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NASA NTRS1992

Large-scale sparse singular value computations

Four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture are presented. Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left and right-singular vectors) for sparse matrices arising from two practical applications: information retrieval and seismic reflection tomography are emphasized. The target architectures for implementations are the CRAY-2S/4-128 and Alliant FX/80. The sparse SVD problem is well motivated by recent information-retrieval techniques in which dominant singular values and their corresponding singular vectors of large sparse term-document matrices are desired, and by nonlinear inverse problems from seismic tomography applications which require approximate pseudo-inverses of large sparse Jacobian matrices.

Berry, Michael W.↗

NASA NTRS1994

A visual detection model for DCT coefficient quantization

The discrete cosine transform (DCT) is widely used in image compression and is part of the JPEG and MPEG compression standards. The degree of compression and the amount of distortion in the decompressed image are controlled by the quantization of the transform coefficients. The standards do not specify how the DCT coefficients should be quantized. One approach is to set the quantization level for each coefficient so that the quantization error is near the threshold of visibility. Results from previous work are combined to form the current best detection model for DCT coefficient quantization noise. This model predicts sensitivity as a function of display parameters, enabling quantization matrices to be designed for display situations varying in luminance, veiling light, and spatial frequency related conditions (pixel size, viewing distance, and aspect ratio). It also allows arbitrary color space directions for the representation of color. A model-based method of optimizing the quantization matrix for an individual image was developed. The model described above provides visual thresholds for each DCT frequency. These thresholds are adjusted within each block for visual light adaptation and contrast masking. For given quantization matrix, the DCT quantization errors are scaled by the adjusted thresholds to yield perceptual errors. These errors are pooled nonlinearly over the image to yield total perceptual error. With this model one may estimate the quantization matrix for a particular image that yields minimum bit rate for a given total perceptual error, or minimum perceptual error for a given bit rate. Custom matrices for a number of images show clear improvement over image-independent matrices. Custom matrices are compatible with the JPEG standard, which requires transmission of the quantization matrix.

Ahumada, Albert J., Jr.↗

NASA NTRS1994

A new look at the simultaneous analysis and design of structures

The minimum weight optimization of structural systems, subject to strength and displacement constraints as well as size side constraints, was investigated by the Simultaneous ANalysis and Design (SAND) approach. As an optimizer, the code NPSOL was used which is based on a sequential quadratic programming (SQP) algorithm. The structures were modeled by the finite element method. The finite element related input to NPSOL was automatically generated from the input decks of such standard FEM/optimization codes as NASTRAN or ASTROS, with the stiffness matrices, at present, extracted from the FEM code ANALYZE. In order to avoid ill-conditioned matrices that can be encountered when the global stiffness equations are used as additional nonlinear equality constraints in the SAND approach (with the displacements as additional variables), the matrix displacement method was applied. In this approach, the element stiffness equations are used as constraints instead of the global stiffness equations, in conjunction with the nodal force equilibrium equations. This approach adds the element forces as variables to the system. Since, for complex structures and the associated large and very sparce matrices, the execution times of the optimization code became excessive due to the large number of required constraint gradient evaluations, the Kreisselmeier-Steinhauser function approach was used to decrease the computational effort by reducing the nonlinear equality constraint system to essentially a single combined constraint equation. As the linear equality and inequality constraints require much less computational effort to evaluate, they were kept in their previous form to limit the complexity of the KS function evaluation. To date, the standard three-bar, ten-bar, and 72-bar trusses have been tested. For the standard SAND approach, correct results were obtained for all three trusses although convergence became slower for the 72-bar truss. When the matrix displacement method was used, correct results were still obtained, but the execution times became excessive due to the large number of constraint gradient evaluations required. Using the KS function, the computational effort dropped, but the optimization seemed to become less robust. The investigation of this phenomenon is continuing. As an alternate approach, the code MINOS for the optimization of sparse matrices can be applied to the problem in lieu of the Kreisselmeier-Steinhauser function. This investigation is underway.

