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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 NTRS1985

Probabilistic finite elements for transient analysis in nonlinear continua

The probabilistic finite element method (PFEM), which is a combination of finite element methods and second-moment analysis, is formulated for linear and nonlinear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in nonlinear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem. The moments calculated compare favorably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.

Liu, W. K.↗
NASA NTRS2004

Entangled Biphoton Virtual-State Spectroscopy of the A(exp 2)Sigma(sup +)-X(exp 2)Pi System of OH

This Letter describes the first application of entanglement-induced virtual-state spectroscopy to a molecular system. Non-classical, non-monotonic behavior in a two-photon absorption cross section of the OH A-X system, induced by an entangled biphoton state is theoretically demonstrated. A Fourier transform analysis of the biphoton cross section permits access to the energy eigenvalues of intermediate rovibronic states with a fixed excitation photon energy. The dependence of the Fourier spectrum on the tuning range of the entanglement time (T(sub e)) and the relative path delay (tau(sub e)) is discussed. Our analysis reveals that the implementation of molecular virtual-state spectroscopy for the OH A-X system requires the tuning of tau(sub e) over a pico-second range with femto-second resolution.

Kojima, Jun↗
NASA NTRS2004

Entangled Biphoton Virtual-State Spectroscopy of the A(exp 2)Sigma(+) - X(exp 2)Pi System of OH

This Letter describes the first application of entanglement-induced virtual-state spectroscopy to a molecular system. Non-classical, non-monotonic behavior in a two-photon absorption cross section of the OH A-X system, induced by an entangled biphoton state is theoretically demonstrated. A Fourier transform analysis of the biphoton cross section permits access to the energy eigenvalues of intermediate rovibronic states with a fixed excitation photon energy. The dependence of the Fourier spectrum on the tuning range of the entanglement time T(sub e), and the relative path delay tau(sub e) is discussed. Our analysis reveals that the implementation of molecular virtual-state spectroscopy for the OH A-X system requires the tuning of tau(sub e) over a pico-second range with femto-second resolution.

Kojima, Jun↗
NASA NTRS1976

Eigenvalue properties of structural mean-axis systems

The authors review some of the properties of the pseudoinverse and oblique pseudoinverse of a linear transformation T from one finite-dimensional inner-product space into another, and then use these properties and a theorem of Milne (1968), which states that the oblique pseudoinverse can be expressed in terms of a weak generalized inverse and two projection operators, in order to compute a mean-axis influence coefficient matrix for the dynamic analysis of an elastic body. Some eigenvalue invariance properties of the mean-axis structural dynamics equations are demonstrated, and on the simple example of a uniform beam, it is shown that the finite frequencies and mode shapes of the mean axis structural system are identical to the nonzero frequencies and mode shapes of the free structure.

Cavin, R. K., III↗
NASA NTRS1987

A new method to real-normalize measured complex modes

A time domain subspace iteration technique is presented to compute a set of normal modes from the measured complex modes. By using the proposed method, a large number of physical coordinates are reduced to a smaller number of model or principal coordinates. Subspace free decay time responses are computed using properly scaled complex modal vectors. Companion matrix for the general case of nonproportional damping is then derived in the selected vector subspace. Subspace normal modes are obtained through eigenvalue solution of the (M sub N) sup -1 (K sub N) matrix and transformed back to the physical coordinates to get a set of normal modes. A numerical example is presented to demonstrate the outlined theory.

Wei, Max L.↗
NASA NTRS1974

A fast-initializing digital equalizer with on-line tracking for data communications

A theory is developed for a digital equalizer for use in reducing intersymbol interference (ISI) on high speed data communications channels. The equalizer is initialized with a single isolated transmitter pulse, provided the signal-to-noise ratio (SNR) is not unusually low, then switches to a decision directed, on-line mode of operation that allows tracking of channel variations. Conditions for optimal tap-gain settings are obtained first for a transversal equalizer structure by using a mean squared error (MSE) criterion, a first order gradient algorithm to determine the adjustable equalizer tap-gains, and a sequence of isolated initializing pulses. Since the rate of tap-gain convergence depends on the eigenvalues of a channel output correlation matrix, convergence can be improved by making a linear transformation on to obtain a new correlation matrix.

