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All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Theoretical study of the incompressible Navier-Stokes equations by the least-squares method

Usually the theoretical analysis of the Navier-Stokes equations is conducted via the Galerkin method which leads to difficult saddle-point problems. This paper demonstrates that the least-squares method is a useful alternative tool for the theoretical study of partial differential equations since it leads to minimization problems which can often be treated by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable and by using the least-squares method, this paper shows that the div-curl system is properly determined and elliptic, and has a unique solution. The same technique then is employed to prove that the Stokes equations are properly determined and elliptic, and that four boundary conditions on a fixed boundary are required for three-dimensional problems. This paper also shows that under four combinations of non-standard boundary conditions the solution of the Stokes equations is unique. This paper emphasizes the application of the least-squares method and the div-curl method to derive a high-order version of differential equations and additional boundary conditions. In this paper, an elementary method (integration by parts) is used to prove Friedrichs' inequalities related to the div and curl operators which play an essential role in the analysis.

Jiang, Bo-Nan↗
NASA NTRS1988

Linear Least Squares for Correlated Data

Throughout the literature authors have consistently discussed the suspicion that regression results were less than satisfactory when the independent variables were correlated. Camm, Gulledge, and Womer, and Womer and Marcotte provide excellent applied examples of these concerns. Many authors have obtained partial solutions for this problem as discussed by Womer and Marcotte and Wonnacott and Wonnacott, which result in generalized least squares algorithms to solve restrictive cases. This paper presents a simple but relatively general multivariate method for obtaining linear least squares coefficients which are free of the statistical distortion created by correlated independent variables.

Dean, Edwin B.↗
NASA NTRS1980

Mark 3 interactive data analysis system

The interactive data analysis system, a major subset of the total Mark 3 very long baseline interferometry (VLBI) software system is described. The system consists of two major and a number of small programs. These programs provide for the scientific analysis of the observed values of delay and delay rate generated by the VLBI data reduction programs and product the geophysical and astrometric parameters which are among the ultimate products of VLBI. The two major programs are CALC and SOLVE. CALC generates the theoretical values of VLBI delay rate as well as partial derivatives based on apriori values of the geophysical and astronometric parameters. SOLVE is a least squares parameters estimation program which yields the geophysical and astrometric parameters using the observed values by the data processing system and theoretical values and partial derivatives provided by CALC. SOLVE is a highly interactive program in which the user selects the exact form of the recovered parameters and the data to be accepted into the solution.

Ryan, J. W.↗
NASA NTRS1985

A Galerkin method for the estimation of parameters in hybrid systems governing the vibration of flexible beams with tip bodies

An approximation scheme is developed for the identification of hybrid systems describing the transverse vibrations of flexible beams with attached tip bodies. In particular, problems involving the estimation of functional parameters are considered. The identification problem is formulated as a least squares fit to data subject to the coupled system of partial and ordinary differential equations describing the transverse displacement of the beam and the motion of the tip bodies respectively. A cubic spline-based Galerkin method applied to the state equations in weak form and the discretization of the admissible parameter space yield a sequence of approximating finite dimensional identification problems. It is shown that each of the approximating problems admits a solution and that from the resulting sequence of optimal solutions a convergent subsequence can be extracted, the limit of which is a solution to the original identification problem. The approximating identification problems can be solved using standard techniques and readily available software.

Banks, H. T.↗
NASA NTRS1984

Acoustooptic linear algebra processors - Architectures, algorithms, and applications

Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.

Casasent, D.↗
NASA NTRS1999

Investigation of Point Doppler Velocimetry (PDV) for Transition Detection in Boundary Layers

A two-component Point Doppler Velocimeter (PDV) system has been improved through the use of vapor-limited iodine cells that have responses that are insensitive to temperature variations. Two-component PDV velocity measurements have been obtained for a 1 inch diameter uniform circular jet flow at a nominal exit velocity of 60 m/sec, corresponding to a Reynolds number of 100,000. Similar data have also been obtained for an annular jet and a swirling jet, These PDV data runs have been duplicated to judge the repeatability of these measurements, and also have been compared with hot wire anemometer data for the same flow conditions. PDV mean velocity results are repeatable to within approximately 1-2 meters per second; the PDV RMS velocity results are also quite repeatable. Exit profiles of PDV mean axial velocity data generally agree with hot wire anemometer results to within about 2 meters per second as well. However, the PDV RMS velocity results are consistently lower than the hot wire results everywhere but at the exit of the standard jet, where they are too high relative to the hot wire data. This is believed to at least be partially due to the method used to compute the RMS (Root Mean Square).

