Showing papers on "Linear approximation published in 1991"

Exact Constants in Approximation Theory

Abstract: This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are based on deep facts from analysis and function theory, such as duality theory and comparison theorems; these are presented in chapters 1 and 3. In keeping with the author's intention to make the book as self-contained as possible, chapter 2 contains an introduction to polynomial and spline approximation. Chapters 4 to 7 apply the theory to specific classes of functions. The last chapter deals with n-widths and generalises some of the ideas of the earlier chapters. Each chapter concludes with commentary, exercises and extensions of results. A substantial bibliography is included. Many of the results collected here have not been gathered together in book form before, so it will be essential reading for approximation theorists.

Combining global and local approximations

Abstract: A method based on a linear approximation to a scaling factor, designated the 'global-local approximation' (GLA) method, is presented and shown capable of extending the range of usefulness of derivative-based approximations to a more refined model The GLA approach refines the conventional scaling factor by means of a linearly varying, rather than constant, scaling factor The capabilities of the method are demonstrated for a simple beam example with a crude and more refined FEM model

Linear approximation for measurement errors in phase shifting interferometry

Abstract: This paper shows how measurement errors in phase shifting interferometry (PSI) can be described to a high degree of accuracy in a linear approximation. System error sources considered here are light source instability, imperfect reference phase shifting, mechanical vibrations, nonlinearity of the detector, and quantization of the detector signal. The measurement inaccuracies resulting from these errors are calculated in linear approximation for several formulas commonly used for PSI. The results are presented in tables for easy calculation of the measurement error magnitudes for known system errors. In addition, this paper discusses the measurement error reduction which can be achieved by choosing an appropriate phase calculation formula.

Solution of porous medium type systems by linear approximation schemes

Abstract: The convergence of semi-discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type The enthalpy formulation and variational technique are used The semi-discretization used reduces the original parabolic PDE to linear elliptic PDE The algebraic correction arising from nonlinearities is treated by Newton-like iterations in finite steps Some numerical experiments are discussed and compared with the analytical solutions

Variance reduction by simultaneous multi-exponential analysis of data sets from different experiments

Abstract: The analysis of experimental data from the photocycle of bacteriorhodopsin (bR) as sums of exponentials has accumulated a large amount of information on its kinetics which is still controversial. One reason for ambiguous results can be found in the inherent instabilities connected with the fitting of noisy data by sums of exponentials. Nevertheless, there are strategies to optimize the experiments and the data analysis by a proper combination of well known techniques. This paper describes an applicable approach based on the correct weighting of the data, a separation of the linear and the non-linear parameters in the process of the least squares approximation, and a statistical analysis applying the correlation matrix, the determinant of Fisher's information matrix, and the variance of the parameters as a measure of the reliability of the results. In addition, the confidence regions for the linear approximation of the non-linear model are compared with confidence regions for the true non-linear model. Evaluation techniques and rules for an optimum experimental design are mainly exemplified by the analysis of numerically generated model data with increasing complexity. The estimation of the number of exponentials significant for the interpretation of a given set of data is demonstrated by using records from eight absorption and photocurrent experiments on the photocycle of bacteriorhodopsin.

A 30-b integrated logarithmic number system processor

Abstract: The authors describe an integrated processor that performs addition and subtraction of 30-b numbers in the logarithmic number system (LNS). This processor offers 5-MOPS performance in 3- mu m CMOS technology, and is implemented in a two-chip set comprising 170 K transistors. Two techniques are used to achieve this precision in a moderate circuit area. Linear approximation of the LNS arithmetic functions using logarithmic arithmetic is shown to be simple due to the particular functions involved. A segmented approach to linear approximation minimizes the amount of table space required. Subsequent nonlinear compression of each lookup table leads to a further reduction in table size. The result is that a factor of 285 reduction in table size is achieved, compared to previous techniques. The circuit area of the implementation is minimized by optimizing the table parameters, using a computer program that accurately models ROM area. The implementation is highly pipelined, and produces one result per clock cycle using a ten-stage pipeline. >

