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NASA NTRS1985

Application of singular value decomposition to structural dynamics systems with constraints

Singular value decomposition is used to construct a coordinate transformation for a linear dynamic system subject to linear, homogeneous constraint equations. The method is compared with two commonly used methods, namely classical Gaussian elimination and Walton-Steeves approach. Although the classical method requires fewer numerical operations, the singular value decomposition method is more accurate and convenient in eliminating the dependent coordinates. Numerical examples are presented to demonstrate the application of the method.

Juang, J.-N.↗

NASA NTRS1984

Singular value decomposition with systolic arrays

Systolic arrays for determining the singular value decomposition of a mxn, m n, matrix A of bandwidth w are presented. After A has been reduced to bidiagonal form B by means of Givens plane rotations, the singular values of B are computed by the Golub-Reinsch iteration. The products of plane rotations form the matrices of left and right singular vectors. Assuming each processor can compute or supply a plane rotation, O(wn) processors accomplish the reduction to bidiagonal form in O(np) steps, where p is the number of superdiagonals. A constant number of processors then determines each singular value in about 6n steps. The singular vectors are computed by rerouting the rotations through the arrays used for the reduction to bidiagonal form, or else along the way by employing another rectangular array of O(wm) processors.

Ipsen, I. C. F.↗

NASA NTRS1984

Singular value decomposition with systolic arrays

Systolic arrays for determining the singular value decomposition of a mxn, m greater than or equal to n, matrix A of bandwidth w are presented. After A has been reduced to bidiagonal form B by means of Givens plane rotations, the singular values of B are computed by the Golub-Reinsch iteration. The products of plane rotations form the matrices of left and right singular vectors. Assuming each processor can compute or supply a plane rotation, O(wn) processors accomplish the reduction to bidiagonal form in O(np) steps, where p is the number of superdiagonals. A constant number of processors then determines each singular value in about 6n steps. The singular vectors are computed by rerouting the rotations through the arrays used for the reduction to bidiagonal form, or else along the way by employing another rectangular array of O(wm) processors.

Ipsen, I.↗

NASA NTRS2018

A Singular Value Decomposition Framework for Retrievals with Vertical Distribution Information from Greenhouse Gas Column Absorption Spectroscopy Measurements

We review the singular value decomposition (SVD) framework and use it for quantifying and discerning vertical information in greenhouse gas retrievals from column integrated absorption measurements. While the commonly used traditional Bayesian optimal estimation (OE) assumes a prior distribution in order to regularize the inversion problem, the SVD approach identifies principal components that can be retrieved from the measurement without explicitly specifying a prior mean and prior covariance matrix. We review the SVD method, explicitly recognize the use of an uninformative prior and show it to incur no bias from the choice of the prior. We also make the connection between the SVD method and the pseudo-inverse, which makes it more intuitive and easy to understand. We illustrate the use of the SVD method on an integrated path differential absorption CO2 lidar measurement model and verify our derivations and bias-free properties versus optimal estimation using numerical simulations. In contrast, traditional OE retrievals exhibit bias when the prior mean used in the retrieval differs from the true mean. Hence, the SVD method is particularly useful for situations in which knowledge of the prior mean and prior covariance of the true state (e.g., greenhouse gas profiles) is inadequate.

Ramanathan, Anand K.↗

NASA NTRS2005

A Survey of Singular Value Decomposition Methods and Performance Comparison of Some Available Serial Codes

This contractor report describes a performance comparison of available alternative complete Singular Value Decomposition (SVD) methods and implementations which are suitable for incorporation into point spread function deconvolution algorithms. The report also presents a survey of alternative algorithms, including partial SVD's special case SVD's, and others developed for concurrent processing systems.

Plassman, Gerald E.↗

NASA NTRS1988

Force prediction using singular-value decomposition

Results of a preliminary investigation into a method which can be used to improve the results of an experimental force prediction analysis is presented. The method uses a singular value decomposition technique to reduce measurement noise present in the measured input data matrix. The fundamental theory for the method along with an experimental example is shown. The method is shown to provide encouraging force prediction improvements when the input data matrix contains high levels of measurement noise.

