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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1984

Efficient matrix partitioning for optical computing

Techniques for partitioning optical linear algebra problems to make them amenable to solution using optical processors programmed with simple algorithms are explored. Generalized methods for splitting a linear algebra matrix into a series of submatrices are reviewed, showing that simple forms can be pipelined smoothly and that parallel accumulation can be achieved by beam combining on detectors or by summing electronically. The techniques offer simplified bookkeeping, algorithmic independence, and high efficiency. The computational speed will depend on the number of multiplier-accumulators devoted to the task.

Caulfield, H. J.↗
NASA NTRS1978

ORACLS: A system for linear-quadratic-Gaussian control law design

A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.

Armstrong, E. S.↗
NASA NTRS1996

A Note on Multigrid Theory for Non-nested Grids and/or Quadrature

We provide a unified theory for multilevel and multigrid methods when the usual assumptions are not present. For example, we do not assume that the solution spaces or the grids are nested. Further, we do not assume that there is an algebraic relationship between the linear algebra problems on different levels. What we provide is a computationally useful theory for adaptively changing levels. Theory is provided for multilevel correction schemes, nested iteration schemes, and one way (i.e., coarse to fine grid with no correction iterations) schemes. We include examples showing the applicability of this theory: finite element examples using quadrature in the matrix assembly and finite volume examples with non-nested grids. Our theory applies directly to other discretizations as well.

Douglas, C. C.↗
NASA NTRS2023

Another Geometric Interpretation of Cramer’s Rule

We develop a geometric interpretation of Cramer’s rule as a generalization of projection onto orthogonal basis vectors using the rows of the adjugate. This interpretation makes connections between elementary linear algebra concepts like the solution to linear equations, inner products, and projections. Such connections are useful for introducing broader concepts related to Hilbert spaces and geometric algebras like Grassman algebra. Such connections were essential for the author’s mathematical education as an engineer.

linear algebra↗
NASA NTRS1998

Analysis of Information Content in High-Spectral Resolution Sounders using Subset Selection Analysis

In this paper, we summarize the results of the sensitivity analysis and data reduction carried out to determine the information content of AIRS and IASI channels. The analysis and data reduction was based on the use of subset selection techniques developed in the linear algebra and statistical community to study linear dependencies in high dimensional data sets. We applied the subset selection method to study dependency among channels by studying the dependency among their weighting functions. Also, we applied the technique to study the information provided by the different levels in which the atmosphere is discretized for retrievals and analysis. Results from the method correlate well with intuition in many respects and point out to possible modifications for band selection in sensor design and number and location of levels in the analysis process.

Velez-Reyes, Miguel↗
NASA NTRS1999

A Textbook for a First Course in Computational Fluid Dynamics

This paper describes and discusses the textbook, Fundamentals of Computational Fluid Dynamics by Lomax, Pulliam, and Zingg, which is intended for a graduate level first course in computational fluid dynamics. This textbook emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. Its underlying philosophy is that the theory of linear algebra and the attendant eigenanalysis of linear systems provides a mathematical framework to describe and unify most numerical methods in common use in the field of fluid dynamics. Two linear model equations, the linear convection and diffusion equations, are used to illustrate concepts throughout. Emphasis is on the semi-discrete approach, in which the governing partial differential equations (PDE's) are reduced to systems of ordinary differential equations (ODE's) through a discretization of the spatial derivatives. The ordinary differential equations are then reduced to ordinary difference equations (O(Delta)E's) using a time-marching method. This methodology, using the progression from PDE through ODE's to O(Delta)E's, together with the use of the eigensystems of tridiagonal matrices and the theory of O(Delta)E's, gives the book its distinctiveness and provides a sound basis for a deep understanding of fundamental concepts in computational fluid dynamics.

Zingg, D. W.↗
NASA NTRS1993

Least-Squares Deconvolution of Compton Telescope Data with the Positivity Constraint

We describe a Direct Linear Algebraic Deconvolution (DLAD) approach to imaging of data from Compton gamma-ray telescopes. Imposition of the additional physical constraint, that all components of the model be non-negative, has been found to have a powerful effect in stabilizing the results, giving spatial resolution at or near the instrumental limit. A companion paper (Dixon et al. 1993) presents preliminary images of the Crab Nebula region using data from COMPTEL on the Compton Gamma-Ray Observatory.