Houts, R. C.↗
NASA NTRS1983

Feature selection methodologies using simulated Thematic Mapper data

The present investigation is concerned with the determination of the intrinsic dimensionality of a simulated Thematic Mapper data set. In addition, the effectiveness and sensitivity of 'standard' statistics separability measures (i.e., transformed divergence) is evaluated in comparison to eigenvectors for identifying the optimum subset of the original Thematic Mapper Simulator (TMS) bands for classifying the various cover types. TMS data were collected on May 2, 1979 by NASA's NS001 aircraft multispectral scanner over a bottomland forested area in South Carolina near the city of Camden. It is found that the eigenvectors and eigenvalues of a covariance matrix from a multispectral scanner system (MSS) data set can be obtained without having to actually transform the data.

Dean, M. E.↗
NASA NTRS1989

A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

In recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices appeared in the literature. We study restricted congruence transformation of Hermitian matrices M which, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain short proofs of most of the known inertia theorems and also derive some new results of this type. For some applications, a special class of almost unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form which, in particular, displays the eigenvalues of A. Finally, applications of this quasi-spectral decomposition to generalize inverses and Hermitian matrix pencils are discussed.

Freund, Roland W.↗
NASA NTRS2013

Mixture-Tuned, Clutter Matched Filter for Remote Detection of Subpixel Spectral Signals

Mapping localized spectral features in large images demands sensitive and robust detection algorithms. Two aspects of large images that can harm matched-filter detection performance are addressed simultaneously. First, multimodal backgrounds may thwart the typical Gaussian model. Second, outlier features can trigger false detections from large projections onto the target vector. Two state-of-the-art approaches are combined that independently address outlier false positives and multimodal backgrounds. The background clustering models multimodal backgrounds, and the mixture tuned matched filter (MT-MF) addresses outliers. Combining the two methods captures significant additional performance benefits. The resulting mixture tuned clutter matched filter (MT-CMF) shows effective performance on simulated and airborne datasets. The classical MNF transform was applied, followed by k-means clustering. Then, each cluster s mean, covariance, and the corresponding eigenvalues were estimated. This yields a cluster-specific matched filter estimate as well as a cluster- specific feasibility score to flag outlier false positives. The technology described is a proof of concept that may be employed in future target detection and mapping applications for remote imaging spectrometers. It is of most direct relevance to JPL proposals for airborne and orbital hyperspectral instruments. Applications include subpixel target detection in hyperspectral scenes for military surveillance. Earth science applications include mineralogical mapping, species discrimination for ecosystem health monitoring, and land use classification.

Thompson, David R.↗
NASA NTRS1972

A fast numerical method for determining the optimum SNR of an array subject to a Q factor constraint.

The problem of finding the optimum current excitations for a point-detector array which extracts a signal from background noise using the criterion of maximizing the SNR subject to a constraint on the supergain ratio (or Q factor) has recently been solved. The formulation reduced the problem to one of solving for the roots of a polynomial whose coefficients are, in general, complex. We demonstrate that by making a certain transformation, it becomes possible to employ a state variable technique which changes the problem into one of finding the eigenvalues of a real matrix which is considerably faster to do.

Winkler, L. P.↗
NASA NTRS1990

An efficient design sensitivity analysis of eigenvectors

Subspace iteration has been a major advance in solving large eigen problems when only a subset of eigen-pairs is required. The essence of this method is a transformation from displacement coordinates of an n-th order eigensystem to generalized coordinates of a smaller q-th order. The eigenvalue problem is then solved in the reduced space. The method was first developed by Clint and Jennings for real symmetric systems and was then called 'simultaneous iteration'. The success of the method prompted further research along this line and there have been many improved algorithms developed. This approach has been widely used by structural engineers for extracting the most useful natural frequencies and mode shapes of large-scale dynamic systems. This paper exploits into a new direction which is in the form of iterative process for simultaneously calculating eigenvector derivatives of many eigenvectors with respect to multi-variables. The method fully uses all the available information from preceding eigenvalue solution and, thus, effectively economizes computational efforts. It iterates through two equations derived from the first variation of the two fundamental equations used in subspace iteration method. There is no expensive large matrix decomposition required and the process converges to acceptable solution in a finite number of iterations. Therefore, the procedure increases its efficiency superiority over the others as the system size or the number of interested eigenvectors become larger and larger.

Ting, T.↗
NASA NTRS1976

Stiffness coupling application to modal synthesis program, users guide

A FORTRAN IV computer program used to perform modal synthesis of structures by stiffness coupling, using the dynamic transformation method is described. The program was named SCAMP (Stiffness Coupling Approach Modal-Synthesis Program). The program begins with the entry of a substructure's physical mode shapes and eigenvalues or a substructure's mass and stiffness matrix. If the mass and stiffness matrices are entered, the eigen problem for the individual substructure is solved. Provisions are included for a maximum of 20 substructures which are coupled by stiffness matrix springs. Each substructure has a number degrees of freedom (DOF), except that for DOF greater than 100; vector sets having maximum row and column size of 100 were generated prior to entering SCAMP. The substructures are then coupled together via coupling springs, and the dynamic transformation is used to reduce the size of the eigen problem.