A refinement of the combination equations for evaporation

Abstract: Most combination equations for evaporation rely on a linear expansion of the saturation vapor-pressure curve around the air temperature. Because the temperature at the surface may differ from this temperature by several degrees, and because the saturation vapor-pressure curve is nonlinear, this approximation leads to a certain degree of error in those evaporation equations. It is possible, however, to introduce higher-order polynomial approximations for the saturation vapor-pressure curve and to derive a family of explicit equations for evaporation, having any desired degree of accuracy. Under the linear approximation, the new family of equations for evaporation reduces, in particular cases, to the combination equations of H. L. Penman (Natural evaporation from open water, bare soil and grass, Proc. R. Soc. London, Ser. A 193, 120–145, 1948) and of subsequent workers. Comparison of the linear and quadratic approximations leads to a simple approximate expression for the error associated with the linear case. Equations based on the conventional linear approximation consistently underestimate evaporation, sometimes by a substantial amount.

An application of the boundary element method to the magnetic field integral equation

Abstract: A boundary element method (BEM) for the solution of electromagnetic scattering problems using the magnetic field integral equation (MFIE) is discussed. The discretized form of the MFIE is written in indicial notation with no limitations placed on the order of either the geometric or functional approximation. By considering several different types of boundary elements, it is determined that geometric errors can be significant and degrade the accuracy of the numerical solution. It is shown that a higher-order approximation for the current could significantly improve the accuracy of the numerical solution. The superparametric boundary element in which the geometry was given quadratic approximation and the current was given linear approximation was more efficient than elements using lower-order approximations. The BEM results are compared to the results obtained using the dielectric bodies of revolution (DBR) code. >

A characterization of the approximation order for multivariate spline spaces

Abstract: We analyze the approximation order associated with a directed set of spaces, {Sh}h>0, each of which spanned by the hZZ-translates of one compactly supported function φh : IR s → C. Under a regularity condition on the sequence {φh}h, we show that the optimal approximation order (in the ∞-norm) is always realized by quasi-interpolants, hence in a linear way. These quasi-interpolants provide the best approximation rates from {Sh}h to an exponential space of good approximation order at the origin. As for the case when each Sh is obtained by scaling S1, under the assumption ∑ α∈ZZ φ1(· − α) 6≡ 0, (∗) the results here provide an unconditional characterization of the best approximation order in terms of the polynomials in S1. The necessity of (∗) in this characterization is demonstrated by a counterexample. AMS (MOS) Subject Classifications: primary 41A15, 41A25, 41A40, 41A63; secondary 65D15.

Approximation to the three-dimensional optical transfer function

Abstract: The approximation to the defocused optical transfer function for an aberration-free system of circular aperture leads to an approximation to the corresponding three-dimensional optical transfer function. Although the former approximation is in good agreement with exact calculations, there is considerable discrepancy for the latter.

A general formulation of applied potential tomography.

Abstract: Reconstruction of the conductivity of a medium from a set of known values of the boundary potential and boundary normal current is described. The geometry is general, two or three dimensional. The method is applied to two cases, a circular medium and an infinite medium under a plane. In both cases the conductivity is found as an expansion on a system of orthogonal functions. In a linear approximation the conductivity may explicitly be reconstructed. The problem is an ill posed one, the condition numbers are found. Some numerical examples are included.

A new learning control scheme for robots

Abstract: For the trajectory following problem of a robot manipulator, a novel linear learning control law, consisting of the conventional proportional-integral-differential (PID) control law, with respect to position tracking error, and an iterative learning term, is provided. The learning part is a linear feedback control of position, velocity, and acceleration errors. It is shown that this learning control can make the position, velocity, and acceleration tracking errors asymptotically stable in the presence of highly nonlinear dynamics. It is also shown that the proposed control is robust in the sense that no knowledge about nonlinear dynamics is required except for its bounding function. As a consequence, neither linear approximation of nonlinear dynamics nor repeatability of robot motion is required. A constructive algorithm for choosing the control gains is also provided. A simple two-degrees-of-freedom manipulator was simulated to test the proposed learning control law. >

An efficient method for estimating neighboring steady-state numerical solutions to the Euler equations

Abstract: The paper concentrates on a linear approximation method for predicting the changes occurring in steady-state numerical solutions of the Euler equations as a consequence of small changes in the independent variables which control the problem. The importance of proper boundary-condition treatment and other issues concerning the problem are covered along with the importance of proper algorithm selection for a fully supersonic inviscid flow. The method is applied to a subsonic nozzle involving variation of the pressure on the outflow boundary and to a supersonic inlet involving variation of the inflow Mach number. In the subsonic test case, the comparisons between the predicted and conventional numerical solutions are shown to be good, while in the supersonic test case, the agreement between the approximation method and conventional numerical solution starts out well but rapidly degenerates at some point in the flowfield as the perturbation of the boundary conditions is increased.

