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All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Application of vector-valued rational approximations to the matrix eigenvalue problem and connections with Krylov subspace methods

Let F(z) be a vectored-valued function F: C approaches C sup N, which is analytic at z=0 and meromorphic in a neighborhood of z=0, and let its Maclaurin series be given. We use vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series in conjunction with power iterations to develop bona fide generalizations of the power method for an arbitrary N X N matrix that may be diagonalizable or not. These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and the corresponding invariant subspaces, and present a detailed convergence theory for them. In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos. Thus, the theory provides a set of completely new results and constructions for these Krylov subspace methods. This theory suggests at the same time a new mode of usage for these Krylov subspace methods that were observed to possess computational advantages over their common mode of usage.

Sidi, Avram↗
NASA NTRS1992

Upper bounds for convergence rates of vector extrapolation methods on linear systems with initial iterations

The application of the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) to a vector sequence obtained by the linear iterative technique x(sub j) + 1 = Ax(sub j) = b,j = 1,2,..., is considered. Both methods produce a two dimensional array of approximations s(sub n,k) to the solution of the system (I - A)x = b. Here, s(sub n,k) is obtained from the vectors x(sub j), n is less than or equal to j is less than or equal to n + k + 1. It was observed in an earlier publication by the first author that the sequence s(sub n,k), k = 1,2,..., for n greater than 0, but fixed, possesses better convergence properties than the sequence s(sub 0,k), k = 1,2,.... A detailed theoretical explanation for this phenomenon is provided in the present work. This explanation is heavily based on approximations by incomplete polynomials. It is demonstrated by numerical examples when the matrix A is sparse that cycling with s(sub n,k) for n greater than 0, but fixed, produces better convergence rates and costs less computationally than cycling with s(sub 0,k). It is also illustrated numerically with a convection-diffusion problem that the former may produce excellent results where the latter may fail completely. As has been shown in an earlier publication, the results produced by s(sub 0,k) are identical to the corresponding results obtained by applying the Arnoldi method or generalized minimal residual scheme (GMRES) to the system (I - A)x = b.

Sidi, Avram↗
NASA NTRS1993

Computation of selected eigenvalues of generalized eigenvalue problems

The Arnoldi (1951) and Lanczos (1950) iterative algorithms are used to obtain a few selected eigenvalues of the generalized eigenvalue problem A(x) = lambdaB(x), where A and B are n x n nonsymmetric banded complex matrices. A shift-and-invert strategy is used to increase the rate of convergence toward the desired eigenvalues, and the two approaches are evaluated in view of their performance in a model problem concerning the linear stability of compressible boundary layers. A general scheme for computing the eigenvalues that lie within a 'box' in the complex plane is presented.

Nayar, Narinder↗
NASA NTRS1999

Localized Ionospheric Particle Acceleration and Wave Acceleration of Auroral Ions: Amicist Data Set

Research supported by this grant covered two main topics: auroral ion acceleration from ELF-band wave activity, and from VLF-spikelet (lower hybrid solitary structure) wave activity. Recent auroral sounding rocket data illustrate the relative significance of various mechanisms for initiating auroral ion outflow. Two nightside mechanisms are shown in detail. The first mechanism is ion acceleration within lower hybrid solitary wave events. The new data from this two payload mission show clearly that: (1) these individual events are spatially localized to scales approximately 100 m wide perpendicular to B, in agreement with previous investigations of these structures, and (2) that the probability of occurrence of the events is greatest at times of maximum VLF wave intensity. The second mechanism is ion acceleration by broadband, low frequency electrostatic waves, observed in a 30 km wide region at the poleward edge of the arc. The ion fluxes from the two mechanisms are compared and it is shown that while lower hybrid solitary structures do indeed accelerate ions in regions of intense VLF waves, the outflow from the electrostatic ion wave acceleration region is dominant for the aurora investigated by this sounding rocket, AMICIST. The fluxes are shown to be consistent with DE-1 and Freja outflow measurements, indicating that the AMICIST observations show the low altitude, microphysical signatures of nightside auroral outflow. In this paper, we present a review of sounding rocket observations of the ion acceleration seen nightside auroral zone lower hybrid solitary structures. Observations from Topaz3, Amicist, and Phaze2 are presented on various spatial scales, including the two-point measurements of the Amicist mission. From this collection of observations, we will demonstrate the following characteristics of transverse ion acceleration (TAI) in LHSS. The ion acceleration process is narrowly confined to 90 degrees pitch angle, in spatially confined regions of up to a few hundred meters across B. The acceleration process does not affect the thermal core of the ambient distribution, and does not directly create a measurable effect on the ambient ion population outside the LHSS themselves. Within the LHSS region, it creates a high energy tail beginning at a few times the thermal ion speed. The ion acceleration events are closely associated with localized wave events. Accelerated ions bursts are also seen without a concurrent observation of a localized wave event, for two possible reasons. In some cases, the pitch angles of the accelerated tail ions are elevated above perpendicular; that is to say, the acceleration occurred below the observer and the mirror force has begun to act upon the distribution, moving it upward from the source. In other cases, the accelerated ion structure is spatially larger than the wave event structure, and the observation catches only the ion event. The occurrence rate of these ion acceleration events is related to the ambient environment in two ways: (1) its altitude dependence can be modelled with the parameter B(exp2)/n(sub e), (2) and it is highest in regions of intense VLF activity. The end result of this study of auroral ion acceleration processes is a manuscript entitled "Auroral Ion Acceleration from Lower Hybrid Solitary Structures: A Summary of Sounding Rocket Observations", by Lynch, Arnoldy, Kintner, Schuck, Bonnell, and Coffey. This manuscript has been submitted for publication to the Journal of Geophysical Research.

