Showing papers on "Basis function published in 1984"

Unified synthetic discriminant function computational formulation.

Abstract: A most attractive approach to distortion-invariant pattern recognition uses a synthetic discriminant function (SDF) as the matched spatial filter in a correlator. In this paper, we (1) provide a general basis function and hyperspace description of SDFs, (2) advance a derivation showing the generality of the correlation matrix observation space that we use in our filter synthesis, and (3) detail a unified SDF filter synthesis technique for five different types of pattern recognition problem.

Electromagnetic modeling of three‐dimensional bodies in layered earths using integral equations

Abstract: We have developed an algorithm based on the method of integral equations to simulate the electromagnetic responses of three‐dimensional bodies in layered earths. The inhomogeneities are replaced by an equivalent current distribution which is approximated by pulse basis functions. A matrix equation is constructed using the electric tensor Green’s function appropriate to a layered earth, and it is solved for the vector current in each cell. Subsequently, scattered fields are found by integrating electric and magnetic tensor Green’s functions over the scattering currents. Efficient evaluation of the tensor Green’s functions is a major consideration in reducing computation time. We find that tabulation and interpolation of the six electric and five magnetic Hankel transforms defining the secondary Green’s functions is preferable to any direct Hankel transform calculation using linear filters. A comparison of responses over elongate three‐dimensional (3-D) bodies with responses over two‐dimensional (2-D) bodie...

Theory of electronically inelastic scattering of electrons by molecules

Abstract: We discuss a multichannel formulation of the Schwinger and a related variational principle (of one order higher than the Schwinger principle) in a form suitable for application to the scattering of low-energy electrons by both linear and nonlinear molecules. The theory includes the effects of polarization straightforwardly and should be particularly useful for obtaining electronically inelastic cross sections. An expansion of the trial scattering wave function in a discrete basis is possible. With certain choices for these basis functions this feature can be particularly advantageous.

Variational calculations on the helium isoelectronic sequence

Abstract: We have performed variational calculations on the helium isoelectronic sequence for values of the nuclear charge $Z$ ranging from 1 to 10. The basis used is a modification of that employed by Frankowski and Pekeris in 1966, whose calculation has not been superseded before now. Using 230-term wave functions, we obtain for $Z=2$ through 10 variational energies accurate to better than a few parts in ${10}^{13}$. Our results illustrate the importance of using basis functions which have the same analytic structure as the exact wave function being approximated.

Orthogonal Function Series Expansions and the Null Space of the Radon Transform

Abstract: The range of the Radon transform for compactly supported functions is spanned by products of Gegenbauer polynomials and spherical harmonics. The inverse transform of those basis functions is given for arbitrary dimensions and arbitrary Gegenbauer polynomials. The resulting inversion formula is applied to study the “ghosts”, i.e. the functions in the null space of the transform for finitely many projections. They are characterized by their series expansions and results concerning the optimal choice of directions for the data acquisition are deduced. The application to nuclear magnetic resonance zeugmatography is discussed.

Triangle and pyramid signal transforms and apparatus

Abstract: Transform techniques for transforming numerical signal data into a transform domain and subsequent reconstruction, for purposes such as compression (bandwidth reduction) for communication or storage. The subject transforms involve several defined basis functions which operate on input data points. The basis functions of the invention are essentially weighting functions such that terms and coefficients (in the transform domain) calculated in accordance with the basis functions are each a particularly weighted average of the values of a selected consecutive plurality of input data points. An important basis function is a triangular weighting function. Successive terms and coefficients generated in accordance with each of the defined basis functions are calculated from successive consecutive pluralities of the input data points, with overlap of input data points depending on the particular basis function. The transforms are organized into a plurality of bands or levels N. Band N is the highest, and Band 1 the lowest. For forward transformation, calculation begins with the highest band, Band N, and works down. For inverse transformation (reconstruction), calculation begins with the lowest band, Band 1, and works up. Results of processing in each band are then employed as inputs for processing in the next lower band until the last band is reached. Many of the calculated coefficients have zero values, and there is also disclosed an efficient coding technique which permits the transmission of only the non-zero coefficients.

Limitations of the Cubical Block Model of Man in Calculating SAR Distributions

Abstract: Block models of man which consist of a limited number of cubical cells are commonly used to predict the internal electromagnetic (EM) fields and specific absorption rate (SAR) distributions inside the human body. Numerical results, for these models, are obtained based on moment-method solutions of the electric-field integral equation (EFIE) with a pulse function being used as the basis for expanding the unknown internal field. In this paper, we first examine the adequacy of the moment-method procedure, with pulse basis functions, to determine SAR distributions in homogeneous models. Calculated results for the SAR distributions in some block models are presented, and the stability of the solutions is discussed. It is shown that, while the moment-method, using pulse basis functions, gives good values for whole-body average SAR, the convergence of the solutions for SAR distributions is questionable. A new technique for improving the spatial resolution of SAR distribution calculations using a different EFIE and Galerkin's method with linear basis functions and polyhedral mathematical cells is also described.

