Abstract: This paper presents a computational solution to an important optimization problem arising in optimal sensitivity theory. The approach is to treat the multivariable problem exactly as the scalar problem in that stability constraints are handled via interpolation. The resulting computations are easily implemented using existing methods.

Abstract: Discrete-time analysis of two schemes for multiplexing voice and data is presented. In each scheme voice and data are multiplexed using the movable boundary frame allocation scheme. In the first scheme, speech activity detectors (SAD's) are not used, and hence, the variations in the voice traffic are only due to the on/off characteristics of voice. In the second scheme, SAD's are employed so that talker silences can he utilized for transmission of additional voice and/or data. In this scheme, the multiplexer performs digital speech interpolation as well as movable boundary frame allocation. The performance measures considered are probability of loss for voice calls, probability of speech clipping, speech packet rejection ratio, and the expected data message delay. In the case of the multiplexer with SAD, a tradeoff exists between data message delay and speech interpolation advantage. Some numerical examples are presented which illustrate the performance of the two multiplexers.

Abstract: This chapter discusses interpolation and approximation. Interpolation is one of the most basic and most often used numerical techniques. Through the years many interpolation methods have been invented with the aim of simplifying the computations when performed by hand or with a desk calculator. However, a general program based on a Newton divided difference formula is usually sufficient for computer applications. A polynomial of degree n is defined by its n + 1 coefficients. Thus, it is natural to expect that an interpolating polynomial of degree n would be completely determined by ( n + 1) function values f k . The number of operations needed to evaluate the Lagrange formula at a single point makes it impractical for many applications. Truncation error and rounding errors are reviewed in the chapter. Some methods for deriving special-purpose interpolation formulas are also discussed. The technique for deriving special types of approximations is called the method of undetermined coefficients. It is a useful device provided the system of equations for determining the coefficients can be solved.

Abstract: L'approche de dualite pour trouver des solutions de ∂F=μ est suffisante pour plusieurs problemes apparaissant en theorie H P . On donne des operateurs solution qui permettent de resoudre des problemes non resolubles par la dualite

Abstract: The use of interpolations in time, rather than the more widely used spatial interpolations, demonstrates several benefits in the application of the method of characteristics to wave problems in hydraulics. Two such time-line schemes are presented and analyzed in the solution of a linearized water hammer problem. Reachback time-line interpolations, where the characteristic lines are projected back before the current time step, demonstrate less damping than the corresponding spatial interpolation scheme at the same discretization. Implicit time-line interpolations, where the characteristic line is projected into the current time step, permit relaxation of the restrictive time step required by the Courant condition. An error analysis and numerical experiments demonstrate the degree of damping and dispersion introduced by both reachback and implicit time-line interpolation methods. The error analyses and their verification may encourage further application of these methods in appropriate problem domains.

Abstract: What is noteworthy here is that any of the usual covering arguments lead only to a weak-type (1,1) bound which grows exponentially in n, and thus by interpolation one obtains by this method (1.1) with Ap replaced by a bound which increases exponentially in n. Thus the foUowing further questions now present themselves: (1) Does M (") have a weak-type (1, 1) bound independent of n? (2) What can be said when the usual balls are replaced by dilates of more general sets?, We give here some partial answers to these questions:

Abstract: The use of rotationally symmetric operators in vision is reviewed and conditions for rotational symmetry are derived for linear and quadratic forms in the first and second partial directional derivatives of a function f ( x , y ). Surface interpolation is considered to be the process of computing the most conservative solution consistent with boundary conditions. The “most conservative” solution is modeled using the calculus of variations to find the minimum function that satisfies a given performance index. To guarantee the existence of a minimum function, Grimson (W. E. L. Grimson, From Images to Surfaces: A Computational Study of the Human Early Visual System , MIT Press: Cambridge, Mass., 1981.) has recently suggested that the performance index should be a seminorm. It is shown that all quadratic forms in the second partial derivatives of the surface satisfy this criterion. The seminorms that are, in addition, rotationally symmetric form a vector space whose basis is the square Laplacian and the quadratic variation. Whereas both seminorms give rise to the same Euler condition in the interior, the quadratic variation offers the tighter constraint at the boundary and is to be preferred for surface interpolation.