Striz, Alfred G.↗

NASA NTRS1995

Thermomechanical Fatigue Behavior of Three Cfcc's

The thermomechanical fatigue (TMF) behavior of three continuous-fiber ceramic composites (CFCC's) was examined. The three matrices consisted of two different glass-ceramics and silicon carbide, respectively. The matrices of some of the specimens with the silicon carbide matrix were enhanced to improve oxidation resistance. All three were reinforced with Nicalon fibers with various fiber architectures. The thermoelastic properties of the matrix relative to the fibers, the elastic moduli and coefficients of thermal expansion (CTE), were different in each composite, providing a comparison of the effects of these critical properties. The specimens were tested under in-phase (IP) and out-of-phase (OP) cyclic loadings with respect to thermal cycling between 600 C and 1100 C. One of the glass-ceramic composites had greatly increased TMF lives compared to the other glass-ceramic matrix composites. This was probably caused by the reversal of the CTE mismatch between the fibers and the matrices and caused by the different oxidation resistances of the composites. For the same TMF lives a cross-ply reinforced composite had a maximum cyclic stress half as great, as that of unidirectionally reinforced composite. This indicated that the (0 deg) ply fibers had a strong influence of TMF life. Both the glass ceramic matrix composites had shallow-slope stress-life plots that indicated sensitivity to damage and therefore low damage tolerance. The composites with an unenhanced silicon carbide matrix experienced the shortest TMF lives of all the CFCC's tested. However, the enhanced composite had the longest. Also, the enhanced composite demonstrated the best damage tolerance as evidenced by a stress-life curve that curves sharply upward to the left. In all the composites, out-of-phase cyclic loading was worse than in-phase cyclic loadings despite the CTE mismatch between fiber and matrix. A damage investigation and a preliminary analysis of the stresses in the fibers and matrices taking into account their respective thermoelastic properties provided insight into the relative behavior of the glass-ceramic matrix composites under IP and OP TMF.

Worthem, Dennis W.↗

NASA NTRS1996

Wavelet Sparse Approximate Inverse Preconditioners

There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle and Chow and Saad also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. Harwell-Boeing collections. Nonetheless a drawback is that it requires rapid decay of the inverse entries so that sparse approximate inverse is possible. However, for the class of matrices that, come from elliptic PDE problems, this assumption may not necessarily hold. Our main idea is to look for a basis, other than the standard one, such that a sparse representation of the inverse is feasible. A crucial observation is that the kind of matrices we are interested in typically have a piecewise smooth inverse. We exploit this fact, by applying wavelet techniques to construct a better sparse approximate inverse in the wavelet basis. We shall justify theoretically and numerically that our approach is effective for matrices with smooth inverse. We emphasize that in this paper we have only presented the idea of wavelet approximate inverses and demonstrated its potential but have not yet developed a highly refined and efficient algorithm.

Chan, Tony F.↗

NASA NTRS1986

An improved approach to the analysis of drug-protein binding by distance geometry

The calculation of side chain centers of coordinates and the subsequent generation of side chain-side chain and side chain-backbone distance matrices is suggested as an improved method for viewing interactions inside proteins and for the comparison of protein structures. The use of side chain distance matrices is demonstrated with free PTI, and the use of difference distance matrices for side chains is shown for free and trypsin-bound PTI as well as for the X-ray structures of trypsin complexes with PTI and with benzamidine. It is found that conformational variations are reflected in the side chain distance matrices much more than in the standard C-C distance representations.

NASA Discipline Exobiology↗

NASA NTRS2005

The Mid-Infrared Absorption Spectra of Neutral PAHs in Dense Interstellar Clouds

Polycyclic aromatic hydrocarbons (PAHs) are common throughout the universe and are expected to be present in dense interstellar clouds. In these environments, some P.4Hs may be present in the gas phase, but most should be frozen into ice mantles or adsorbed onto dust grains and their spectral features are expected to be seen in absorption. Here we extend our previous work on the infrared spectral properties of the small PAH naphthalene (C10H8) in several media to include the full mid-infrared laboratory spectra of 11 other PAHs and related aromatic species frozen in H2O ices. These include the molecules 1,2-dihydronaphthalene, anthracene, 9,1O-dihydroanthracene, phenanthrene, pyrene, benzo[e]pyrene, perylene, benzo(k)fluoranthene, pentacene, benzo[ghi]perylene, and coronene. These results demonstrate that PAHs and related molecules, as a class, show the same spectral behaviors as naphthalene when incorporated into H2O-rich matrices. When compared to the spectra of these same molecules isolated in inert matrices (e.g., Ar or N2), the absorption bands produced when they are frozen in H2O matrices are broader (factors of 3-10), show small position shifts in either direction (usually < 4/cm, always < 10/cm), and show variable changes in relative band strengths (typically factors of 1-3). There is no evidence of systematic increases or decreases in the absolute strengths of the bands of these molecules when they are incorporated in H2O matrices. In H2O-rich ices, their absorption bands are relatively insensitive to concentration over the range of 10 < H2O/PAH < 200): The absorption bands of these molecules are also insensitive to temperature over the 10 K < T < 125 K range, although the spectra can show dramatic changes as the ices are warmed through the temperature range in which amorphous H2O ice converts to its cubic and hexagonal crystalline forms (T > 125 Kj. Given the small observed band shifts cause by H2O, the current database of spectra from Ar matrix-isolated neutral PAHs and related molecules should be useful for the search for these species in dense clouds on the basis of observed absorption band positions. Furthermore, these data permit determination of column densities to better than a factor of 3 for PAHs in dense clouds. Column density determination of detected aromatics to better than a factor of 3 will, however, require good knowledge about the nature of the matrix in which the PAH is embedded and laboratory studies of relevant samples.

