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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces

The approximation of unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft are discussed. Two methods of formulating these approximations are extended to include the same flexibility in constraining the approximations and the same methodology in optimizing nonlinear parameters as another currently used extended least-squares method. Optimal selection of nonlinear parameters is made in each of the three methods by use of the same nonlinear, nongradient optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is lower order than that required when no optimization of the nonlinear terms is performed. The free linear parameters are determined using the least-squares matrix techniques of a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from different approaches are described and results are presented that show comparative evaluations from application of each of the extended methods to a numerical example.

Tiffany, Sherwood H.↗
NASA NTRS1998

A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models

Two methods of creating approximation models are compared through the calculation of the modeling accuracy on test problems involving one, five, and ten independent variables. Here, the test problems are representative of the modeling challenges typically encountered in realistic engineering optimization problems. The first approximation model is a quadratic polynomial created using the method of least squares. This type of polynomial model has seen considerable use in recent engineering optimization studies due to its computational simplicity and ease of use. However, quadratic polynomial models may be of limited accuracy when the response data to be modeled have multiple local extrema. The second approximation model employs an interpolation scheme known as kriging developed in the fields of spatial statistics and geostatistics. This class of interpolating model has the flexibility to model response data with multiple local extrema. However, this flexibility is obtained at an increase in computational expense and a decrease in ease of use. The intent of this study is to provide an initial exploration of the accuracy and modeling capabilities of these two approximation methods.

Giunta, Anthony A.↗
DOE OSTI2023

Robust scalable initialization for Bayesian variational inference with multi-modal Laplace approximations

Predictive modeling typically relies on Bayesian model calibration to provide uncertainty quantification. Variational inference utilizing fully independent (“mean-field”) Gaussian distributions are often used as approximate probability density functions. This simplification is attractive since the number of variational parameters grows only linearly with the number of unknown model parameters. However, the resulting diagonal covariance structure and unimodal behavior can be too restrictive to provide useful approximations of intractable Bayesian posteriors that exhibit highly non-Gaussian behavior, including multimodality. High-fidelity surrogate posteriors for these problems can be obtained by considering the family of Gaussian mixtures. Gaussian mixtures are capable of capturing multiple modes and approximating any distribution to an arbitrary degree of accuracy, while maintaining some analytical tractability. Unfortunately, variational inference using Gaussian mixtures with full-covariance structures suffers from a quadratic growth in variational parameters with the number of model parameters. The existence of multiple local minima due to strong nonconvex trends in the loss functions often associated with variational inference present additional complications, These challenges motivate the need for robust initialization procedures to improve the performance and computational scalability of variational inference with mixture models. In this work, we propose a method for constructing an initial Gaussian mixture model approximation that can be used to warm-start the iterative solvers for variational inference. The procedure begins with a global optimization stage in model parameter space. In this step, local gradient-based optimization, globalized through multistart, is used to determine a set of local maxima, which we take to approximate the mixture component centers. Around each mode, a local Gaussian approximation is constructed via the Laplace approximation. Finally, the mixture weights are determined through constrained least squares regression. The robustness and scalability of the proposed methodology is demonstrated through application to an ensemble of synthetic tests using high-dimensional, multimodal probability density functions. Here, the practical aspects of the approach are demonstrated with inversion problems in structural dynamics.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1972

Smoothing of functions of range and range rate measurements from earth orbiting satellites

It is shown that for satellites in circular Earth orbits with altitudes of 500 kilometers to 1500 kilometers, and for satellites in elliptical orbits with an approximate 4000 kilometer height of perigee, a high degree least squares polynomial (e.g., a 9th or 10th degree in some cases) is required to smooth both range and range rate data for purposes of input to orbit determination programs. In order to circumvent this problem, functions of range and range rate are smoothed with lower degree least squares polynomials (e.g., 3rd and 4th degree) and it is shown that under the above geometric constraints the standard deviation of fit can be reduced to levels commensurate with typical S-band tracking system resolution which is 1 to 2 meters in range and 0.005 meters/second in range rate for a 1 per second data rate. Also shown are the effects of Gaussian random noise, biases, and periodic noise. This analysis includes numerous examples applied to the 44 point data smoothing interval currently used in much of the operational preprocessing at the Goddard Space Flight Center.

Grenchik, T. J.↗
NASA NTRS1980

Minimum Bayes risk image correlation

In this paper, the problem of designing a matched filter for image correlation will be treated as a statistical pattern recognition problem. It is shown that, by minimizing a suitable criterion, a matched filter can be estimated which approximates the optimum Bayes discriminant function in a least-squares sense. It is well known that the use of the Bayes discriminant function in target classification minimizes the Bayes risk, which in turn directly minimizes the probability of a false fix. A fast Fourier implementation of the minimum Bayes risk correlation procedure is described.

