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All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

Implicit application of polynomial filters in a k-step Arnoldi method

The Arnoldi process is a well known technique for approximating a few eigenvalues and corresponding eigenvectors of a general square matrix. Numerical difficulties such as loss of orthogonality and assessment of the numerical quality of the approximations as well as a potential for unbounded growth in storage have limited the applicability of the method. These issues are addressed by fixing the number of steps in the Arnoldi process at a prescribed value k and then treating the residual vector as a function of the initial Arnoldi vector. This starting vector is then updated through an iterative scheme that is designed to force convergence of the residual to zero. The iterative scheme is shown to be a truncation of the standard implicitly shifted QR-iteration for dense problems and it avoids the need to explicitly restart the Arnoldi sequence. The main emphasis of this paper is on the derivation and analysis of this scheme. However, there are obvious ways to exploit parallelism through the matrix-vector operations that comprise the majority of the work in the algorithm. Preliminary computational results are given for a few problems on some parallel and vector computers.

Sorensen, D. C.↗
NASA NTRS1991

Acceleration of convergence by shifting the spectrum of implicit finite difference operators associated with the equations of gas dynamics

Eigensystem analysis techniques are applied to finite difference formulations of the Navier-Stokes equations in one dimension. Spectra of the resulting implicit difference operators are computed. The largest eigenvalues are calculated by using a combination of the Frechet derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the rate of convergence of the iterative method with the dominant eigenvalue of the original iteration matrix. On the basis of the pattern of eigenvalue distributions for various flow configurations, a shifting of the implicit operators in question is devised. This procedure has improved the rates of convergence of CFD codes by 20 - 50 percent.

Cheer, A.↗
NASA NTRS1996

Implicity restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations

Eigenvalues and eigenfunctions of linear operators are important to many areas of applied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.

Sorensen, Danny C.↗
NASA NTRS2003

Eigensolver for a Sparse, Large Hermitian Matrix

A parallel-processing computer program finds a few eigenvalues in a sparse Hermitian matrix that contains as many as 100 million diagonal elements. This program finds the eigenvalues faster, using less memory, than do other, comparable eigensolver programs. This program implements a Lanczos algorithm in the American National Standards Institute/ International Organization for Standardization (ANSI/ISO) C computing language, using the Message Passing Interface (MPI) standard to complement an eigensolver in PARPACK. [PARPACK (Parallel Arnoldi Package) is an extension, to parallel-processing computer architectures, of ARPACK (Arnoldi Package), which is a collection of Fortran 77 subroutines that solve large-scale eigenvalue problems.] The eigensolver runs on Beowulf clusters of computers at the Jet Propulsion Laboratory (JPL).

Tisdale, E. Robert↗
NASA NTRS1985

The energization of electrons and ions by electron beams injected in the ionosphere

This paper represents a continuation of an investigation of the nature and origin of the particle populations associated with an electron beam-emitting rocket experiment in the ionosphere. In this investigation, the intense flux of suprathermal electrons created around such a rocket has been studied by several groups. Arnoldy and Winckler (1981) have discussed the vehicle floating potential problem. The present paper is concerned with results which have been obtained from the Echo 4 and 5 experiments considered by Swanson (1983). Attention is given to instrumentation, suprathermal electrons, thermal ions, energetic ions, and ion resonances.

Arnoldy, R. L.↗
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 disk 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.↗
NASA NTRS1988

Numerical solution of large nonsymmetric eigenvalue problems

Several methods are discribed for combinations of Krylov subspace techniques, deflation procedures and preconditionings, for computing a small number of eigenvalues and eigenvectors or Schur vectors of large sparse matrices. The most effective techniques for solving realistic problems from applications are those methods based on some form of preconditioning and one of several Krylov subspace techniques, such as Arnoldi's method or Lanczos procedure. Two forms of preconditioning are considered: shift-and-invert and polynomial acceleration. The latter presents some advantages for parallel/vector processing but may be ineffective if eigenvalues inside the spectrum are sought. Some algorithmic details are provided that improve the reliability and effectiveness of these techniques.

Saad, Youcef↗
NASA NTRS1990

Search for lightning-induced electron precipitation with rocket-borne photometers

Photometers at 3914 A and 5577 A and an optical imager were part of an experimental package launched on a sounding rocket in the 1987 Wave Induced Particle Precipitation campaign at Wallops Island, Virginia. The objective was to measure lightning-induced electron precipitation (LEP) by means of its optical signature. This was the first attempt to measure LEP using rocket-borne optical instrumentation. Launch criteria included nearby thunderstorm activity and ground-based observations of Trimpi events. Lightning flashes are clearly discernible in the data. The photometer data was also characterized by large spin and precession modulations in the photon count rate, consistent with elevated steady particle fluxes in the northern portion of the instrument field of view. No evidence of LEP was observed by the photometers or onboard particle detectors (Arnoldy and Kintner, 1989). Analysis of the data has made it possible to place an upper limit of 0.0008 ergs/sq cm per sec on any burst precipitation energy flux that may have occurred during the rocket flight in the regions explored by the photometers.

Massey, R. D.↗
NASA NTRS1989

Projection methods for the numerical solution of Markov chain models

Projection methods for computing stationary probability distributions for Markov chain models are presented. A general projection method is a method which seeks an approximation from a subspace of small dimension to the original problem. Thus, the original matrix problem of size N is approximated by one of dimension m, typically much smaller than N. A particularly successful class of methods based on this principle is that of Krylov subspace methods which utilize subspaces of the form span(v,av,...,A(exp m-1)v). These methods are effective in solving linear systems and eigenvalue problems (Lanczos, Arnoldi,...) as well as nonlinear equations. They can be combined with more traditional iterative methods such as successive overrelaxation, symmetric successive overrelaxation, or with incomplete factorization methods to enhance convergence.

Saad, Youcef↗
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.↗