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Results for “Nonlinear least-squares problem”

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NASA NTRS1992

A quadratic-tensor model algorithm for nonlinear least-squares problems with linear constraints

A new algorithm for solving nonlinear least-squares and nonlinear equation problems is proposed which is based on approximating the nonlinear functions using the quadratic-tensor model by Schnabel and Frank. The algorithm uses a trust region defined by a box containing the current values of the unknowns. The algorithm is found to be effective for problems with linear constraints and dense Jacobian matrices.

Hanson, R. J.↗

NASA NTRS1983

A new algorithm for constrained nonlinear least-squares problems, part 1

A Gauss-Newton algorithm is presented for solving nonlinear least squares problems. The problem statement may include simple bounds or more general constraints on the unknowns. The algorithm uses a trust region that allows the objective function to increase with logic for retreating to best values. The computations for the linear problem are done using a least squares system solver that allows for simple bounds and linear constraints. The trust region limits are defined by a box around the current point. In its current form the algorithm is effective only for problems with small residuals, linear constraints and dense Jacobian matrices. Results on a set of test problems are encouraging.

Hanson, R. J.↗

NASA NTRS1990

Photometric theory for wide-angle phenomena

An examination is made of the problem posed by wide-angle photographic photometry, in order to extract a photometric-morphological history of Comet P/Halley. Photometric solutions are presently achieved over wide angles through a generalization of an assumption-free moment-sum method. Standard stars in the field allow a complete solution to be obtained for extinction, sky brightness, and the characteristic curve. After formulating Newton's method for the solution of the general nonlinear least-square problem, an implementation is undertaken for a canonical data set. Attention is given to the problem of random and systematic photometric errors.

Usher, Peter D.↗

NASA NTRS1991

Least-squares/parabolized Navier-Stokes procedure for optimizing hypersonic wind tunnel nozzles

A new procedure is demonstrated for optimizing hypersonic wind-tunnel-nozzle contours. The procedure couples a CFD computer code to an optimization algorithm, and is applied to both conical and contoured hypersonic nozzles for the purpose of determining an optimal set of parameters to describe the surface geometry. A design-objective function is specified based on the deviation from the desired test-section flow-field conditions. The objective function is minimized by optimizing the parameters used to describe the nozzle contour based on the solution to a nonlinear least-squares problem. The effect of the changes in the nozzle wall parameters are evaluated by computing the nozzle flow using the parabolized Navier-Stokes equations. The advantage of the new procedure is that it directly takes into account the displacement effect of the boundary layer on the wall contour. The new procedure provides a method for optimizing hypersonic nozzles of high Mach numbers which have been designed by classical procedures, but are shown to produce poor flow quality due to the large boundary layers present in the test section. The procedure is demonstrated by finding the optimum design parameters for a Mach 10 conical nozzle and a Mach 6 and a Mach 15 contoured nozzle.

Korte, John J.↗

NASA NTRS1992

CAN-DO, CFD-based Aerodynamic Nozzle Design and Optimization program for supersonic/hypersonic wind tunnels

A design program is developed which incorporates a modern approach to the design of supersonic/hypersonic wind-tunnel nozzles. The approach is obtained by the coupling of computational fluid dynamics (CFD) with design optimization. The program can be used to design a 2D or axisymmetric, supersonic or hypersonic, wind-tunnel nozzles that can be modeled with a calorically perfect gas. The nozzle design is obtained by solving a nonlinear least-squares optimization problem (LSOP). The LSOP is solved using an iterative procedure which requires intermediate flowfield solutions. The nozzle flowfield is simulated by solving the Navier-Stokes equations for the subsonic and transonic flow regions and the parabolized Navier-Stokes equations for the supersonic flow regions. The advantages of this method are that the design is based on the solution of the viscous equations eliminating the need to make separate corrections to a design contour, and the flexibility of applying the procedure to different types of nozzle design problems.

Korte, John J.↗

NASA NTRS1992

Reconstruction of internal density distributions in porous bodies from laser ultrasonic data

It is presently shown that, for density-reconstruction problems in which information about the inhomogeneity is known a priori, the nonlinear least-squares algorithm yields satisfactory results on the basis of limited projection data. The back-projection algorithm, which obviates assumptions about the objective function to be reconstructed, does not recover the boundary of the inhomogeneity when the number of projections is limited and ray-bending is ignored.