Minter, T. C., Jr.↗
NASA NTRS1989

Numerical algorithms for steady and unsteady incompressible Navier-Stokes equations

The numerical analysis of the incompressible Navier-Stokes equations are becoming important tools in the understanding of some fluid flow problems which are encountered in research as well as in industry. With the advent of the supercomputers, more realistic problems can be studied with a wider choice of numerical algorithms. An alternative formulation is presented for viscous incompressible flows. The incompressible Navier-Stokes equations are cast in a velocity/vorticity formulation. This formulation consists of solving the Poisson equations for the velocity components and the vorticity transport equation. Two numerical algorithms for the steady two-dimensional laminar flows are presented. The first method is based on the actual partial differential equations. This uses a finite-difference approximation of the governing equations on a staggered grid. The second method uses a finite element discretization with the vorticity transport equation approximated using a Galerkin approximation and the Poisson equations are obtained using a least squares method. The equations are solved efficiently using Newton's method and a banded direct matrix solver (LINPACK). The method is extended to steady three-dimensional laminar flows and applied to a cubic driven cavity using finite difference schemes and a staggered grid arrangement on a Cartesian mesh. The equations are solved iteratively using a plane zebra relaxation scheme. Currently, a two-dimensional, unsteady algorithm is being developed using a generalized coordinate system. The equations are discretized using a finite-volume approach. This work will then be extended to three-dimensional flows.

Hafez, Mohammed↗
NASA NTRS1992

Ellipsometric study of oxide films formed on LDEF metal samples

The optical constants of samples of six different metals (Al, Cu, Ni, Ta, W, and Zr) exposed to space on the Long Duration Exposure Facility (LDEF) were studied by variable angle spectroscopic ellipsometry. Measurements were also carried out on portions of each sample which were shielded from direct exposure by a metal bar. A least-squares fit of the data using an effective medium approximation was then carried out, with thickness and composition of surface films formed on the metal substrates as variable parameters. The analysis revealed that exposed portions of the Cu, Ni, Ta, and Zr samples are covered with porous oxide films ranging in thickness from 500 to 1000 A. The 410 A thick film of Al2O3 on the exposed Al sample is practically free of voids. Except for Cu, the shielded portions of these metals are covered by thin non-porous oxide films characteristic of exposure to air. The shielded part of the Cu sample has a much thicker porous coating of Cu2O. The tungsten data could not be analyzed.

Franzen, W.↗
NASA NTRS1994

X-ray spectrum and variability of the quasar PG 1211+143

We present preliminary results of an ASCA observation of the classic soft-excess quasar PG 1211+143. The overall ASCA spectrum can be characterized by a blackbody with a temperature of approximately 125 eV (quasar frame) and a power law with photon index of approximately 2. Simultaneous ROSAT data are suggestive of further steepening of the spectrum just below the ASCA band. Comparison with previous observations shows that the soft flux in the 0.1-2 keV band varies by at least a factor of approximately 16, scaling roughly as the square of the hard flux in the 2-10 keV band over a timescale of approximately 13.5 yr. We also find evidence of short-term amplitude variability of up to a factor of approximately 2 on a timescale of approximately 2 x 10(exp 4) sec, in both the soft and hard flux so that the soft and hard photons are likely to originate from the same, compact, region. The data rule out variable absorption (cold or ionized) as the origin of the soft excess, favoring an intrinsic emission component. However, we argue against optically thin emission for the 'blue bump' in PG 1211+143. The large amplitude soft X-ray variability may be indicative of variations in the effective temperature, or peak, of the soft component. There is only marginal evidence for Fe K line emission between 6-7 keV in the quasar frame.

Yaqoob, Tahir↗
NASA NTRS1998

Approximation Of Multi-Valued Inverse Functions Using Clustering And Sugeno Fuzzy Inference

Finding the inverse of a continuous function can be challenging and computationally expensive when the inverse function is multi-valued. Difficulties may be compounded when the function itself is difficult to evaluate. We show that we can use fuzzy-logic approximators such as Sugeno inference systems to compute the inverse on-line. To do so, a fuzzy clustering algorithm can be used in conjunction with a discriminating function to split the function data into branches for the different values of the forward function. These data sets are then fed into a recursive least-squares learning algorithm that finds the proper coefficients of the Sugeno approximators; each Sugeno approximator finds one value of the inverse function. Discussions about the accuracy of the approximation will be included.

