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Results for “Eigenvalues and eigenvectors”

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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301 records · Page 17

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NASA NTRS1987

Vibration analysis of quasi-symmetric structures

A computational technique is developed to reduce the size of the dynamical model of a large complex quasi-symmetric structure prior to free vibration analysis. The structure is discretized using a two-field mixed FEM model; the response vectors are decomposed into symmetric and antisymmetric components; the eigenvectors of the quasi-symmetric structure are approximated by a linear combination of a few preselected sets of symmetric and antisymmetric modes generated by the reduced-size model; global approximation vectors are selected and generated; and the resulting eigenvalue problem is significantly reduced via a classical Bubnov-Galerkin procedure. Numerical results for a shallow laminated anisotropic panel of quadrilateral planform are presented in extensive tables and graphs, demonstrating the effectiveness of the method.

Noor, Ahmed K.↗

NASA NTRS1978

Eigenvalue/eigenvector assignment using output feedback

The problem of eigenvalue assignment in a linear time-invariant system using output feedback is considered. New sufficient conditions are derived to assign an almost arbitrary set of minimum (n,m + r - 1) distinct eigenvalues where n, m, and r are the number of states, inputs, and outputs, respectively. These conditions precisely identify the class of systems where such an assignment is impossible. The synthesis technique also highlights the freedom in selection of closed-loop eigenvectors under output feedback. The utility of eigenvalue/eigenvector assignment in transient response shaping is illustrated by the design of a controller for the lateral dynamics of an aircraft.

Srinathkumar, S.↗

DOE OSTI2023

Hybrid eigensolvers for nuclear configuration interaction calculations

We examine and compare several iterative methods for solving large-scale eigenvalue problems arising from nuclear structure calculations. In particular, we discuss the possibility of using block Lanczos method, a Chebyshev filtering based subspace iterations and the residual minimization method accelerated by direct inversion of iterative subspace (RMM-DIIS) and describe how these algorithms compare with the standard Lanczos algorithm and the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm. Although the RMM-DIIS method does not exhibit rapid convergence when the initial approximations to the desired eigenvectors are not sufficiently accurate, it can be effectively combined with either the block Lanczos or the LOBPCG method to yield a hybrid eigensolver that has several desirable properties. We will describe a few practical issues that need to be addressed to make the hybrid solver efficient and robust.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Material coherence from trajectories via Burau eigenanalysis of braids

In this paper, we provide a numerical tool to study a material’s coherence from a set of 2D Lagrangian trajectories sampling a dynamical system, i.e., from the motion of passive tracers. We show that eigenvectors of the Burau representation of a topological braid derived from the trajectories have levelsets corresponding to components of the Nielsen–Thurston decomposition of the dynamical system. One can thus detect and identify clusters of space–time trajectories corresponding to coherent regions of the dynamical system by solving an eigenvalue problem. Unlike previous methods, the scalable computational complexity of our braid-based approach allows the analysis of large amounts of trajectories.

36 MATERIALS SCIENCE↗

NASA NTRS1991

Sequential design of linear quadratic state regulators with prescribed eigenvalues and specified relative stability

This paper considers the problem of optimal regulator design of linear multivariable systems with prescribed pole locations and/or poles corresponding to specified relative stability. A sequential method based on the frequency-domain optimality condition is proposed for achieving the desired pole assignment and determination of the corresponding quadratic performance index. This design method enables the retention of some stable open-loop poles and the associated eigenvectors in the closed-loop system. An illustrative example is provided to demonstrate the effectiveness of the proposed method.

Ganesan, Sekar↗

NASA NTRS1982

Sampling errors in the estimation of empirical orthogonal functions

Empirical Orthogonal Functions (EOF's), eigenvectors of the spatial cross-covariance matrix of a meteorological field, are reviewed with special attention given to the necessary weighting factors for gridded data and the sampling errors incurred when too small a sample is available. The geographical shape of an EOF shows large intersample variability when its associated eigenvalue is 'close' to a neighboring one. A rule of thumb indicating when an EOF is likely to be subject to large sampling fluctuations is presented. An explicit example, based on the statistics of the 500 mb geopotential height field, displays large intersample variability in the EOF's for sample sizes of a few hundred independent realizations, a size seldom exceeded by meteorological data sets.