Bernstein, M. P.↗

NASA NTRS2012

Constructing an Efficient Self-Tuning Aircraft Engine Model for Control and Health Management Applications

Self-tuning aircraft engine models can be applied for control and health management applications. The self-tuning feature of these models minimizes the mismatch between any given engine and the underlying engineering model describing an engine family. This paper provides details of the construction of a self-tuning engine model centered on a piecewise linear Kalman filter design. Starting from a nonlinear transient aerothermal model, a piecewise linear representation is first extracted. The linearization procedure creates a database of trim vectors and state-space matrices that are subsequently scheduled for interpolation based on engine operating point. A series of steady-state Kalman gains can next be constructed from a reduced-order form of the piecewise linear model. Reduction of the piecewise linear model to an observable dimension with respect to available sensed engine measurements can be achieved using either a subset or an optimal linear combination of "health" parameters, which describe engine performance. The resulting piecewise linear Kalman filter is then implemented for faster-than-real-time processing of sensed engine measurements, generating outputs appropriate for trending engine performance, estimating both measured and unmeasured parameters for control purposes, and performing on-board gas-path fault diagnostics. Computational efficiency is achieved by designing multidimensional interpolation algorithms that exploit the shared scheduling of multiple trim vectors and system matrices. An example application illustrates the accuracy of a self-tuning piecewise linear Kalman filter model when applied to a nonlinear turbofan engine simulation. Additional discussions focus on the issue of transient response accuracy and the advantages of a piecewise linear Kalman filter in the context of validation and verification. The techniques described provide a framework for constructing efficient self-tuning aircraft engine models from complex nonlinear simulations.Self-tuning aircraft engine models can be applied for control and health management applications. The self-tuning feature of these models minimizes the mismatch between any given engine and the underlying engineering model describing an engine family. This paper provides details of the construction of a self-tuning engine model centered on a piecewise linear Kalman filter design. Starting from a nonlinear transient aerothermal model, a piecewise linear representation is first extracted. The linearization procedure creates a database of trim vectors and state-space matrices that are subsequently scheduled for interpolation based on engine operating point. A series of steady-state Kalman gains can next be constructed from a reduced-order form of the piecewise linear model. Reduction of the piecewise linear model to an observable dimension with respect to available sensed engine measurements can be achieved using either a subset or an optimal linear combination of "health" parameters, which describe engine performance. The resulting piecewise linear Kalman filter is then implemented for faster-than-real-time processing of sensed engine measurements, generating outputs appropriate for trending engine performance, estimating both measured and unmeasured parameters for control purposes, and performing on-board gas-path fault diagnostics. Computational efficiency is achieved by designing multidimensional interpolation algorithms that exploit the shared scheduling of multiple trim vectors and system matrices. An example application illustrates the accuracy of a self-tuning piecewise linear Kalman filter model when applied to a nonlinear turbofan engine simulation. Additional discussions focus on the issue of transient response accuracy and the advantages of a piecewise linear Kalman filter in the context of validation and verification. The techniques described provide a framework for constructing efficient self-tuning aircraft engine models from complex nonlinear simulatns.

Armstrong, Jeffrey B.↗

NASA NTRS2013

Evidence for Extended Aqueous Alteration in CR Carbonaceous Chondrites

We are currently studying the chemical interrelationships between the main rockforming components of carbonaceous chondrites (hereafter CC), e.g. silicate chondrules, refractory inclusions and metal grains, and the surrounding meteorite matrices. It is thought that the fine-grained materials that form CC matrices are representing samples of relatively unprocessed protoplanetary disk materials [1-3]. In fact, modern non-destructive analytical techniques have shown that CC matrices host a large diversity of stellar grains from many distinguishable stellar sources [4]. Aqueous alteration has played a role in homogeneizing the isotopic content that allows the identification of presolar grains [5]. On the other hand, detailed analytical techniques have found that the aqueously-altered CR, CM and CI chondrite groups contain matrices in which the organic matter has experienced significant processing concomitant to the formation of clays and other minerals. In this sense, clays have been found to be directly associated with complex organics [6, 7]. CR chondrites are particularly relevant in this context as this chondrite group contains abundant metal grains in the interstitial matrix, and inside glassy silicate chondrules. It is important because CR are known for exhibiting a large complexity of organic compounds [8-10], and only metallic Fe is considered essential in Fischer-Tropsch catalysis of organics [11-13]. Therefore, CR chondrites can be considered primitive materials capable to provide clues on the role played by aqueous alteration in the chemical evolution of their parent asteroids.

