Showing papers on "Basis function published in 1991"
TL;DR: In this article, a new method is presented for flexible regression modeling of high dimensional data, which takes the form of an expansion in product spline basis functions, where the number of basis functions as well as the parameters associated with each one (product degree and knot locations) are automatically determined by the data.
Abstract: A new method is presented for flexible regression modeling of high dimensional data. The model takes the form of an expansion in product spline basis functions, where the number of basis functions as well as the parameters associated with each one (product degree and knot locations) are automatically determined by the data. This procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties. Unlike recursive partitioning, however, this method produces continuous models with continuous derivatives. It has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables. In addition, the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions.
6,651 citations
TL;DR: The authors propose an alternative learning procedure based on the orthogonal least-squares method, which provides a simple and efficient means for fitting radial basis function networks.
Abstract: The radial basis function network offers a viable alternative to the two-layer neural network in many applications of signal processing. A common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular-value decomposition to solve for the weights of the network. Such a procedure has several drawbacks, and, in particular, an arbitrary selection of centers is clearly unsatisfactory. The authors propose an alternative learning procedure based on the orthogonal least-squares method. The procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed. In the algorithm, each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill-conditioning problems. The orthogonal least-squares learning strategy provides a simple and efficient means for fitting radial basis function networks. This is illustrated using examples taken from two different signal processing applications. >
3,414 citations
Book•
29 Mar 1991
TL;DR: In this article, the authors present a general solution based on the principle of virtual work for two-dimensional linear elasticity problems and their convergence rates in one-dimensional dimensions. But they do not consider the case of three-dimensional LEL problems.
Abstract: Mathematical Models and Engineering Decisions. Generalized Solutions Based on the Principle of Virtual Work. Finite Element Discretizations in One Dimension. Extensions and Their Convergence Rates in One Dimension. Two-Dimensional Linear Elastostatic Problems. Element-Level Basis Functions in Two Dimensions. Computation of Stiffness Matrices and Load Vectors for Two Dimensional Elastostatic Problems. Potential Flow Problems. Assembly, Constraint Enforcement, and Solution. Extensions and Their Convergence Rates in Two Dimensions. Computation of Displacements, Stresses and Stress Resultants. Computation of the Coefficients of Asymptotic Expansions. Three-Dimensional Linear Elastostatic Problems. Models for Plates and Shells. Miscellaneous Topics. Estimation and Control of Errors of Discretization. Mathematical Models. Appendices. Index.
2,748 citations
TL;DR: In this paper, a modified Wang basis function is proposed which has the property of making all operators which are even powers of angular momentum pure real and all odd powers pure imaginary, and a generalized direction cosine operator is described, which can be calculated in a Wang basis using spherical tensor formalism.
1,985 citations
TL;DR: These methods provide a compact and accurate anatomic description of the ventricles suitable for use in finite element stress analysis, simulation of cardiac electrical activation, and other cardiac field modeling problems.
Abstract: We developed a mathematical representation of ventricular geometry and muscle fiber organization using three-dimensional finite elements referred to a prolate spheroid coordinate system. Within elements, fields are approximated using basis functions with associated parameters defined at the element nodes. Four parameters per node are used to describe ventricular geometry. The radial coordinate is interpolated using cubic Hermite basis functions that preserve slope continuity, while the angular coordinates are interpolated linearly. Two further nodal parameters describe the orientation of myocardial fibers. The orientation of fibers within coordinate planes bounded by epicardial and endocardial surfaces is interpolated linearly, with transmural variation given by cubic Hermite basis functions. Left and right ventricular geometry and myocardial fiber orientations were characterized for a canine heart arrested in diastole and fixed at zero transmural pressure. The geometry was represented by a 24-element ensemble with 41 nodes. Nodal parameters fitted using least squares provided a realistic description of ventricular epicardial [root mean square (RMS) error less than 0.9 mm] and endocardial (RMS error less than 2.6 mm) surfaces. Measured fiber fields were also fitted (RMS error less than 17 degrees) with a 60-element, 99-node mesh obtained by subdividing the 24-element mesh. These methods provide a compact and accurate anatomic description of the ventricles suitable for use in finite element stress analysis, simulation of cardiac electrical activation, and other cardiac field modeling problems.
705 citations
TL;DR: In this paper, the Navier-Stokes equations in boundary layers and mixing layers are solved by two numerical methods which employ rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions.
674 citations
TL;DR: In this article, a density-functional method for calculations on periodic systems (periodicity in one, two, or three dimensions) is presented in which all aspects of numerical precision are efficiently controlled.