Wavelet-Galerkin approximation of linear translation invariant operators

Abstract: It is shown that the wavelet-Galerkin discretization of linear translation invariant (LTI) operators has good numerical properties, arising from the vanishing moments property of wavelets. If a wavelet has M vanishing moments, then it can have at most M-1 continuous derivatives, and hence operators of the form d/sup p//dx/sup p/, where p>M, have to be considered as generalized derivatives. Even in this case the approximation results derived hold. Also, the virtual expansion theorem is useful in the sense that there is no need to compute the expansion coefficients of the function at some level V/sub triangle x/. >

The Bernstein filter-a new class of linear phase filter approximation

Abstract: A class of linear phase analog filter approximation based on the Bernstein polynomials is presented. The advantages obtained using this approximation when compared to the Thomson approximation include the fact that the group delay can be varied by varying one of the two parameters available, and in the stopband the magnitude may or may not approach zero as frequency increases. This class of low-pass linear phase transfer functions gives a magnitude which has quasi-maximally flat magnitude. The function has several parameters that allow the function to change the slope in the phase function, the slope in the transition band, the cut-off frequency, and the magnitude value at infinite frequency. >

Profile summaries for arima time series model parameters

Abstract: . Autoregressive intergrated moving average (ARIMA) times series models are nonlinear in the parameters and so summarizing the inferential results for such models can be difficult. A common approach is to present parameter joint and marginal inference regions based on the linear approximation, and although such approximate regions are easy to calculate, it is not known generally whether they approximate the true regions adequately. In this paper we present exact approaches for summarizing the inferential results for time series model parameters, called profile t and profile trace plots, which are based on the work of Bates and Watts. Calculations for the profile plots are simple and can be used to determine exact regions, and so can be used to assess the accuracy of linear approximation regions. In addition to developing the profile plots for time series models, the main finding of the paper is that, for ARIMA model parameters, linear approximation regions are very satisfactory except when a parameter estimate is within about two standard errors of the stationarity or invertibility region boundary.

Statistical Linearization for Multi-Input/Multi-Output Nonlinearities

Abstract: Formulas are derived for the computation of the random input-describing functions for MIMO nonlinearities; these straightforward and rigorous derivations are based on the optimal mean square linear approximation. The computations involve evaluations of multiple integrals. It is shown that, for certain classes of nonlinearities, multiple-integral evaluations are obviated and the computations are significantly simplified.

Solution approaches for multimode multiproduct assignment problems

Abstract: This paper deals with the solution of multimode-multiproduct assignment models which are used to analyze freight transportation networks at regional and national levels. We evaluate the efficiency of some variants of the linear approximation algorithm, where at each iteration and for each product, a few extremal solutions are generated by the linear approximation approach, and then, an approximate search is performed on the convex hull of the extremal solutions and the current solution. The variant where two extremal solutions are generated, and the approximate search consists of two Partan iterations, appears to be the most efficient. Finally, this paper points out some costly computations that may be avoided, in order to reduce the total execution times.