COMPTEL↗
NASA NTRS1982

On conforming mixed finite element methods for incompressible viscous flow problems

The application of conforming mixed finite element methods to obtain approximate solutions of linearized Navier-Stokes equations is examined. Attention is given to the convergence rates of various finite element approximations of the pressure and the velocity field. The optimality of the convergence rates are addressed in terms of comparisons of the approximation convergence to a smooth solution in relation to the best approximation available for the finite element space used. Consideration is also devoted to techniques for efficient use of a Gaussian elimination algorithm to obtain a solution to a system of linear algebraic equations derived by finite element discretizations of linear partial differential equations.

Gunzburger, M. D↗
NASA NTRS2012

Description and User Instructions for the Quaternion_to_Orbit_v3 Software

For a given inertial frame of reference, the software combines the spacecraft orbits with the spacecraft attitude quaternions, and rotates the body-fixed reference frame of a particular spacecraft to the inertial reference frame. The conversion assumes that the two spacecraft are aligned with respect to the mutual line of sight, with a parameterized time tag. The software is implemented in Python and is completely open source. It is very versatile, and may be applied under various circumstances and for other related purposes. Based on the solid linear algebra analysis, it has an extra option for compensating the linear pitch. This software has been designed for simulation of the calibration maneuvers performed by the two spacecraft comprising the GRAIL mission to the Moon, but has potential use for other applications. In simulations of formation flights, one needs to coordinate the spacecraft orbits represented in an appropriate inertial reference frame and the spacecraft attitudes. The latter are usually given as the time series of quaternions rotating the body-fixed reference frame of a particular spacecraft to the inertial reference frame. It is often desirable to simulate the same maneuver for different segments of the orbit. It is also useful to study various maneuvers that could be performed at the same orbit segment. These two lines of study are more timeand labor-efficient if the attitude and orbit data are generated independently, so that the part of the data that has not been changed can be recycled in the course of multiple simulations.

Strekalov, Dmitry V.↗
NASA NTRS1990

Dynamical basis sets for algebraic variational calculations in quantum-mechanical scattering theory

New basis sets are proposed for linear algebraic variational calculations of transition amplitudes in quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained, even for basis sets with the majority of the members independent of energy.

Sun, Yan↗
NASA NTRS2007

Analytic Guidance for the First Entry in a Skip Atmospheric Entry

This paper presents an analytic method to generate a reference drag trajectory for the first entry portion of a skip atmospheric entry. The drag reference, expressed as a polynomial function of the velocity, will meet the conditions necessary to fit the requirements of the complete entry phase. The generic method proposed to generate the drag reference profile is further simplified by thinking of the drag and the velocity as density and cumulative distribution functions respectively. With this notion it will be shown that the reference drag profile can be obtained by solving a linear algebraic system of equations. The resulting drag profile is flown using the feedback linearization method of differential geometric control as guidance law with the error dynamics of a second order homogeneous equation in the form of a damped oscillator. This approach was first proposed as a revisited version of the Space Shuttle Orbiter entry guidance. However, this paper will show that it can be used to fly the first entry in a skip entry trajectory. In doing so, the gains in the error dynamics will be changed at a certain point along the trajectory to improve the tracking performance.

Garcia-Llama, Eduardo↗
NASA NTRS1991

Improved solution for ill-conditioned algebraic equations by epsilon decomposition

Matrix eigenvalue theory is presently used to examine the source of ill-conditioning in linear algebraic equations; the approach highlights the critical role played by the zero and near-zero eigenvalues and corresponding eigenvectors of poorly-conditioned systems. Insights derived from this approach are used to improve the recently developed epsilon-decomposition (E-D) solution procedure. The efficiency of E-D is significant for large matrices possessing small rank deficiency.

