global Positioning Satellite System
This investigation considers the problem of autonomous satellite navigation using the NAVSTAR Global Positioning System (GPS). The major topics covered include the design, implementation, and validation of onboard navigation filter algorithms by means of computer simulations. The primary errors that the navigation filter design must minimize are computational effects and modeling inaccuracies due to limited capability of the onboard computer. The minimization of the effect of these errors is attained by applying the sequential extended Kalman filter using a factored covariance implementation with Q-matrix or dynamical model compensations. Peformance evaluation of the navigation filter design is carried out using both the CDC Cyber 170/750 computer and the PDP-11/60 computer. The results are obtained assuming the Phase I GPS constellation, consisting of six satellites, and a Landsat-D type spacecraft as the model for the user satellite orbit.
Murata, M.↗
Spinor representations for fields with any spin: Lorentz tensor basis for operators and covariant multipole decomposition
This paper discusses a framework to parametrize and decompose operator matrix elements for particles with higher spin (j > 1/2) using chiral representations of the Lorentz group, i.e. the (j, 0) and (0, j) representations and their parity-invariant direct sum. Unlike traditional approaches that require imposing constraints to eliminate spurious degrees of freedom, these chiral representations contain exactly the 2j + 1 components needed to describe a spin-j particle. The central objects in the construction are the t-tensors, which are generalizations of the Pauli four-vector σ μ for higher spin. For the generalized spinors of these representations, we demonstrate how the algebra of the t-tensors allows to formulate a generalization of the Dirac matrix basis for any spin. For on-shell bilinears, we show that a set consisting exclusively of covariant multipoles of order 0 ≤ m ≤ 2j forms a complete basis. We provide explicit expressions for all bilinears of the generalized Dirac matrix basis, which are valid for any spin value. As a byproduct of our derivations we present an efficient algorithm to compute the t-tensor matrix elements. The formalism presented here paves the way to use a more unified approach to analyze the non-perturbative QCD structure of hadrons and nuclei across different spin values, with clear physical interpretation of the resulting distributions as covariant multipoles.
Angular momentum of light↗
Optimal inputs for system identification
Identification criteria are presented for linear dynamic systems with and without process noise. With process noise, the state equations are replaced by the Kalman filter equations. If the identification performance index is expanded in a Taylor's series with respect to the parameters to be identified, then maximizing the weighting factor of the quadratic term with respect to the inputs will insure that an identification algorithm will converge more rapidly and to a more accurate result than with nonoptimal inputs. The expectation of this weighting factor is the Fisher information matrix, and its inverse is a lower bound for the covariance of the parameters. Direct and indirect methods of calculating the information matrix are presented for systems with and without process noise.
Reid, D. B.↗
140,142 Ce Neutron Cross Section Resolved Resonance Region Evaluation
A resolved resonance region evaluation of 140,142 Ce was conducted by Oak Ridge National Laboratory. Requested by the US Nuclear Criticality Safety Program, this evaluation is based on recent high-resolution transmission and capture high-resolution measurements of nat Ce and 142 Ce conducted at JRC-Geel at the Geel Linear Accelerator facility. It is also based on recently measured thermal constants available from the EX FOR database. Starting from the resonance parameters from the ENDF/B-VIII.0 library and following a preliminary R-matrix analysis, an updated set of resonance parameters and corresponding covariance in formation was derived by the fit of these experimental datasets using the Reich–Moore approximation of the R-matrix theory, as implemented in the SAMMY code system. The resolved resonance region upper energy limit for 140 Ce was kept at 200 keV, whereas the 142 Ce resonance region was extended from 13 to 26 keV. This new evaluation was found to be in good agreement not only with several integral quantities of interest to the reactor physics community, but also with the stellar Maxwellian-averaged cross section.