Neural networks for extracting unsymmetric principal components

Abstract: The authors introduce two forms of unsymmetric principal component analysis (UPCA), namely the cross-correlation UPCA and the linear approximation UPCA problem. Both are concerned with the SVD of the input-teacher cross-correlation matrix itself (first problem) or after prewhitening (second problem). The second problem is also equivalent to reduced-rank Wiener filtering. For the former problem, the authors propose an unsymmetric linear model for extracting one or more components using lateral inhibition connections in the hidden layer. The numerical convergence properties of the model are theoretically established. For the linear approximation UPCA problem, one can apply back-propagation extended either using a straightforward deflation procedure or with the use of lateral orthogonalizing connections in the hidden layer. All proposed models were tested and the simulation results confirm the theoretical expectations. >

A generalized reduced‐gravity ocean model

Abstract: A major difficulty with two‐layer ocean models, except in the linear approximation, is encountered in situations where the interface intersects the seabed. For the single mode, reduced‐gravity version it is assumed that a passive deep lower layer exists everywhere, which in fact precludes any influence by bathymetry. These restrictions can be removed by employing a realistic continuous density stratification. Discussed here is a generalized, equivalent barotropic model in isopycnic coordinates that is based on the gravest vertical structure function E(σ) for current determined by the global, levelled‐state density distribution. The eigenvalue problem employs a rigid‐lid condition at the sea surface and requires E = 0 (a vanishing current) at the seabed for each local depth h. The gravest mode has monotonie E(σ) and an associated eigencelerity that is typical of a first baroclinic mode. Both the structure function and the celerity (hence the Rossby deformation radius) depend on the local depth h. ...

Nuclear response function at finite temperature.

Abstract: The static-path approximation is applied to the real-time response function of a nucleus at finite temperature. This approximation is shown to become accurate in the high-temperature limit for an appropriately energy-smoothed strength function. An additional local approximation to overlap integrals yields the familiar adiabatic approximation. These two approximations are compared with exact results for the Lipkin model. For a sufficiently large ratio of two-body interaction to single-particle energy, the static-path approximation accurately describes the exact strength function. For parameters characteristic of nuclear shape transitions, the static-path approximation is not reliable and the adiabatic approximation is qualitatively in error.

Algorithm for modification of parameters in vibrating systems

Abstract: Concluding Remarks A global-local approximation based on a linear approximation to a scaling factor has been presented. The approximation permits us to use a global approximation based on a simple model of our problem to extend the range of usefulness of derivative-based approximations to a more refined model. The method was demonstrated for a simple beam example with a crude and more refined finite element model.

Symmetry in partially coherent imaging of semi-transparent edges

Abstract: The extended linear approximation [1] is applied to defocused imaging of semi-transparent edges. The approximated intensity distributions are given and the influence of the following parameters are studied: edge transparency and phase shift, coherence parameter and defocusing. For an opaque edge and a phase edge approximated intensity curves are compared to curves obtained by the exact calculation with the ‘bilinear transfer function’. It is shown that characteristic effects of the image intensity can be referred to the two transfer functions of the extended linear approximation and that it is possible to separate the influence of the object and the system, which is the principal advantage of the use of transfer functions.

Nonlinear model of thermal effects in YAG:Nd laser crystals

Abstract: A nonlinear (steady-state) model of thermal effects in YAG:Nd is proposed: it takes into account the real temperature dependence of the thermal conductivity of the active medium. Equations are derived for the stresses, the maximum theoretical heat release, and the optical power of the thermal lens for an active element in the form of a long circular rod. It is shown that these equations contain new dependences and describe the experiments considerably more accurately than the relationships of the linear model. When the cooling rate is increased, the nonlinear equations yield the familiar relationships of the linear approximation.

Ω-convergence, a criterion for linear approximation

Abstract: The h-convergence test is well established as a measure for approximation schemes. This paper describes an alternative view of h-convergence which provides more information. Although the name ω-convergence is (probably) new, the method uses well-established principles, and so the claim for novelty here is more on the relevance and significance of the approach than on the technique used.

Amplification of neutron star magnetic fields by thermoelectric effects. II - Linear approximation

Abstract: In the first paper (Geppert & Wiebicke 1990), a formalism for investigating the influence of thermoelectric effects on the magnetic field in the envelope of neutron stars, based on the simultaneous solution of the time-dependent nonlinear heat transport equation and induction equation, was presented. Here, the linear approximation is considered, which is justified for moderate magnetic field strengths. Numerically calculated growth and decay rates of magnetic field components with different multipolarities n are presented

Study of the superiority of generalized linear interpolation approximation

Abstract: Consider a set of waveforms whose Fourier spectra are band-limited within a certain frequency band. We assume that the weighted Lp integrals (1