North, G. R.↗

NASA NTRS1984

Matrix partitioning and EOF/principal component analysis of Antarctic Sea ice brightness temperatures

A field of measured anomalies of some physical variable relative to their time averages, is partitioned in either the space domain or the time domain. Eigenvectors and corresponding principal components of the smaller dimensioned covariance matrices associated with the partitioned data sets are calculated independently, then joined to approximate the eigenstructure of the larger covariance matrix associated with the unpartitioned data set. The accuracy of the approximation (fraction of the total variance in the field) and the magnitudes of the largest eigenvalues from the partitioned covariance matrices together determine the number of local EOF's and principal components to be joined by any particular level. The space-time distribution of Nimbus-5 ESMR sea ice measurement is analyzed.

Murray, C. W., Jr.↗

NASA NTRS1979

Principal component analysis of solar flares in the soft X-ray flux

The paper considers principal component analysis of solar flares in the soft X-ray flux, a technique for extracting the salient features from a mass of data. The method applies particularly to the analysis of nonstationary ensembles, and its computations require the evaluation of eigenvalues of matrices. The Eispack matrix eigen system routines were used to analyze full-disk proportional-counter data from the X-ray event analyzer which was part of the Skylab experiment. Empirical orthogonal functions were derived for events in the soft X-ray spectrum between 2.5 and 20 A during different time periods, indicating that about 90% of the cumulative power of each analyzed flare is contained in the largest eigenvector. The first two largest eigenvectors are sufficient for an empirical curve fit through the raw data and a characterization of solar flares in the soft X-ray flux, and power spectra of two largest eigenvectors reveal a reported periodicity of about 5 min.

Teuber, D. L.↗

NASA NTRS1984

Design Spectrum Analysis in NASTRAN

The utility of Design Spectrum Analysis is to give a mode by mode characterization of the behavior of a design under a given loading. The theory of design spectrum is discussed after operations are explained. User instructions are taken up here in three parts: Transient Preface, Maximum Envelope Spectrum, and RMS Average Spectrum followed by a Summary Table. A single DMAP ALTER packet will provide for all parts of the design spectrum operations. The starting point for getting a modal break-down of the response to acceleration loading is the Modal Transient rigid format. After eigenvalue extraction, modal vectors need to be isolated in the full set of physical coordinates (P-sized as opposed to the D-sized vectors in RF 12). After integration for transient response the results are scanned over the solution time interval for the peak values and for the times that they occur. A module called SCAN was written to do this job, that organizes these maxima into a diagonal output matrix. The maximum amplifier in each mode is applied to the eigenvector of each mode which then reveals the maximum displacements, stresses, forces and boundary reactions that the structure will experience for a load history, mode by mode. The standard NASTRAN output processors have been modified for this task. It is required that modes be normalized to mass.

Butler, T. G.↗

NASA NTRS1984

Eigenspace design of an active flutter suppression system

An active control system is designed for the suppression of wing flutter in a flight test vehicle. Eigenvalue placement is used to synthesize a full state controller which satisfies performance specifications on control surface activity and which exhibits excellent gain and phase margins. The use of limited state feedback is examined; however, it is found that a simple frequency response matching technique can be used to design a realizable compensator which reproduces the feedback properties of the full state controller. The performance of the control system using this compensator is evaluated at various flight conditions and found to be satisfactory. In addition eigenvector shaping is used to enhance the gust load alleviation capabilities of the flutter control system.