Effective basis sets for calculations of exchange‐repulsion energy

Abstract: Usefulness of different Gaussian basis sets for reproducing the “tail” region of the SCF wavefunctions employed in calculations of the exchange-repulsion effect is investigated for the model He-He interaction. It has been shown that extension of the monomer-centered basis set in the scheme of regularized even-tempered basis sets [M. W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71, 3951 (1979)] can be more efficient than augmentation of the fully energy-optimized basis set with diffuse basis functions. It has been also found that Landshoff term vanishes and the “tail” region is well reproduced if monomer wavefunctions are calculated with the basis set of the dimer.

Spectral-domain analysis of circular patch frequency selective surfaces

Abstract: A computationally efficient method is presented for analyzing the scattering from frequency selective surfaces (FSS) comprised of circular metal patches. The formulation is carried out in the spectral domain where the convolution form of the integral equation for the induced current reduces to an algebraic one and the spectral-Galerkin technique is used to solve the resulting equation. Entire-domain basis functions that satisfy the edge condition are introduced to expand the unknown induced current on the metal patches. Calculated results using this procedure show good agreement with data reported by other authors.

The Dirac equation in the algebraic approximation. II. Extended basis set calculations for hydrogenic atoms

Abstract: For pt.I see ibid., vol.17, p.L45 (1984b). The solution of the Dirac equation for hydrogen-like atoms within the algebraic approximation, that is, by using a finite basis set, is considered. It is shown that by making an appropriate choice of basis functions the problem which has been termed 'variational collapse' can be avoided. Applications using a systematic sequence of even-tempered basis sets are presented and convergence of the calculated energies within the algebraic approximation to the exact energies with increasing size of basis set is investigated.

Fourier transforms of atomic orbitals. I. Reduction to four-dimensional harmonics and quadratic transformations

Abstract: A general expression for the Fourier transform of the basis functions of exponential class has been derived. Particular cases of Slater functions, hydrogen-like functions, Shull and Lowdin functions, Shavitt, Filter, and Steinborn functions have been considered. In many particular cases the Fourier transforms have been shown to reveal some important special properties (reduction to four-dimensional harmonics, quadratic transformations, etc.) which considerably simplify the mathematical treatment of these functions and lead to new possibilities in the development of calculation methods for multicenter integrals.

The Dirac equation in the algebraic approximation. I. Criteria for the choice of basis functions and minimum basis set calculations for hydrogenic atoms

Abstract: Criteria for the choice of basis functions for the solution of the Dirac equation in a finite basis set are derived from considerations of matrix products involving the Dirac kinetic energy. Minimum basis set calculations for the ground state of the hydrogen atom are described.

Accurate Hartree–Fock wave functions without exponent optimization

Abstract: Basic functions with singularities matching those of the actual orbitals have been tested in analytical Hartree–Fock calculations. Such functions should provide the most rapidly convergent basis set expansions. Exponential singularities at r=∞, characterized by certain ‘‘asymptotic exponents,’’ have been identified by an asymptotic analysis of the Fock equation. Basis sets of Slater functions with these exponents give atomic energies and properties comparable to the most accurate existing analytical calculations, without significantly increasing the number of basis functions. No nonlinear optimizations were required. Calculations of the orbital moments 〈rn〉 show that only moments with n≤N, the number of Slater basis functions, can be evaluated with accuracy, whether or not the exponents are optimized. This effect appears to be caused by the neglect of certain irrational powers in asymptotic forms of the orbitals. The results for molecules suggest that basis functions which more adequately describe the nuc...

Variational calculation of lower vibrational energy levels of formaldehyde X̃ 1A1

Abstract: The energies of lower lying vibrational states (J=0) of formaldehyde in its ground electronic state are calculated variationally for H2CO and D2CO. The functions of the basis set correspond to products of harmonic oscillator functions. The full Watson Hamiltonian is used and integrals are evaluated by Gauss–Hermite numerical quadrature. Two different potential functions in internal displacement coordinates, which have been given in the literature, are employed in the calculations. Calculations are carried out for A1, B2, and B1 symmetries with 196, 165, and 108 basis functions, respectively. The forms of some of the eigenvectors are reported.

Adaptivity and Error Estimation for the Finite Element Method Applied to Convection Diffusion Problems

Abstract: A detailed analysis is performed for a finite element method applied to the general one-dimensional convection diffusion problem. Piecewise polynomials are used for the trial space. The test space is formed by locally projecting L-spline basis functions onto “upwinded” polynomials. The error is measured in the $L_p$ mesh dependent norm. The method is shown to be quasi-optimal, provided that the input data is piecewise smooth—a reasonable assumption in practice. A posteriors error estimates are derived having the property that the effectivity index $\theta = $ (error estimate/true error) converges to one as the maximum mesh size goes to zero. These error estimates are composed of locally computable error indicators, providing for an adaptive mesh refinement strategy. Numerical results show that $\theta $ is nearly one even on coarse meshes, and optimal rates of convergence are attained by the adaptive procedure. The robustness of the algorithm is tested on a nonlinear turning point problem modeling flow th...