Elliott, K. B.↗

NASA NTRS2018

A Singular Value Decomposition Framework for Retrievals with Vertical Distribution Information from Greenhouse Gas Column Absorption Spectroscopy Measurements

We describe a variation of the Optimal Estimation (OE) method for greenhouse gas remote sensing retrievals using a singular value decomposition (SVD) and an uninformative prior. The SVD method is capable of discerning vertical information in column integrated absorption measurements. While traditional Bayesian optimal estimation (OE) assumes a prior distribution in order to regularize the inversion problem, the SVD approach identifies principal components that can be retrieved from the measurement without explicitly specifying a prior mean and prior covariance matrix. We discuss the method, illustrate its use on an integrated path differential absorption CO2 lidar measurement model, and compare it to traditional optimal estimation using numerical simulations. In the absence of forward model error, the mathematics behind the SVD method guarantee it to be bias-free, which is confirmed by the numerical simulations. In contrast, traditional OE retrievals exhibit bias when the prior mean used in the retrieval differs from the true mean. While the SVD approach can be used for most trace gas retrievals, it is particularly useful for situations where prior knowledge of the trace gas profile is poor. The SVD analysis is also useful in illustrating how vertical information is treated by the traditional OE approach.

Ramanathan, Anand K.↗

NASA NTRS1989

Inversion of imaging spectrometry data using singular value decomposition

The use of imaging spectrometers, which acquire data that are both spectrally contiguous images and spatially contiguous spectra, for quantitative remote sensing of the earth is addressed. Such data sets cannot be analyzed fully using either existing spectroscopic or image techniques. Singular value decomposition (SVD) is used here for spectral unmixing and determination of the spatial scales of mixing. It is shown that when it is used to invert the mixing endmember library, SVD allows more insight into library characteristics and more control of the inversion process than other commonly used matrix inversion techniques.

Boardman, Joe W.↗

NASA NTRS1989

Eigenvector derivatives of repeated eigenvalues using singular value decomposition

An explicit formula is obtained for the first-order eigenvector derivative that corresponds to the eigenvector of a repeated eigenvalue, in the case of the nonself-adjoint eigenvalue problem. This method applies to the class of nondefective problems whose first eigenvalue derivatives of the repeated eigenvalues are distinct. A singular-value decomposition approach is used to compute four requisite bases for eigenspaces, as well as to keep track of the dimensions of state variables and the conditioning of the state equations.

Lim, Kyong B.↗

NASA NTRS1988

Efficient load measurements using singular value decomposition

Various basic research was performed on efficient load measurement estimation techniques for aircraft structure analysis. An overview is presented of the load measurement problem. Two basic equivalent approaches to load measurement evaluations were considered. Under approach 1, the load values are modeled as depending linearly on the measured values. Under approach 2, the measured values depend linearly on the load values. By using the modern Singular Value Decomposition method, it was shown that under all conditions of the number of loads and number of gages, approach 1 is equivalent to approach 2. By using the conventional normal equation (linear regression) approach, approach 1 is only valid when the number of loads is equal to or greater than the number of gages, while approach 2 is the reverse. Furthermore, except for the case of the number of loads equals the number of gages, the load prediction formulas under the two approaches are not equivalent.

Yao, Kung↗

NASA NTRS1989

A multi-point force appropriation method based upon a singular value decomposition approach

Force appropriation methods aim to provide an estimate for the force pattern required to excite a 'pure' normal mode in a vibration test where multiple exciters are available. In this paper, a new method based upon a Singular Value Decompositon approach is presented. The appropriated force vector which minimizes the in-plane response components while maximizing the required quadrature components is found, subject to a fixed force norm. The method can accommodate more transducers than exciters and also more exciters than effective degrees of freedom. The method is demonstrated upon a six degree of freedom theoretical model and it is shown that an estimate for the effective degrees of freedom can be obtained.

Juang, Jer-Nan↗

NASA NTRS1992

Regularizing the satellite temperature-retrieval problem through singular-value decomposition of the radiative transfer physics

A new technique is derived for retrieving atmospheric temperature profiles from satellite-measured spectral radiance that appears, in first tests, to effectively circumvent certain difficulties of other well-known and implemented techniques. This method provides an effective manner to avoid numerical instability without having to force the algorithm toward adherence to a priori statistics not dependent on the radiance measurements. This provides the opportunity to obtain more information from satellite sounding instruments, without encountering instabilities owing to overlapping weighting functions.