Lynch, Kristina A.↗
NASA NTRS1999

On the Stability of Three-Dimensional Boundary Layers: Secondary Instability - Pt. 2

The secondary instability of three-dimensional incompressible boundary layers is studied using Floquet theory. Starting from the equilibrium solutions that we obtained from the PSE computations documented in Part 1, we investigate the region where a purely stationary crossflow disturbance saturates for its secondary instability characteristics utilizing developed global and local eigenvalue solvers that are based on the Implicitly Restarted Arnoldi Method, and a Newton-Raphson technique, respectively. The main focuses of this study are on the existence of multiple roots in the eigenvalue spectrum that could explain experimental observations of time-dependent occurrences of an explosive growth of traveling disturbances, on the routes by which high-frequency disturbances enter the boundary layer, as well as on gaining more information about threshold amplitudes for the growth of secondary disturbances.

Janke, Erik↗
NASA NTRS1975

Thickness Noise of Helicopter Rotors at High Tip Speeds

A new formulation of helicopter rotor thickness, noise for hover and forward flight, is discussed. The parameters required for this formulation are rotor motion, planform and airfoil thickness distribution. A computer program has been developed to calculate the pressure signature due to blade thickness for a helicopter in arbitrary motion. Comparison with high-speed helicopter tests shows good agreement with calculations when the observer is in or near the horizontal plane in which the rotor disc lies. Characteristics of thickness noise are illustrated by numerical examples indicating strongly that the high-speed blade slap may be due primarily to the thickness effect. The methods of Deming and Arnoldi are discussed as the special cases of this technique.

Farassat, F.↗
DOE OSTI2024

The NanoSIMS-HR: The Next Generation of High Spatial Resolution Dynamic SIMS

The high lateral resolution and sensitivity of the NanoSIMS 50 and 50L series of dynamic SIMS instruments have enabled numerous scientific advances over the past 25 years. Here, in this study, we report on the NanoSIMS-HR, the first major upgrade to the series, and analytical tests in a suite of sample types, including an aluminum sample containing silicon crystals, microalgae, and plant roots colonized with a symbiotic fungus. Significant improvements have been made in the Cs + ion source, high voltage (HV) control, stage reproducibility, and other aspects of the instrument that affect performance. The modified design of the NanoSIMS-HR thermal-ionization Cs + source enables a 5 pA primary ion beam to be focused into a 100 nm spot, a ~2.5-fold increase compared to Cs + sources on previous instruments (~2 pA at 100 nm). The brightness of the new Cs + source enables an ultimate lateral resolution as high as 30 nm and improved detection limits for a given analysis area. Sample stage movement accuracy is higher than 500 nm, enabling many-fold higher throughput automated analyses. With the new HV control, the primary ion beam impact energy can be reduced from 16 to 2 keV, which enables higher depth resolution during depth profiling (a 2-fold improvement), albeit with a 5-fold decrease in lateral resolution. In the NanoSIMS-HR, the secondary ion column and detection system are identical to those used in the previous series, and the isotopic analysis performance is as precise as in previous NanoSIMS instruments.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1960

Observations of the Van Allen Radiation Regions During August and September 1959

An integrating ionization chamber and a single Geiger counter were flown on United States satellite Explorer VI in an elliptical orbit extending to 48,000 km. In addition to the Van Allen inner zone and the great outer zone, a stable and distinct intermediate zone was detected throughout August and September 1959. The outer-zone intensity showed a large decrease following the sudden commencement of a geomagnetic storm. Later in the storm the outer zone increased to much in excess of its prestorm level. During stable periods the outer zone was fairly constant and less intense than it had been observed to be with Pioneer III or Pioneer IV or the first Soviet cosmic rocket. Cosmic-ray background counting rates were reached on most passes in August and Septmeber near apogee of the satellite. The radiation ?dumped? from the outer zone during the geomagnetic storm fits very well with the intensity and latitude distribution required to account for balloon observations of auroral X rays made during the IGY period. This paper is based on preliminary analysis of Explorer VI data.

Arnoldy, R. L.↗