46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
Developement of an Optimum Interpolation Analysis Method for the CYBER 205
A state-of-the-art technique to assimilate the diverse observational database obtained during FGGE, and thus create initial conditions for numerical forecasts is described. The GLA optimum interpolation (OI) analysis method analyzes pressure, winds, and temperature at sea level, mixing ratio at six mandatory pressure levels up to 300 mb, and heights and winds at twelve levels up to 50 mb. Conversion to the CYBER 205 required a major re-write of the Amdahl OI code to take advantage of the CYBER vector processing capabilities. Structured programming methods were used to write the programs and this has resulted in a modular, understandable code. Among the contributors to the increased speed of the CYBER code are a vectorized covariance-calculation routine, an extremely fast matrix equation solver, and an innovative data search and sort technique.
Nestler, M. S.↗
Integration of a Decentralized Linear-Quadratic-Gaussian Control into GSFC's Universal 3-D Autonomous Formation Flying Algorithm
A decentralized control is investigated for applicability to the autonomous formation flying control algorithm developed by GSFC for the New Millenium Program Earth Observer-1 (EO-1) mission. This decentralized framework has the following characteristics: The approach is non-hierarchical, and coordination by a central supervisor is not required; Detected failures degrade the system performance gracefully; Each node in the decentralized network processes only its own measurement data, in parallel with the other nodes; Although the total computational burden over the entire network is greater than it would be for a single, centralized controller, fewer computations are required locally at each node; Requirements for data transmission between nodes are limited to only the dimension of the control vector, at the cost of maintaining a local additional data vector. The data vector compresses all past measurement history from all the nodes into a single vector of the dimension of the state; and The approach is optimal with respect to standard cost functions. The current approach is valid for linear time-invariant systems only. Similar to the GSFC formation flying algorithm, the extension to linear LQG time-varying systems requires that each node propagate its filter covariance forward (navigation) and controller Riccati matrix backward (guidance) at each time step. Extension of the GSFC algorithm to non-linear systems can also be accomplished via linearization about a reference trajectory in the standard fashion, or linearization about the current state estimate as with the extended Kalman filter. To investigate the feasibility of the decentralized integration with the GSFC algorithm, an existing centralized LQG design for a single spacecraft orbit control problem is adapted to the decentralized framework while using the GSFC algorithm's state transition matrices and framework. The existing GSFC design uses both reference trajectories of each spacecraft in formation and by appropriate choice of coordinates and simplified measurement modeling is formulated as a linear time-invariant system. Results for improvements to the GSFC algorithm and a multiple satellite formation will be addressed. The goal of this investigation is to progressively relax the assumptions that result in linear time-invariance, ultimately to the point of linearization of the non-linear dynamics about the current state estimate as in the extended Kalman filter. An assessment will then be made about the feasibility of the decentralized approach to the realistic formation flying application of the EO-1/Landsat 7 formation flying experiment.
Folta, David C.↗
Mapping the Milky Way Galaxy with LISA
Gravitational wave detectors in the mHz band (such as the Laser Interferometer Space Antenna, or LISA) will observe thousands of compact binaries in the galaxy which can be used to better understand the structure of the Milky Way. To test the effectiveness of LISA to measure the distribution of the galaxy, we simulated the Close White Dwarf Binary (CWDB) gravitational wave sky using different models for the Milky Way. To do so, we have developed a galaxy density distribution modeling code based on the Markov Chain Monte Carlo method. The code uses different distributions to construct realizations of the galaxy. We then use the Fisher Information Matrix to estimate the variance and covariance of the recovered parameters for each detected CWDB. This is the first step toward characterizing the capabilities of space-based gravitational wave detectors to constrain models for galactic structure, such as the size and orientation of the bar in the center of the Milky Way
McKinnon, Jose A.↗
Minimum-Variance Control Allocation Considering Parametric Model Uncertainty
The control allocation problem was investigated for linear dynamical systems with known parametric uncertainty. Minimizing a cost function that penalizes the variance of the error in achieving commanded forces and moments on the vehicle resulted in a special case of the weighted pseudo-inverse allocator. Rather than an engineer designing the weighting matrix, it is computed from the covariances of the control effectiveness parameters. This minimum-variance allocator balances the effectiveness of the control inputs against the corresponding levels of uncertainty. The approach was demonstrated using simulations of aircraft with realistic uncertainty levels operating in open-loop and closed-loop configurations. Results showed that when model uncertainty is known, significant, and unevenly distributed amongst the controls, the minimum-variance allocator more often achieves the intended forces and moments on the vehicle in comparison to other allocators, which can lead to increased performance, reliability, and safety during flight tests. The cost for this robustness is a diminished achievable moment space for the vehicle.