Kuhlman, John↗
NASA NTRS1986

The identification of a distributed parameter model for a flexible structure

A computational method is developed for the estimation of parameters in a distributed model for a flexible structure. The structure we consider (part of the RPL experiment) consists of a cantilevered beam with a thruster and linear accelerometer at the free end. The thruster is fed by a pressurized hose whose horizontal motion effects the transverse vibration of the beam. The Euler-Bernoulli theory is used to model the vibration of the beam and treat the hose-thruster assembly as a lumped or point mass-dashpot-spring system at the tip. Using measurements of linear acceleration at the tip, it is estimated that the parameters (mass, stiffness, damping) and a Voight-Kelvin viscoelastic structural damping parameter for the beam using a least squares fit to the data. Spline based approximations to the hybrid (coupled ordinary and partial differential equations) system are considered; theoretical convergence results and numerical studies with both simulation and actual experimental data obtained from the structure are presented and discussed.

Banks, H. T.↗
NASA NTRS1990

The augmented Lagrangian method for parameter estimation in elliptic systems

In this paper a new technique for the estimation of parameters in elliptic partial differential equations is developed. It is a hybrid method combining the output-least-squares and the equation error method. The new method is realized by an augmented Lagrangian formulation, and convergence as well as rate of convergence proofs are provided. Technically the critical step is the verification of a coercivity estimate of an appropriately defined Lagrangian functional. To obtain this coercivity estimate a seminorm regularization technique is used.

Ito, Kazufumi↗
NASA NTRS1993

Investigation for improving Global Positioning System (GPS) orbits using a discrete sequential estimator and stochastic models of selected physical processes

GEODYNII is a conventional batch least-squares differential corrector computer program with deterministic models of the physical environment. Conventional algorithms were used to process differenced phase and pseudorange data to determine eight-day Global Positioning system (GPS) orbits with several meter accuracy. However, random physical processes drive the errors whose magnitudes prevent improving the GPS orbit accuracy. To improve the orbit accuracy, these random processes should be modeled stochastically. The conventional batch least-squares algorithm cannot accommodate stochastic models, only a stochastic estimation algorithm is suitable, such as a sequential filter/smoother. Also, GEODYNII cannot currently model the correlation among data values. Differenced pseudorange, and especially differenced phase, are precise data types that can be used to improve the GPS orbit precision. To overcome these limitations and improve the accuracy of GPS orbits computed using GEODYNII, we proposed to develop a sequential stochastic filter/smoother processor by using GEODYNII as a type of trajectory preprocessor. Our proposed processor is now completed. It contains a correlated double difference range processing capability, first order Gauss Markov models for the solar radiation pressure scale coefficient and y-bias acceleration, and a random walk model for the tropospheric refraction correction. The development approach was to interface the standard GEODYNII output files (measurement partials and variationals) with software modules containing the stochastic estimator, the stochastic models, and a double differenced phase range processing routine. Thus, no modifications to the original GEODYNII software were required. A schematic of the development is shown. The observational data are edited in the preprocessor and the data are passed to GEODYNII as one of its standard data types. A reference orbit is determined using GEODYNII as a batch least-squares processor and the GEODYNII measurement partial (FTN90) and variational (FTN80, V-matrix) files are generated. These two files along with a control statement file and a satellite identification and mass file are passed to the filter/smoother to estimate time-varying parameter states at each epoch, improved satellite initial elements, and improved estimates of constant parameters.

Goad, Clyde C.↗
NASA NTRS1993

What determines the stellar mass functions in globular clusters?