Walden, Maria A.↗
DOE OSTI2026

Interpretable and flexible non-intrusive reduced-order models using reproducing kernel Hilbert spaces

This paper develops an interpretable, non-intrusive reduced-order modeling technique using regularized kernel interpolation. Existing non-intrusive approaches approximate the dynamics of a reduced-order model (ROM) by solving a data-driven least-squares regression problem for low-dimensional matrix operators. Our approach instead leverages regularized kernel interpolation, which yields an optimal approximation of the ROM dynamics from a user-defined reproducing kernel Hilbert space. We show that our kernel-based approach can produce interpretable ROMs whose structure mirrors full-order model structure by embedding judiciously chosen feature maps into the kernel. The approach is flexible and allows a combination of informed structure through feature maps and closure terms via more general nonlinear terms in the kernel. We also derive a computable a posteriori error bound that combines standard error estimates for intrusive projection-based ROMs and kernel interpolants. In conclusion, the approach is demonstrated in several numerical experiments that include comparisons to operator inference using both proper orthogonal decomposition and quadratic manifold dimension reduction.

Data-driven model reduction↗
NASA NTRS1989

Performance of FFT methods in local gravity field modelling

Fast Fourier transform (FFT) methods provide a fast and efficient means of processing large amounts of gravity or geoid data in local gravity field modelling. The FFT methods, however, has a number of theoretical and practical limitations, especially the use of flat-earth approximation, and the requirements for gridded data. In spite of this the method often yields excellent results in practice when compared to other more rigorous (and computationally expensive) methods, such as least-squares collocation. The good performance of the FFT methods illustrate that the theoretical approximations are offset by the capability of taking into account more data in larger areas, especially important for geoid predictions. For best results good data gridding algorithms are essential. In practice truncated collocation approaches may be used. For large areas at high latitudes the gridding must be done using suitable map projections such as UTM, to avoid trivial errors caused by the meridian convergence. The FFT methods are compared to ground truth data in New Mexico (xi, eta from delta g), Scandinavia (N from delta g, the geoid fits to 15 cm over 2000 km), and areas of the Atlantic (delta g from satellite altimetry using Wiener filtering). In all cases the FFT methods yields results comparable or superior to other methods.

Forsberg, Rene↗
NASA NTRS1983

Experimental determination of the particle motions associated with the low order acoustic modes in enclosures

A procedure for experimentally determining, in terms of the particle motions, the shapes of the low order acoustic modes in enclosures is described. The procedure is based on finding differentiable functions which approximate the shape functions of the low order acoustic modes when these modes are defined in terms of the acoustic pressure. The differentiable approximating functions are formed from polynomials which are fitted by a least squares procedure to experimentally determined values which define the shapes of the low order acoustic modes in terms of the acoustic pressure. These experimentally determined values are found by a conventional technique in which the transfer functions, which relate the acoustic pressures at an array of points in the enclosure to the volume velocity of a fixed point source, are measured. The gradient of the function which approximates the shape of a particular mode in terms of the acoustic pressure is evaluated to give the mode shape in terms of the particle motion. The procedure was tested by using it to experimentally determine the shapes of the low order acoustic modes in a small rectangular enclosure.

Byrne, K. P.↗
NASA NTRS1992

Size-reduction techniques for the determination of efficient aeroservoelastic models

Size-reduction techniques for determining efficient time-domain state-space aeroservoelastic models are presented. Various rational function approximation methods of the unsteady aerodynamic force coefficients are brought to a common motion, emphasizing their differences. Among these, the classic Roger's method is the easier to apply but its resulting number of aerodynamic states is typically equal to or larger than the number of structural states. On the other hand, the minimum-state (MS) method, which typically reduces the number of aerodynamic states by 70 percent or more, requires the solution of an iterative nonlinear least-square solution. The MS computational efforts are reduced significantly when three approximation constraints are applied.

Karpel, Mordechay↗
NASA NTRS1997

Exponential Approximations Using Fourier Series Partial Sums

The problem of accurately reconstructing a piece-wise smooth, 2(pi)-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to approximate the locations and magnitudes of the discontinuities in f and its first M derivatives. This is accomplished by first finding initial estimates of these quantities based on certain properties of Gibbs phenomenon, and then refining these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least-squares approach. It is conjectured that the locations of the singularities are approximated to within O(N(sup -M-2), and the associated jump of the k(sup th) derivative of f is approximated to within O(N(sup -M-l+k), as N approaches infinity, and the method is robust. These estimates are then used with a class of singular basis functions, which have certain 'built-in' singularities, to construct a new sequence of approximations to f. Each of these new approximations is the sum of a piecewise smooth function and a new Fourier series partial sum. When N is proportional to M, it is shown that these new approximations, and their derivatives, converge exponentially in the maximum norm to f, and its corresponding derivatives, except in the union of a finite number of small open intervals containing the points of singularity of f. The total measure of these intervals decreases exponentially to zero as M approaches infinity. The technique is illustrated with several examples.