Liebst, B. S.↗

DOE OSTI2024

Demonstration of the rodeo algorithm on a quantum computer

The rodeo algorithm is an efficient algorithm for eigenstate preparation and eigenvalue estimation for any observable on a quantum computer. This makes it a promising tool for studying the spectrum and structure of atomic nuclei as well as other fields of quantum many-body physics. The only requirement is that the initial state has sufficient overlap probability with the desired eigenstate. While it is exponentially faster than well-known algorithms such as phase estimation and adiabatic evolution for eigenstate preparation, it has yet to be implemented on an actual quantum device. In this work, we apply the rodeo algorithm to determine the energy levels of a random one-qubit Hamiltonian, resulting in a relative error of 0.08% using mid-circuit measurements on the IBM Q device Casablanca. This surpasses the accuracy of directly-prepared eigenvector expectation values using the same quantum device. We take advantage of the high-accuracy energy determination and use the Hellmann-Feynman theorem to compute eigenvector expectation values for a different random one-qubit observable. For the Hellmann-Feynman calculations, we find a relative error of 0.7%. Here, we conclude by discussing possible future applications of the rodeo algorithm for multi-qubit Hamiltonians.

algorithm↗

NASA NTRS1985

Optimization of cascade blade mistuning. II - Global optimum and numerical optimization

The values of the mistuning which yield the most stable eigenvectors are analytically determined, using the simplified equations of motion which were developed in Part I of this work. It is shown that random mistunings, if large enough, may lead to the maximal stability, whereas the alternate mistunings cannot. The problem of obtaining maximum stability for minimal mistuning is formulated, based on numerical optimization techniques. Several local minima are obtained using different starting mistuning vectors. The starting vectors which lead to the global minimum are identified. It is analytically shown that all minima appear in multiplicities which are equal to the number of compressor blades. The effect of mistuning on the flutter speed is studied using both an optimum mistuning vector and an alternate mistuning vector. Effects of mistunings in elastic axis locations are shown to have a negligible effect on the eigenvalues. Finally, it is shown that any general two-dimensional bending-torsion system can be reduced to an equivalent uncoupled torsional system.

Nissim, E.↗

DOE OSTI2026

Optimal Transfer Operators in Algebraic Two-Level Methods for Nonsymmetric and Indefinite Problems

Consider an algebraic two-level method applied to the 𝑛-dimensional linear system 𝐴⁢𝒙 = 𝒃 using fine-space preconditioner (i.e., “relaxation” or “smoother”) 𝑀, with 𝑀 ≈ 𝐴, restriction and interpolation 𝑅 and 𝑃, and algebraic coarse-space operator 𝐴 𝑐 : = 𝑅 ∗ ⁢𝐴⁢𝑃. Then, what are the best possible transfer operators 𝑅 and 𝑃 of a given dimension 𝑛 𝑐 < 𝑛? Brannick et al. [12] showed that when 𝐴 and 𝑀 are Hermitian positive definite (HPD), the optimal interpolation is such that its range contains the 𝑛 𝑐 smallest generalized eigenvectors of the matrix pencil (𝐴, 𝑀). Recently, in Ali et al. [5] we generalized this framework to the non-HPD setting, by considering both right (interpolation) and left (restriction) generalized eigenvectors of (𝐴, 𝑀) and defining corresponding nonsymmetric transfer operators {𝑅#, 𝑃#}. Tight convergence bounds for {𝑅#, 𝑃#} are derived in spectral radius, as well as a proof of pseudo-optimality. Note, {𝑅#, 𝑃#} are typically complex valued, which is not practical for real-valued problems. Here, in this work, we build on [5], first characterizing all inner products in which the coarse-space correction defined by {𝑅#, 𝑃#} is orthogonal. We then develop tight two-level convergence bounds in these norms, and prove that the underlying transfer operators {𝑅#, 𝑃#} are genuinely optimal. As a special case, our theory both recovers and extends the HPD results from [12]. Finally, we show how to construct optimal, real-valued transfer operators in the case of that 𝐴 and 𝑀 are real valued, but are not HPD. Numerical examples arising from a discretized advection-reaction equation, wave-equation, and Stokes equations are used to verify and illustrate the theory.

97 MATHEMATICS AND COMPUTING↗