We analyze the dependence of stellar mass function slopes for a sample of 17 globular clusters on a variety of cluster parameters. The principal novelty of our approach is the use of appropriate multivariate statistical methods to disentangle the complex situation which is present in this problem: the slopes depend simultaneously on more than one variable, and many cluster parameters are mutually correlated. We find that the mass function slopes in the range M/M(solar) = 0.5-0.8 are largely determined by the position in the Galaxy and to a lesser extent by the cluster metallicity. Clusters closer to the Galactic center or plane have shallower mass function slopes. At a given distance to the Galactic center, clusters closer to the Galactic plane have shallower mass function slopes. At a given R(GC) and/or Z(GP), more metal-rich clusters have shallower mass function slopes. Thus, the monovariate correlations with the position or metallicity are both correct, but only partial, and in terms of slopes, biased descriptions of the situation. We present trivariate least-squares solutions where the mass function slopes can be predicted within the measurement accuracy. This relation can serve as a powerful observational constraint for theories of globular cluster formation and evolution, and it is one of the tightest correlations between globular cluster properties now known.

Djorgovski, S.↗
NASA NTRS1979

Solution of a few nonlinear problems in aerodynamics by the finite elements and functional least squares methods

The numerical simulation of the transonic flows of idealized fluids and of incompressible viscous fluids, by the nonlinear least squares methods is presented. The nonlinear equations, the boundary conditions, and the various constraints controlling the two types of flow are described. The standard iterative methods for solving a quasi elliptical nonlinear equation with partial derivatives are reviewed with emphasis placed on two examples: the fixed point method applied to the Gelder functional in the case of compressible subsonic flows and the Newton method used in the technique of decomposition of the lifting potential. The new abstract least squares method is discussed. It consists of substituting the nonlinear equation by a problem of minimization in a H to the minus 1 type Sobolev functional space.

Periaux, J.↗
NASA NTRS1981

Incorporating partially identified sample segments into acreage estimation procedures: Estimates using only observations from the current year

Several methods of estimating individual crop acreages using a mixture of completely identified and partially identified (generic) segments from a single growing year are derived and discussed. A small Monte Carlo study of eight estimators is presented. The relative empirical behavior of these estimators is discussed as are the effects of segment sample size and amount of partial identification. The principle recommendations are (1) to not exclude, but rather incorporate partially identified sample segments into the estimation procedure, (2) try to avoid having a large percentage (say 80%) of only partially identified segments, in the sample, and (3) use the maximum likelihood estimator although the weighted least squares estimator and least squares ratio estimator both perform almost as well. Sets of spring small grains (North Dakota) data were used.

Sielken, R. L., Jr.↗
NASA NTRS1988

The identification of a distributed parameter model for a flexible structure

A computational method is developed for the estimation of parameters in a distributed model for a flexible structure. The structure we consider (part of the RPL experiment) consists of a cantilevered beam with a thruster and linear accelerometer at the free end. The thruster is fed by a pressurized hose whose horizontal motion effects the transverse vibration of the beam. The Euler-Bernoulli theory is used to model the vibration of the beam and treat the hose thruster assembly as a lumped or point mass dashpot spring system at the tip. Measurements of linear acceleration at the tip are used to estimate the hose parameters (mass, stiffness, damping) and a Voigt-Kelvin viscoelastic structural damping parameter for the beam using a least squares fit to the data. Spline based approximations are considered to the hybrid (coupled ordinary and partial differential equations) systems; theoretical convergence results and numerical studies with both simulation and actual experimental data obtained from the structure are presented and discussed.

Banks, H. T.↗
NASA NTRS1984

A numerical scheme for the identification of hybrid systems describing the vibration of flexible beams with tip bodies

A cubic spline based Galerkin-like method is developed for the identification of a class of hybrid systems which describe the transverse vibration to flexible beams with attached tip bodies. The identification problem is formulated as a least squares fit to data subject to the system dynamics given by a coupled system of ordnary and partial differential equations recast as an abstract evolution equation (AEE) in an appropriate infinite dimensional Hilbert space. Projecting the AEE into spline-based subspaces leads naturally to a sequence of approximating finite dimensional identification problems. The solutions to these problems are shown to exist, are relatively easily computed, and are shown to, in some sense, converge to solutions to the original identification problem. Numerical results for a variety of examples are discussed.

Rosen, I. G.↗