Kuhar, E. J.↗
NASA NTRS1982

Numerical computation of exponential matrices using the Cayley-Hamilton theorem

A method for computing exponential matrices, which often arise naturally in the solution of systems of linear differential equations, is developed. An exponential matrix is generated as a linear combination of a finite number (equal to the matrix order) of matrices, the coefficients of which are scalar infinite sums. The method can be generalized to apply to any formal power series of matrices. Attention is focused upon the exponential function, and the matrix exponent is assumed tri-diagonal in form. In such cases, the terms in the coefficient infinite sums can be extracted, as recursion relations, from the characteristic polynomial of the matrix exponent. Two numerical examples are presented in some detail: (1) the three dimensional infinitesimal rotation rate matrix, which is skew symmetric, and (2) an N-dimensional tri-diagonal and symmetric finite difference matrix which arises in the numerical solution of the heat conduction partial differential equation. In the second example, the known eigenvalues and eigenvectors of the finite difference matrix permit an analytical solution for the exponential matrix, through the theory of diagonalization and similarity transformations, which is used for independent verification. The convergence properties of the scalar infinite summations are investigated for finite difference matrices of various orders up to ten, and it is found that the number of terms required for convergence increases slowly with the order of the matrix.

Walden, H.↗
NASA NTRS1990

A Shifted Block Lanczos Algorithm 1: The Block Recurrence

In this paper we describe a block Lanczos algorithm that is used as the key building block of a software package for the extraction of eigenvalues and eigenvectors of large sparse symmetric generalized eigenproblems. The software package comprises: a version of the block Lanczos algorithm specialized for spectrally transformed eigenproblems; an adaptive strategy for choosing shifts, and efficient codes for factoring large sparse symmetric indefinite matrices. This paper describes the algorithmic details of our block Lanczos recurrence. This uses a novel combination of block generalizations of several features that have only been investigated independently in the past. In particular new forms of partial reorthogonalization, selective reorthogonalization and local reorthogonalization are used, as is a new algorithm for obtaining the M-orthogonal factorization of a matrix. The heuristic shifting strategy, the integration with sparse linear equation solvers and numerical experience with the code are described in a companion paper.

Grimes, Roger G.↗
NASA NTRS1977

The role of eigenvalues in linear feature selection theory

The analysis concerns the role of eigenvalues in determining a particular measure of pattern class distinction called the divergence, which is the pairwise average of the expected interclass divergence derived from Hajek's two-class divergence. Decel and Quirein (1973) showed that there always exists a k x n real matrix B such that the transformation determined by B maximizes divergence in k-dimensional space, and that B can be written as a product involving an orthogonal n x n matrix U. In the present paper it is shown that divergence measure of pattern class distinction does not depend on the eigenvalues of U.

Brown, D. R.↗
NASA NTRS1986

Wave-packet evolution and quantization

A general relationship is established between wave-packet evolution and quantization in classically integrable systems. Because of wave-packet spreading, one cannot simply take the Fourier transform of the time evolution of a wave-packet. Instead, one must propagate the wave-packet using the actions as Hamiltonians. The energy eigenvalues which result are the Einstein-Brillouin-Keller values, and new forms for the eigenfunctions appear. These are free of caustic singularities, and represent averages of wave packets over the invariant torus.

Littlejohn, R. G.↗
NASA NTRS1987

Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

Hunt, L. R.↗
NASA NTRS1988

The optimal polarizations for achieving maximum contrast in radar images

There is considerable interest in determining the optimal polarizations that maximize contrast between two scattering classes in polarimetric radar images. A systematic approach is presented for obtaining the optimal polarimetric matched filter, i.e., that filter which produces maximum contrast between two scattering classes. The maximization procedure involves solving an eigenvalue problem where the eigenvector corresponding to the maximum contrast ratio is an optimal polarimetric matched filter. To exhibit the physical significance of this filter, it is transformed into its associated transmitting and receiving polarization states, written in terms of horizontal and vertical vector components. For the special case where the transmitting polarization is fixed, the receiving polarization which maximizes the contrast ratio is also obtained. Polarimetric filtering is then applies to synthetic aperture radar images obtained from the Jet Propulsion Laboratory. It is shown, both numerically and through the use of radar imagery, that maximum image contrast can be realized when data is processed with the optimal polarimeter matched filter.

Swartz, A. A.↗