Abstract: This paper describes a method for connecting an MOSFET 2-D device simulator to a circuit simulator via a 3-D table look-up MOSFET model. The computational cost of the device simulator is drastically reduced by a proposed monotonic piecewise cubic interpolation technique. With this technique, the device simulator needs to calculate only 100 ~ 200 points to make up an accurate 3-D table look-up MOSFET model. The computational time necessary for the interpolation is only about one third of the time for calculating one current point by the device simulator.

Abstract: A new interpolation-based order preserving hashing algorithm suitable for on-line maintenance of large dynamic external files under sequences of four kinds of operationsinsertion, update, deletion, andorthogonal range query is proposed. The scheme, an adaptation of linear hashing, requires no index or address directory structure and utilizesO(n) space for files containingn records; all of the benefits of linear hashing are inherited by this new scheme. File implementations yielding average successful search lengths much less than 2 and average unsuccessful search lengths much less than 4 for individual records are obtainable; the actual storage required is controllable by the implementor.

Abstract: A computational procedure for generating three-dimensional nonorthogonal surface-fitted mesh systems is presented. The method is based on the concept of transfinite interpolation and makes use of normal derivatives of the mapping function at the boundaries to obtain the desired mesh control. A brief description of the theory is presented together with 2D examples to demonstrate the general principles. The generation of 3D meshes of $O-O$ type around wings and schematic wing-fuselage configurations is described in detail and several computed examples are shown.

Abstract: A systematic study of interpolation of Fourier transform (FT) spectra is reported. Interpolation errors are examined for both frequency determination and intensity determination for different interpolation procedures for both absorption mode and magnitude mode FT spectra. The errors are presented in both analytical and graphical form as functions of the number of zero-fillings and (T/τ), the ratio of the acquisition time to the relaxation time of the time domain signal. For interpolation of absorption mode spectra, parabolic interpolation is superior to Lorentzian interpolation if T/τ 2, Lorentzian interpolation is superior. For small values of T/τ, both parabolic interpolation and Lorentzian interpolation of the absorption line shape give greater errors than no interpolation. For interpolation of the magnitude lineshape, interpolation with the

Abstract: Disclosed is hardware for providing pixel data by interpolation. In the hardware previously memorized weight factors corresponding to the particular site are retrieved under the influence of the outputs of addressing circuits, multiplied with the original pixel data corresponding to the site prior to convolution and the products added together to derive the new pixel data which is subsequently stored in a memory. Also disclosed is hardware for rapidly enlarging an image. In this hardware, four lines, for example, of original pixel data are read out and, according to the degree of enlargement required, a predetermined sequence of weight factors are simultaneously applied to sixteen original pixel data (four from each line) to determine the data for the site or sites in question.

Abstract: Numerical noise has been a problem with finite element solutions to the shallow water equations. Two methods used to reduce the noise level are evaluated, and these results are compared with published results for equal-order interpolations. The two methods are mixed-interpolation (quadratic interpolation for velocity and linear interpolation for sea level) and a spectral form of the wave equation. Whereas mixed interpolation removes the troublesome sea level mode, it can still have considerable noise in velocity. The spectral wave equation is efficient and does not contain the spurious eigenmodes which contribute to high noise levels.

Abstract: This paper is the first of three dealing with the three-dimensional wind field analysis from dual-Doppler radar data. Here we deal with the first step of the analysis which consists in interpolating and filtering the raw radial velocity fields within each coplane (or common plane simultaneously scanned by the two radars). To carry out such interpolation and filtering, a new method is proposed based on the principles of numerical variational analysis described by Sasaki (1970): the “filtered” representation of the observed field should be both “close” to the data points (in a least-squares sense) and verify some imperative of mathematical regularity. Any method for interpolating and smoothing data is inherently a filtering process. The proposed variational method enables this filtering to be controlled. The presented method is developed for any function of two variables but could be extended to the case of three or more variables. Numerical simulations substantiate the theoretically predicted filt...

Abstract: The concept of the sampling window is introduced for the central interpolation of finite energy band-limited functions. The sampling window does not increase the rate of convergence of the truncation error series, as do various convergence factors, but does significantly reduce the truncation-error bound.