Lu, Yichi↗

NASA NTRS1990

Constrained optimization using design of experiment surfaces

An algorithm for solving constrained optimization problems is presented. First, design of experiment techniques are used to survey the design space. After evaluating the objective and constraint functions, as specified by Taguchi orthogonal arrays, analytical models of these functions are generated using a least-squares regression analysis. Next, a nonlinear programming package is used to optimize the analytical model. Based on the optimization information, the design space is reduced so as to close in around the minimum, and the entire procedure is repeated until convergence. An important feature of the algorithm is that function gradients are not required; therefore, for problems in which gradients would have to be estimated using finite-differences the number of function evaluations required for the optimization is significantly reduced, when compared with traditional nonlinear programming techniques. In addition, there is no requirement that the gradients must be smooth and continuous.

Bolt, Marvin Vance↗

NASA NTRS1984

Absolute intensities of spectral lines in carbon dioxide bands near 2050/cm

The absolute intensities of individual lines of eleven relatively weak bands of (C-12)(O-16)2, (C-13)(O-16)2, (O-16)(C-12)(O-18), and (O-16)(C-12)(O-17) near 2050/cm were measured and recorded at room temperature and 0.01/cm resolution with the Fourier transform spectrometer located in the McMath solar telescope complex located at the National Solar Observatory on Kitt Peak. Each spectrum was obtained with an integration time of one hour, the temperature and pressure were monitored during this time, and the signal-to-rms noise in the 5 micron region was approximately 4000 for all scans. The intensities were determined from analysis of 1-2/cm segments of spectra using nonlinear least-squares spectral curve fitting, and analyzed to determine the vibrational band intensity and F-factor coefficients for each of the bands. The results are pertinent to a wide variety of problems in atmospheric physics and for the analysis of remote-sensing data.

Rinsland, C. P.↗

NASA NTRS1982

Least-squares sequential parameter and state estimation for large space structures

This paper presents the formulation of simultaneous state and parameter estimation problems for flexible structures in terms of least-squares minimization problems. The approach combines an on-line order determination algorithm, with least-squares algorithms for finding estimates of modal approximation functions, modal amplitudes, and modal parameters. The approach combines previous results on separable nonlinear least squares estimation with a regression analysis formulation of the state estimation problem. The technique makes use of sequential Householder transformations. This allows for sequential accumulation of matrices required during the identification process. The technique is used to identify the modal prameters of a flexible beam.

Thau, F. E.↗

NASA NTRS1987

Nonlinear programming extensions to rational function approximations of unsteady aerodynamics

This paper deals with approximating unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these approximations are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is of lower order than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.

Tiffany, Sherwood H.↗

NASA NTRS1994

Measurement uncertainty and feasibility study of a flush airdata system for a hypersonic flight experiment

Presented is a feasibility and error analysis for a hypersonic flush airdata system on a hypersonic flight experiment (HYFLITE). HYFLITE heating loads make intrusive airdata measurement impractical. Although this analysis is specifically for the HYFLITE vehicle and trajectory, the problems analyzed are generally applicable to hypersonic vehicles. A layout of the flush-port matrix is shown. Surface pressures are related airdata parameters using a simple aerodynamic model. The model is linearized using small perturbations and inverted using nonlinear least-squares. Effects of various error sources on the overall uncertainty are evaluated using an error simulation. Error sources modeled include boundarylayer/viscous interactions, pneumatic lag, thermal transpiration in the sensor pressure tubing, misalignment in the matrix layout, thermal warping of the vehicle nose, sampling resolution, and transducer error. Using simulated pressure data for input to the estimation algorithm, effects caused by various error sources are analyzed by comparing estimator outputs with the original trajectory. To obtain ensemble averages the simulation is run repeatedly and output statistics are compiled. Output errors resulting from the various error sources are presented as a function of Mach number. Final uncertainties with all modeled error sources included are presented as a function of Mach number.

Whitmore, Stephen A.↗

NASA NTRS1973

An application of a linear programing technique to nonlinear minimax problems

A differential correction technique for solving nonlinear minimax problems is presented. The basis of the technique is a linear programing algorithm which solves the linear minimax problem. By linearizing the original nonlinear equations about a nominal solution, both nonlinear approximation and estimation problems using the minimax norm may be solved iteratively. Some consideration is also given to improving convergence and to the treatment of problems with more than one measured quantity. A sample problem is treated with this technique and with the least-squares differential correction method to illustrate the properties of the minimax solution. The results indicate that for the sample approximation problem, the minimax technique provides better estimates than the least-squares method if a sufficient amount of data is used. For the sample estimation problem, the minimax estimates are better if the mathematical model is incomplete.