Trigo-Rodriquez, J. M.↗

NASA NTRS2019

Observations on the Computation of Eigenvalue and Eigenvector Jacobians

Many scientific and engineering problems benefit from analytic expressions for eigenvalue and eigenvector derivatives with respect to the elements of the parent matrix. While there exists extensive literature on the calculation of these derivatives, which take the form of Jacobian matrices, there are a variety of deficiencies that have yet to be addressed — including the need for both left and right eigenvectors, limitations on the matrix structure, and issues with complex eigenvalues and eigenvectors. This work addresses these deficiencies by proposing a new analytic solution for the eigenvalue and eigenvector derivatives. The resulting analytic Jacobian matrices are numerically efficient to compute and are valid for the general complex case. It is further shown that this new general result collapses to previously known relations for the special cases of real symmetric matrices and real diagonal matrices. Finally, the new Jacobian expressions are validated using forward finite differencing and performance is compared with another technique.

Jacobian↗

DOE OSTI2019

Effect of Microstructure and Properties of Ni-WC Composite Coatings on Their Solid Particle Erosion Behavior

Metal matrix composites (MMCs) of WC in Co or Ni matrices are widely used due to their high wear resistance. Previous studies showed that tailoring the mechanical properties through optimum WC content can significantly extend the life cycle of MMCs. In this study, composite coatings of Ni with various WC content ranging from 10 to 55 vol.% were cold sprayed using agglomerated WCNi powders and two different sizes of Ni powders, − 10 + 4 (d{sub 50} = 7 µm) referred to as Ni(7) and − 35 + 15 (d{sub 50} = 25 µm) to as Ni(25). Microstructural characterization, including the deformed structure of Ni splats and retention and distribution of WC, was performed by scanning electron microscopy (SEM). Composite coatings were subjected to solid particle erosion following ASTM G76-13 standard. At low WC content coatings, erosion occurred primarily by plastic deformation of Ni matrices and knocking out of WC particles. At high WC concentrations, network distribution of compacted WCNi particles in the matrices led to a significant drop in erosion rates of coatings, due to a change in erosion mechanism from severe cutting and plowing of Ni and WC dislodging to WCNi splats spalling. Composite coatings sprayed using finer size Ni powder were usually more erosion resistant.

36 MATERIALS SCIENCE↗

DOE OSTI2018

On polymorphism for discrete evolutionary dynamics driven either by selection or segregation distortion

We revisit some problems arising in the context of multiallelic discrete-time and deterministic evolutionary dynamics driven first by fitness differences and then by segregation distortion. In the model with fitness, we describe classes of fitness matrices exhibiting polymorphism. In the segregation case, still in search for conditions of polymorphism, we focus on a class of skew-symmetric matrices with a unique strictly positive kernel vector. Our main results reduce the study of these cases to the analysis of stochastic matrices.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2010

Maps and inverse maps in open quantum dynamics

Two kinds of maps that describe evolution of states of a subsystem coming from dynamics described by a unitary operator for a larger system, maps defined for fixed mean values and maps defined for fixed correlations, are found to be quite different for the same unitary dynamics in the same situation in the larger system. An affine form is used for both kinds of maps to find necessary and sufficient conditions for inverse maps. All the different maps with the same homogeneous part in their affine forms have inverses if and only if the homogeneous part does. Some of these maps are completely positive; others are not, but the homogeneous part is always completely positive. The conditions for an inverse are the same for maps that are not completely positive as for maps that are. For maps defined for fixed mean values, the homogeneous part depends only on the unitary operator for the dynamics of the larger system, not on any state or mean values or correlations. Necessary and sufficient conditions for an inverse are stated several different ways: in terms of the maps of matrices, basis matrices, density matrices, or mean values. The inverse maps are generally not tied to the dynamics the way the maps forward are. A trace-preserving completely positive map that is unital cannot have an inverse that is obtained from any dynamics described by any unitary operator for any states of a larger system.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2011

Covariances of Evaluated Nuclear Cross Section Data for (232)Th, (180,182,183,184,186)W and (55)Mn