Banerjee, Nana S.↗
NASA NTRS1975

On the cost of approximating and recognizing a noise perturbed straight line or a quadratic curve segment in the plane

Approximation of noisy data in the plane by straight lines or elliptic or single-branch hyperbolic curve segments arises in pattern recognition, data compaction, and other problems. The efficient search for and approximation of data by such curves were examined. Recursive least-squares linear curve-fitting was used, and ellipses and hyperbolas are parameterized as quadratic functions in x and y. The error minimized by the algorithm is interpreted, and central processing unit (CPU) times for estimating parameters for fitting straight lines and quadratic curves were determined and compared. CPU time for data search was also determined for the case of straight line fitting. Quadratic curve fitting is shown to require about six times as much CPU time as does straight line fitting, and curves relating CPU time and fitting error were determined for straight line fitting. Results are derived on early sequential determination of whether or not the underlying curve is a straight line.

Cooper, D. B.↗
NASA NTRS1981

Application of HCMM data to regional geologic analysis for mineral and energy resource evaluation

Experimentation with several potentially promising techniques led to the selection of a fairly simple scheme for registration of data from the HCMM using an affine transformation. A method based solely on remote sensing was developed to estimate those meteorological effects which are required for thermal inertia mapping. It assumes that the atmospheric fluxes are spatially invariant and that the solar, sky and sensible heat fluxes can be approximated by a simple mathematical form. Coefficients are determined from least squares method by fitting observational data to the thermal model.

Watson, K.↗
NASA NTRS1999

Novel, Miniature Multi-Hole Probes and High-Accuracy Calibration Algorithms for their use in Compressible Flowfields

Two new calibration algorithms were developed for the calibration of non-nulling multi-hole probes in compressible, subsonic flowfields. The reduction algorithms are robust and able to reduce data from any multi-hole probe inserted into any subsonic flowfield to generate very accurate predictions of the velocity vector, flow direction, total pressure and static pressure. One of the algorithms PROBENET is based on the theory of neural networks, while the other is of a more conventional nature (polynomial approximation technique) and introduces a novel idea of local least-squares fits. Both algorithms have been developed to complete, user-friendly software packages. New technology was developed for the fabrication of miniature multi-hole probes, with probe tip diameters all the way down to 0.035". Several miniature 5- and 7-hole probes, with different probe tip geometries (hemispherical, conical, faceted) and different overall shapes (straight, cobra, elbow probes) were fabricated, calibrated and tested. Emphasis was placed on the development of four stainless-steel conical 7-hole probes, 1/16" in diameter calibrated at NASA Langley for the entire subsonic regime. The developed calibration algorithms were extensively tested with these probes demonstrating excellent prediction capabilities. The probes were used in the "trap wing" wind tunnel tests in the 14'x22' wind tunnel at NASA Langley, providing valuable information on the flowfield over the wing. This report is organized in the following fashion. It consists of a "Technical Achievements" section that summarizes the major achievements, followed by an assembly of journal articles that were produced from this project and ends with two manuals for the two probe calibration algorithms developed.

Rediniotis, Othon K.↗
NASA NTRS1999

Wind Speed Measurement from Bistatically Scattered GPS Signals

Instrumentation and retrieval algorithms are described which use the forward, or bistatically scattered range-coded signals from the Global Positioning System (GPS) radio navigation system for the measurement of sea surface roughness. This roughness is known to be related directly to the surface wind speed. Experiments were conducted from aircraft along the TOPEX ground track, and over experimental surface truth buoys. These flights used a receiver capable of recording the cross correlation power in the reflected signal. The shape of this power distribution was then compared against analytical models derived from geometric optics. Two techniques for matching these functions were studied. The first recognized the most significant information content in the reflected signal is contained in the trailing edge slope of the waveform. The second attempted to match the complete shape of the waveform by approximating it as a series expansion and obtaining the nonlinear least squares estimate. Discussion is also presented on anomalies in the receiver operation and their identification and correction.

Garrison, James L.↗