Thompson, Owen E.↗

NASA NTRS2003

Evaluation of a Singular Value Decomposition Approach for Impact Dynamic Data Correlation

Impact dynamic tests are used in the automobile and aircraft industries to assess survivability of occupants during crash, to assert adequacy of the design, and to gain federal certification. Although there is no substitute for experimental tests, analytical models are often developed and used to study alternate test conditions, to conduct trade-off studies, and to improve designs. To validate results from analytical predictions, test and analysis results must be compared to determine the model adequacy. The mathematical approach evaluated in this paper decomposes observed time responses into dominant deformation shapes and their corresponding contribution to the measured response. To correlate results, orthogonality of test and analysis shapes is used as a criterion. Data from an impact test of a composite fuselage is used and compared to finite element predictions. In this example, the impact response was decomposed into multiple shapes but only two dominant shapes explained over 85% of the measured response

Horta, Lucas G.↗

NASA NTRS1991

Applications of singular value analysis and partial-step algorithm for nonlinear orbit determination

An adaptive method in which cruise and nonlinear orbit determination problems can be solved using a single program is presented. It involves singular value decomposition augmented with an extended partial step algorithm. The extended partial step algorithm constrains the size of the correction to the spacecraft state and other solve-for parameters. The correction is controlled by an a priori covariance and a user-supplied bounds parameter. The extended partial step method is an extension of the update portion of the singular value decomposition algorithm. It thus preserves the numerical stability of the singular value decomposition method, while extending the region over which it converges. In linear cases, this method reduces to the singular value decomposition algorithm with the full rank solution. Two examples are presented to illustrate the method's utility.

Ryne, Mark S.↗

NASA NTRS2018

Spreading Sequence System for Full Connectivity Relay Network

Fully connected uplink and downlink fully connected relay network systems using pseudo-noise spreading and despreading sequences subjected to maximizing the signal-to-interference-plus-noise ratio. The relay network systems comprise one or more transmitting units, relays, and receiving units connected via a communication network. The transmitting units, relays, and receiving units each may include a computer for performing the methods and steps described herein and transceivers for transmitting and/or receiving signals. The computer encodes and/or decodes communication signals via optimum adaptive PN sequences found by employing Cholesky decompositions and singular value decompositions (SVD). The PN sequences employ channel state information (CSI) to more effectively and more securely computing the optimal sequences.

Kwon, Hyuck M.↗

NASA NTRS1990

Input/output system identification - Learning from repeated experiments

The paper describes three approaches and possible variations for the determination of the Markov parameters for forced response data using general inputs. It is shown that, when the parameters in the solution procedure are bootstrapped, the results can be obtained very efficiently, but the errors propagate throughout all parameters. By arranging the data in a different form and using singular value decomposition, the resulting identified parameters are more accurate, in the least number of successive experiments, at the expense of a large matrix singular value decomposition. When a recursive procedure is employed, the calculations can be performed very efficiently, but the number of repetitions of the experiments is much greater for a given accuracy than for any of the previous approaches. An alternative formulation is proposed to combine the advantages of each of the approaches.

Juang, Jer-Nan↗

NASA NTRS1993

A new technique for deconvolution of data from instruments that make integral measurements, e.g. RIMS on DE-1

A general method for deconvolving an unknown function from integral measurements is described and applied to data from the instrument Retarding Ion Mass Spectrometer (RIMS) aboard the spacecraft Dynamics Explorer 1 (DE-1). The principal features of the method are: (1) it uses objective criteria based upon fundamental statistical principles, i.e. Bayesian statistics; (2) it provides for insertion of prior knowledge in a non-prejudicial, explicit manner through the choice of breakpoints that determine the bicubic spline expansion functions; (3) it prevents random fluctuations from controlling the fit to the data through the use of singular value decomposition and the elimination of small singular values; and (4) it guards agianst the introduction of spurious features into the result by including a penalty function and using the principle of generalized cross validation. Illustrative examples from RIMS data for H(+) and O(+) show that the method provides enhanced accuracy and detail in deconvolving the ion phase space density.

Perez, J. D.↗

NASA NTRS1996

Design of Multi-Parameter Steerable Functions Using Cascade Basis Reduction

A new cascade basis reduction method of computing the optimal least-squares set of basis functions steering a given function is presented. The method combines the Lie group-theoretic and the singular value decomposition approaches in such a way that their respective strengths complement each other. Since the Lie group-theoretic approach is used, the set of basis and steering functions computed can be expressed analytically. Because the singular value decomposition method is used, this set of basis and steering functions is optimal in the least-squares sense. Furthermore, the computational complexity in designing basis functions for transformation groups with large numbers of parameters is significantly reduced. The efficiency of the cascade basis reduction method is demonstrated by designing a set of basis functions that steers a Gabor function under the four-parameter linear transformation group.

Teo, P.↗