Ojalvo, Irving U.↗
NASA NTRS2010

Analytic Development of a Reference Profile for the First Entry in a Skip Atmospheric Entry

This note shows that a feasible reference drag profile for the first entry portion of a skip entry can be generated as a polynomial expression of the velocity. The coefficients of that polynomial are found through the resolution of a system composed of m + 1 equations, where m is the degree of the drag polynomial. It has been shown that a minimum of five equations (m = 4) are required to establish the range and the initial and final conditions on velocity and flight path angle. It has been shown that at least one constraint on the trajectory can be imposed through the addition of one extra equation in the system, which must be accompanied by the increase in the degree of the drag polynomial. In order to simplify the resolution of the system of equations, the drag was considered as being a probability density function of the velocity, with the velocity as a distribution function of the drag. Combining this notion with the introduction of empirically derived constants, it has been shown that the system of equations required to generate the drag profile can be successfully reduced to a system of linear algebraic equations. For completeness, the resulting drag profiles have been flown using the feedback linearization method of differential geometric control as a guidance law with the error dynamics of a second order homogeneous equation in the form of a damped oscillator. Satisfactory results were achieved when the gains in the error dynamics were changed at a certain point along the trajectory that is dependent on the velocity and the curvature of the drag as a function of the velocity. Future work should study the capacity to update the drag profile in flight when dispersions are introduced. Also, future studies should attempt to link the first entry, as presented and controlled in this note, with a more standard control concept for the second entry, such as the Apollo entry guidance, to try to assess the overall skip entry performance. A guidance law that includes an integral feedback term, as is the case in the actual Space Shuttle entry guidance and as is proposed in Ref 29, could be tried in future studies to assess whether its use results in an improvement of the tracking performance, and to evaluate the design needs when determining the control gains.

Garcia-Llama, Eduardo↗
NASA NTRS2010

Regression Model Term Selection for the Analysis of Strain-Gage Balance Calibration Data

The paper discusses the selection of regression model terms for the analysis of wind tunnel strain-gage balance calibration data. Different function class combinations are presented that may be used to analyze calibration data using either a non-iterative or an iterative method. The role of the intercept term in a regression model of calibration data is reviewed. In addition, useful algorithms and metrics originating from linear algebra and statistics are recommended that will help an analyst (i) to identify and avoid both linear and near-linear dependencies between regression model terms and (ii) to make sure that the selected regression model of the calibration data uses only statistically significant terms. Three different tests are suggested that may be used to objectively assess the predictive capability of the final regression model of the calibration data. These tests use both the original data points and regression model independent confirmation points. Finally, data from a simplified manual calibration of the Ames MK40 balance is used to illustrate the application of some of the metrics and tests to a realistic calibration data set.

Ulbrich, Norbert Manfred↗
NASA NTRS1989

Optical laboratory solution and error model simulation of a linear time-varying finite element equation

The use of simplified error models to accurately simulate and evaluate the performance of an optical linear-algebra processor is described. The optical architecture used to perform banded matrix-vector products is reviewed, along with a linear dynamic finite-element case study. The laboratory hardware and ac-modulation technique used are presented. The individual processor error-source models and their simulator implementation are detailed. Several significant simplifications are introduced to ease the computational requirements and complexity of the simulations. The error models are verified with a laboratory implementation of the processor, and are used to evaluate its potential performance.

Taylor, B. K.↗
NASA NTRS1980

Newton's method: A link between continuous and discrete solutions of nonlinear problems

Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions.

Thurston, G. A.↗
NASA NTRS1989

A survey on the structured singular value

The structured singular value, U, is an important linear algebra tool to study a class of matrix perturbation problems. It is useful for analyzing the robustness of stability and performance of uncertain, (nominally) linear systems. Computation of (M) is difficult, and usually, upper and lower bounds are all that can be reliably computed. Upper bounds give conservative estimates of the sizes of allowable perturbations. The maximum singular value of a matrix M is an upper bound for (M). As an upper bound, it can be improved by finding a transformations to the data (i.e. M) which do not change the structured singular value, but do reduce the maximum singular value. Typically, upper bound algorithms involve searches over sets of transformations to yield the tightest bound. Lower bound algorithms are intelligent searches for minimum-norm solutions to multivariable polynomial equations, and are based on various optimality conditions that hold at the global (and, unfortunately, some local) minima. The current methods to compute both of these types of bounds are reviewed. Theoretical justification and extensive numerical experience with the various algorithms are covered.

Packard, Andy↗