Analytical integration formulae for linear isoparametric finite elements

Abstract: Exact and approximate analytical expressions can be derived for integrals arising in finite element methods, employing isoparametric linear quadrilaterals in two space dimensions with bilinear basis functions. The formulae associated with rectangular elements, arbitrarily oriented in space, can be shown to be a special case. The proposed method provides considerable savings in computational effort, in comparison with a numerical method that employs Gaussian quadrature procedures. In addition, the method, when applied to a quadrilateral inscribable in a circle, can be shown to produce better accuracy than the associated (2 × 2) Gaussian quadrature formulae.

Simulation of Two-Dimensional Waterflooding By Using Mixed Finite Elements

Abstract: A new method for the simulation of incompressible diphasic flows in two dimensions is presented, the distinctive features of which are: (1) reformation of the basic equation and specific choices of the finite element approximation of the same; (11) use of a mixed finite elements method, approximating both scalar and vector functions. Several test examples are shown, including gravity and capillary effects. The use of discontinuous basis functions proved successful for an accurate representation of sharp fronts. 16 refs.

Infinitesimal scaling--A new procedure for modeling exterior field problems

Abstract: The ability to treat extended uniform linear regions (particularly unbounded external regions) as single elements is a desirable addition to electromagnetic finite element techniques. The use of extended elements requires basis functions that are better approximations to the solutions of the underlying partial differential equations than the conventional polynomial basis functions. This problem can be circumvented by means of the infinitesimal scaling technique, which enables the coefficient matrix required for assembling the contribution of an extended element into the global functional to be determined without explicit knowledge of internal basis functions.

Experiments with a spectral tropical cyclone model

Abstract: The three-layer balanced axisymmetric tropical cyclone model presented by Ooyama is generalized to dimensions and the resultant primitive equations are solved using the spectral (Galerkin) method with Fourier basis functions on a doubly-periodic midlatitude β-plane. The nonlinear terms are evaluated using the transform method where the necessary transforms are performed using FFT algorithms. The spectral equations are transformed so that the dependent variables represent the normal modes of the linearized equations. For the three-layer model, the normal modes correspond to internal or external gravity or rotational modes or to inertial oscillations associated with the constant depth boundary layer. When the governing equations, are written in terms of the normal modes, the linear terms can be evaluated exactly and the application of the nonlinear normal mode initialization scheme proposed by Machenhauer is straightforward. Results from a simulation with an axisymmetric initial condition on an f-p...

The horizontal loop electromagnetic response of a thin plate in a conductive earth: Part I—Computational method

Abstract: The frequency-domain EM response of a thin vertical tabular conductor in a two-layer earth was obtained by solving the electric field integral equation numerically. The method of point collocation and expansion of the scattering current components in global polynomic basis functions was employed. Selection of polynomials which obey a boundary condition of no current flow from the edges had a beneficial effect on the results. As the number of basis functions increases, the solution converges to a stationary value which is valid over a sufficiently wide range of parameters to permit the computation of horizontal loop (Slingram) profiles over a plate which can be large enough to simulate a semi-infinite half-plane. The solution gradually worsens as the conductivity contrast between the plate and the host increases, and it eventually fails in situations where magnetic induction effects are strong and electric effects are negligible. The solution also worsens as the plate size increases or as the vertical magnetic dipole source approaches the plate.--Modified journal abstract.

Generating differential operators for the basis functions in the variational LCAO‐type methods

Abstract: The analytical expressions for operators, which allow us to generate the basis functions from the simplest (ground) states by means of differentiating by parameters, are obtained. Some important properties of these operators are indicated. This approach allows us to simplify the structure of molecular matrix elements, whose calculation presents considerable difficulties in the variational LCAO‐type methods.

Variational R-matrix calculations of e-+H2 scattering using numerical asymptotic basis functions

Abstract: Variational R-matrix calculations of elastic electron scattering from the ground state of H2 are reported for the 2 Sigma u+ resonant scattering state at R=1.402a0. The orbital basis set includes exponential functions centred on each nucleus, spherical Bessel functions and numerical functions obtained by integrating asymptotic coupled differential equations. Polarisation response is treated in terms of polarisation pseudo-states.

A method for computing scattering by large arrays of narrow strips

Abstract: The electromagnetic scattering characteristics of an array of narrow, conducting strips can he developed readily by extending the work of Butler and Wilton who show that Chebyshev polynomials augmented with the edge condition can be used to solve the narrow-strip/narrow-slot integral equations. The strips reside in a homogeneous medium of infinite extent and are considered narrow relative to wavelength in the medium at the frequency of excitation. The unknown current distributions on the strips are represented as linear combinations of certain basis functions that are exact solutions to the approximate equation for an isolated narrow strip subject to a special excitation. The resulting power-series treatment allows easy calculation of the coupling terms among the strips in the array in a simple matrix equation by which the unknown coefficients in the current distribution expansions may be readily computed. With these coefficients, one can obtain the distribution of current on each strip and the total scattered field. The method is particularly well suited for handling large arrays with more strips than could be accommodated by the usual moment method. Numerical data-currents and scattered fields-are presented for various cases of interest.