Control allocation↗
A comparison of discrete linear filtering algorithms.
Seven filter algorithms were presented in a recent survey paper (Kaminski, 1971), and were compared computationally (operations count) when relatively few observations were to be processed. These algorithms are now elaborated further. Details of the computations are presented, and it is shown that for problems with even moderately large amounts of data, the information matrix and square-root information matrix formulations are computationally more efficient than the other methods considered (conventional Kalman, stabilized Kalman, and square-root covariance mechanizations). It is pointed out that Schmidt's matrix factorization-Householder transformation technique leads to the same equations as those obtained via Potter's method. Several improvements in the equation mechanization are given.
Bierman, G. J.↗
R-matrix Resolved Resonance Region Evaluation of 140,142 Ce
Oak Ridge National Laboratory completed the resolved resonance region (RRR) evaluation of the two most abundant cerium isotopes, 140 Ce (88.45%) and 142 Ce (11.11%), as requested by the US Nuclear Criticality Safety Program. These evaluations are based on recent high-resolution transmission and capture measurements performed on nat Ce and highly enriched 142 Ce samples at the JRC-Geel Linear Accelerator facility, as well as measured thermal constants available from the EXFOR database. Starting from the resonance parameters of the ENDF/B-VIII.0 library followed by a preliminary R-matrix analysis, an updated set of resonance parameters and corresponding covariance information were derived by fitting these measured data using the Reich–Moore approximation of the R -matrix theory, as implemented in the SAMMY code system. The 140 Ce RRR upper energy limit was kept at 200 keV, whereas the 142 Ce resonance region was extended from 13 to 26 keV. Updated statistical properties were obtained for the new evaluations and compared to those derived from the ENDF/B-VIII.0 nuclear data library. The new evaluation work improved some of the discrepancies found in previous work, such as the capture resonance integral and stellar Maxwellian-averaged cross sections. These integral quantities were mainly derived from the fit of the latest measured data, especially the neutron capture yield data for 142 Ce isotope.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Realistic Covariance Generation for the GPM Spacecraft
We present several different methods for generating realistic predictive covariance, including Monte-Carlo simulations and more direct linear methods which require the addition of process noise. The Monte-Carlo simulation starts with an epoch uncertainty sample basis and propagates each trial to a time of interest in the future. The variance-covariance of the state elements as well as other higher order sample statistics can be readily computed from the propagated sample. While this method preserves the nonlinear effects on the propagated uncertainty, it is computationally intensive as a statistically significant sample size must be considered in the propagation process. Moreover, if the epoch covariance is optimistic, this can result in an underestimation of the prediction error. Another method is to propagate a state sensitivity matrix simultaneously with the satellite state, which allows the epoch covariances to be propagated forward in a linear fashion. This method does not preserve the non-linearity of the satellite state uncertainty but is much less computationally intensive. The propagated state covariance is the scaled to represent the realistic level of GPM state uncertainty via a "e-tuning process." The tuning process generates an inflation factor based on the observed error statistics of the predictive satellite trajectories when compared to the definitive ones. Difference tuning strategies are considered and compared via Goodness-of-Fit method testing for the Gaussian properties of the scaled covariance.
prediction error↗