Schiess, J. R.↗

NASA NTRS1991

A new finite element formulation for computational fluid dynamics. X - The compressible Euler and Navier-Stokes equations

A space-time element method is presented for solving the compressible Euler and Navier-Stokes equations. The proposed formulation includes the variational equation, predictor multi-corrector algorithms and boundary conditions. The variational equation is based on the time-discontinuous Galerkin method, in which the physical entropy variables are employed. A least-squares operator and a discontinuity-capturing operator are added, resulting in a high-order accurate and unconditionally stable method. Implicit/explicit predictor multi-corrector algorithms, applicable to steady as well as unsteady problems, are presented; techniques are developed to enhance their efficiency. Implementation of boundary conditions is addressed; in particular, a technique is introduced to satisfy nonlinear essential boundary conditions, and a consistent method is presented to calculate boundary fluxes. Numerical results are presented to demonstrate the performance of the method.

Shakib, Farzin↗

NASA NTRS1992

Frequency domain state-space system identification

An algorithm for identifying state-space models from frequency response data of linear systems is presented. A matrix-fraction description of the transfer function is employed to curve-fit the frequency response data, using the least-squares method. The parameters of the matrix-fraction representation are then used to construct the Markov parameters of the system. Finally, state-space models are obtained through the Eigensystem Realization Algorithm using Markov parameters. The main advantage of this approach is that the curve-fitting and the Markov parameter construction are linear problems which avoid the difficulties of nonlinear optimization of other approaches. Another advantage is that it avoids windowing distortions associated with other frequency domain methods.

Chen, Chung-Wen↗

NASA NTRS1981

Model error estimation for large flexible spacecraft

The basic model obtained for spacecraft control system design applications is used to develop the control and estimation algorithms which constitute integral elements of the control system. The performance of this system is limited on account of modeling errors. The presence of such errors is inevitable in connection with truncated dynamics, parameter uncertainties, neglected nonlinearities, and external disturbances. In many cases, an approximate knowledge of the model errors can only be established by means of an estimation procedure. The present investigation has the objective to outline an approach to estimate errors inherent in a reduced-order model and to illustrate its application to modeling problems arising in spacecraft control system design. The considered procedure makes use of the principle of least-squares. In the discussed application this principle is employed to minimize a quadratic functional of the model errors.

Rodriguez, G.↗

NASA NTRS1988

GaussFit - A system for least squares and robust estimation

GaussFit is a new computer program for solving least-squares and robust estimation problems. It has a number of unique features, including a complete programming language designed especially to formulate estimation problems, a built-in compiler and interpreter to support the programming language, and a built-in algebraic manipulator for calculating the required partial derivatives analytically. These features make GaussFit very easy to use, so that even complex problems can be set up and solved with minimal effort. GaussFit can correctly handle many cases of practical interest: nonlinear models, exact constraints, correlated observations, and models where the equations of condition contain more than one observed quantity. An experimental robust estimation capability is built into GaussFit so that data sets contaminated by outliers can be handled simply and efficiently.

Jefferys, W. H.↗

NASA NTRS1995

Estimating errors in least-squares fitting

While least-squares fitting procedures are commonly used in data analysis and are extensively discussed in the literature devoted to this subject, the proper assessment of errors resulting from such fits has received relatively little attention. The present work considers statistical errors in the fitted parameters, as well as in the values of the fitted function itself, resulting from random errors in the data. Expressions are derived for the standard error of the fit, as a function of the independent variable, for the general nonlinear and linear fitting problems. Additionally, closed-form expressions are derived for some examples commonly encountered in the scientific and engineering fields, namely ordinary polynomial and Gaussian fitting functions. These results have direct application to the assessment of the antenna gain and system temperature characteristics, in addition to a broad range of problems in data analysis. The effects of the nature of the data and the choice of fitting function on the ability to accurately model the system under study are discussed, and some general rules are deduced to assist workers intent on maximizing the amount of information obtained form a given set of measurements.

Richter, P. H.↗

NASA NTRS1972

Numerical method for the solution of large systems of differential equations of the boundary layer type

A numerical method for the solution of large systems of nonlinear differential equations of the boundary-layer type is described. The method is a modification of the technique for satisfying asymptotic boundary conditions. The present method employs inverse interpolation instead of the Newton method to adjust the initial conditions of the related initial-value problem. This eliminates the so-called perturbation equations. The elimination of the perturbation equations not only reduces the user's preliminary work in the application of the method, but also reduces the number of time-consuming initial-value problems to be numerically solved at each iteration. For further ease of application, the solution of the overdetermined system for the unknown initial conditions is obtained automatically by applying Golub's linear least-squares algorithm. The relative ease of application of the proposed numerical method increases directly as the order of the differential-equation system increases. Hence, the method is especially attractive for the solution of large-order systems. After the method is described, it is applied to a fifth-order problem from boundary-layer theory.

Green, M. J.↗