Abstract: Least-square error criteria are used to fit 1-D interference fringe pattern irradiance data to a physically meaningful function of the form I(x) = B(x) + E(x) cos[P(x)], where B(x), E(x), and P(x) are low-order polynomials This procedure is intended to complement digital fringe recognition by providing a method for smoothing and interpolating among fringe position data when the number of fringes is small, there are more than ten irradiance measurements per fringe, and accurate phase values are needed at arbitrary locations in the field

Abstract: A method to evaluate the accuracy of trajectory models for the Long‐Range Transport of Atmospheric Pollutants (LRTAP) is described. The method involves derivation of horizontally non‐divergent wind fields from streamfunctions that are dependent on latitude and longitude. Appropriate choice of seven adjustable parameters enables a fairly realistic simulation of frictionless flow to be made. The differential equations for trajectories in these analytically specified flows are solved numerically with a high degree of accuracy, giving a standard against which results from trajectory models may be compared. Theoretical estimates suggest that an important source of error in LRTAP trajectory models is horizontal interpolation of wind data. This error is significantly reduced through use of a cubic interpolation scheme. Theory also indicates that truncation error in the trajectory equation can be made almost negligible in comparison with observational errors by using a “constant‐acceleration” scheme. A l...

Abstract: Apparatus for measuring the time delay between pulse signals, particularly in conjunction with electro-optical range finders. A coarse measurement counter counts the output of a reference oscillator, while a fine measurement interpolator determines the residual time at the start and finish of a measuring interval. Both residual times are successively determined by the same fine measurement interpolator. A delay circuit only supplies the pulse transmitter with the trigger signal when the interpolation for the start signal in the interpolator is ended. An adjustable series of test pulses is produces by a clock generator, in order to form the average value from such a series of measurements in a logic circuit. Preferably, the clock generator is synchronized with the reference oscillator, which ensures the formation of a mean value with a narrow range of errors.

Abstract: In two recent papers, an exact polar interpolation formula was used to reconstruct computer tomography (CT) imagery by a procedure known as direct Fourier transform inversion. The resulting imagery compared favorably to CT imagery reconstructed by filtered convolution back projection. Strictly speaking, however, the interpolation formula was inappropriately used since it is valid only for an odd number of azimuthal samples while in CT one uses an even number of samples. When the number of samples N is large (say > 200 as in CT) the error is not noticeable and the "appropriateness" of the formula has no practical significance. However, when N is small, a large error can result. We derive, in this paper, exact azimuthal interpolation formulas for N even and arbitrary N.

Abstract: One of the major approaches to numerical grid generation is the explicit algebraic expression of a physical grid as a function of a uniform grid in a rectangular computational coordinate system. The algebraic methods are based on mathematical interpolation, and the primary advantages are speed and directness. The relation between interpolation and grid generation is described. For three-dimensional grid generation, transfinite interpolation using the coordinate control processes developed in the multisurface method and two-boundary technique are advocated. Grid singularities encountered in three dimensions are discussed, and the exploration of multiple overlapping grids is proposed. Some aspects of interactive algebraic grid computation in three dimensions are discussed.

Abstract: A simple procedure for generating diagonal dominance in certain linear systems is presented, and the iterative solution of the transformed systems is discussed. The method is presented for systems encountered in the numerical solution of Fredholm equations of the first kind with singular kernels, and for systems arising in the interpolation of surfaces by certain sets of radial functions. In the second case the method provides new sets of basis functions generalizing in some sense the univariate B-splines.

Abstract: The surface appropriate for a given problem depends on the application; there is no universal surface form. The numerous applications of surface methods include modeling physical phenomena (e.g., combustion) and designing objects such as airplanes and cars. In addition to these 3D surfaces, there are interesting 4D "surfaces" such as temperature as a function of the three spatial variables. Because the geometric information for these problems can be located arbitrarily in 3D or 4-Dimensional space, the schemes must be able to handle arbitrarily located data. The standard (and easier) approach to surfacesusing tensor products of curve methods-restricts the surface method's applicability to rectangularly "gridded" data. Two broad classes of methods suitable for solving these problems (i.e., problems for which simplifying geometric assumptions cannot be made) are (1) surface interpolants defined over triangles or tetrahedra and (2) distanceweighted interpolants. Users ordinarily want smoother surfaces than their data imply directly, so additional information must usually be created. (A notable feature of the methods shown here is that the smoothness of the surface is always greater than or equal to the smoothness of the defining data. The author covers Surface form selection (Interpolation versus approximation; Representation versus design; Smoothness; Shape fidelity; Local versus global methods; and Rendering), and Interpolation surfaces. It is noted that triangular interpolants and distance-weighted interpolants excel as surface methods because of their smooth interpolation of arbitrarily located data.