The EMPIRE code system is a versatile package for nuclear model calculations that is often used for nuclear data evaluation. Its capabilities include random sampling of model parameters, which can be utilized to generate a full covariance matrix of all scattering cross sections, including cross-reaction correlations. The EMPIRE system was used to prepare the prior covariance matrices of reaction cross sections of (232)Th, (180,182,183,184,186)W and (55)Mn nuclei for incident neutron energies up to 60 MeV. The obtained modeling prior was fed to the GANDR system, which is a package for a global assessment of nuclear data, based on the Generalized Least-Squares method. By introducing experimental data from the EXFOR database into GANDR, the constrained covariance matrices and cross section adjustment functions were obtained. Applying the correction functions on the cross sections and formatting the covariance matrices, the final evaluations in ENDF-6 format including covariances were derived. In the resonance energy range, separate analyses were performed to determine the resonance parameters with their respective covariances. The data files thus obtained were then subjected to detailed testing and validation. Described evaluations with covariances of (232)Th, (180,182,183,184,186)W and (55)Mn nuclei are included into the ENDF/B-VII.1 library release.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2012

Dim 6 nucleon decay in anomalous U(1){sub A} SUSY GUT

Anomalous U(1){sub A} SUSY GUT models are atractive because they can solve many difficulties in SUSY GUT models. In anomalous U(1){sub A} SUSY GUT models, nucleon decay amplitudes of dim 6 effective interactions are larger than that of dim 5 effective interactions. Then, I discuss dim 6 effective interactions in some models, and predict proton lifetimes for many decay modes by calculating interactions. First, I fix unitary matrices that make Yukawa coupling matrices diagonalize, and predict proton lifetimes. Second, I study uncertainties of these unitary matrices, and show these effects on proton lifetimes. From these results, I show that we can distinguish some models from by nucleon decay.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2015

Periodic local MP2 method employing orbital specific virtuals

We introduce orbital specific virtuals (OSVs) to represent the truncated pair-specific virtual space in periodic local Møller-Plesset perturbation theory of second order (LMP2). The OSVs are constructed by diagonalization of the LMP2 amplitude matrices which correspond to diagonal Wannier-function (WF) pairs. Only a subset of these OSVs is adopted for the subsequent OSV-LMP2 calculation, namely, those with largest contribution to the diagonal pair correlation energy and with the accumulated value of these contributions reaching a certain accuracy. The virtual space for a general (non diagonal) pair is spanned by the union of the two OSV sets related to the individual WFs of the pair. In the periodic LMP2 method, the diagonal LMP2 amplitude matrices needed for the construction of the OSVs are calculated in the basis of projected atomic orbitals (PAOs), employing very large PAO domains. It turns out that the OSVs are excellent to describe short range correlation, yet less appropriate for long range van der Waals correlation. In order to compensate for this bias towards short range correlation, we augment the virtual space spanned by the OSVs by the most diffuse PAOs of the corresponding minimal PAO domain. The Fock and overlap matrices in OSV basis are constructed in the reciprocal space. The 4-index electron repulsion integrals are calculated by local density fitting and, for distant pairs, via multipole approximation. New procedures for determining the fit-domains and the distant-pair lists, leading to higher efficiency in the 4-index integral evaluation, have been implemented. Generally, and in contrast to our previous PAO based periodic LMP2 method, the OSV-LMP2 method does not require anymore great care in the specification of the individual domains (to get a balanced description when calculating energy differences) and is in that sense a black box procedure. Discontinuities in potential energy surfaces, which may occur for PAO-based calculations if one is not careful, virtually disappear for OSV-LMP2. Moreover, due to much increased compactness of the pair-specific virtual spaces, the OSV-LMP2 calculations are faster and require much less memory than PAO-LMP2 calculations, despite the noticeable overhead of the initial OSV construction procedure.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2011

Framed 4-graphs: Euler tours, Gauss circuits and rotating circuits

We consider connected finite 4-valent graphs with the structure of opposite edges at each vertex (framed 4-graphs). For any of such graphs there exist Euler tours, in travelling along which at each vertex we turn from an edge to a nonopposite one (rotating circuits); and at the same time, it is not true that for any such graph there exists an Euler tour passing from an edge to the opposite one at each vertex (a Gauss circuit). The main result of the work is an explicit formula connecting the adjacency matrices of the Gauss circuit and an arbitrary Euler tour. This formula immediately gives us a criterion for the existence of a Gauss circuit on a given framed 4-graph. It turns out that the results are also valid for all symmetric matrices (not just for matrices realisable by a chord diagram). Bibliography: 24 titles.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2011

Entanglement and separability of two-qubit states on Minkowski space with compact support

The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with 4x4 matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, s{sup 2}=t{sup 2}-V(vector sign){sup 2}, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