Abstract: A density-functional method for calculations on periodic systems (periodicity in one, two, or three dimensions) is presented in which all aspects of numerical precision are efficiently controlled. Highly accurate and rapidly converging strategies have been implemented for (a) the computation of Hamiltonian matrix elements (by a numerical integration method based on a partitioning of space and application of product Gauss rules), (b) the approximation of integrals over the Brillouin zone (by the quadratic tetrahedron method), (c) the evaluation and processing of the Coulomb potential (via a density-fitting procedure), and (d) the expansion of one-particle states in suitable basis functions (numerical atomic orbitals, Slater-type exponential functions, and plane waves). Absolute precision and convergence are demonstrated for all these aspects and show that the method is a well-suited tool for unambiguous investigations of the density-functional approximation itself. Attention is given, in particular, to basis-set questions. Although the method is of the mixed-basis type, it is demonstrated that plane waves are not necessary; this holds for metals as well as for insulators and semiconductors. By a general prescription, sequences of accurate linear-combination-of-atomic-orbital (LCAO) basis sets can be defined that systematically approach the basis-set limit. This enables the routine application of the inherently efficient LCAO method to all kinds of systems. Exemplary calculations are performed on bulk Si-, g-C (graphite), Na, Ni, Cu, and NaCl, and on a hexagonal monolayer of weakly interacting ${\mathrm{O}}_{2}$ molecules.
435 citations
TL;DR: It is demonstrated how the transfer equation in connection with the use of spherical harmonic Gaussians offers a very attractive path to compute the two‐electron integrals of such basis functions.
Abstract: The reduced multiplication scheme of the Rys quadrature is presented. The method is based on new ways in which the Rys quadrature can be developed if it is implemented together with the transfer equation applied to the contracted integrals. In parallel to the new scheme of the Rys quadrature improvements are suggested to the auxiliary function based algorithms. The two new methods have very favorable theoretical floating point operation (FLOP) counts as compared to other methods. It is noted that the only significant difference in performance of the two new methods is due to the vectorizability of the presented algorithms. In order to exhibit this, both methods were implemented in the integral program seward. Timings are presented for comparisons with other implemenations. Finally, it is demonstrated how the transfer equation in connection with the use of spherical harmonic Gaussians offers a very attractive path to compute the two‐electron integrals of such basis functions. It is demonstrated both theoretically and with actual performance that the use of spherical harmonic Gaussians offers a clear advantage over the traditional evaluation of the two‐electron integrals in the Cartesian Gaussian basis.
254 citations
TL;DR: In this article, the equilibrium geometries and relative stabilities of the hypothetical C 60 H 60 and C 60 F 60 molecules are predicted at the Hartree-Fock level of theory employing basis sets of double-zeta plus polarization (DZP) quality.
212 citations
01 Sep 1991
TL;DR: In this article, the tangential vector finite elements are used to eliminate spurious modes in the finite-element solution of the vector wave equation, and the configurations of these vector-valued basis functions are derived explicitly, and interpolatory meanings of the unknowns are derived.
Abstract: One approach to eliminating spurious modes in the finite-element solution of the vector wave equation is the use of tangential vector finite elements. With tangential vector finite elements, only the tangential components of the vector field are made continuous across the element boundaries. Edge-elements are the simplest example of tangential vector finite elements. However, edge-elements provide only the lowest-order of accuracy in numerical computations, since in this approach the tangential component of the field is assumed to be constant along each edge of the element. The configurations of the tangential vector finite elements which are of higher-order approximations on two- and three-dimensional tetrahedral elements are presented. The vector-valued basis functions are written explicitly, and the interpolatory meanings of the unknowns are derived. >
164 citations
TL;DR: This article examines how sensor values are modified in the mutual reflection region and shows that a good approximation of the surface spectral reflectance function for each surface can be recovered by using the extra information from mutual reflection.
Abstract: Mutual reflection occurs when light reflected from one surface illuminates a second surface. In this situation, the color of one or both surfaces can be modified by a color-bleeding effect. In this article we examine how sensor values (e.g., RGB values) are modified in the mutual reflection region and show that a good approximation of the surface spectral reflectance function for each surface can be recovered by using the extra information from mutual reflection. Thus color constancy results from an examination of mutual reflection. Use is made of finite dimensional linear models for ambient illumination and for surface spectral reflectance. If m and n are the number of basis functions required to model illumination and surface spectral reflectance respectively, then we find that the number of different sensor classes p must satisfy the condition p≥(2 n+m)/3. If we use three basis functions to model illumination and three basis functions to model surface spectral reflectance, then only three classes of sensors are required to carry out the algorithm. Results are presented showing a small increase in error over the error inherent in the underlying finite dimension models.
TL;DR: In this paper, a degenerate perturbation theoretic treatment of the helioseismic forward and inverse problems for solar differential rotation is presented, which allows each degree of differential rotation to be estimated independently from all other degrees.
Abstract: A general, degenerate perturbation theoretic treatment of the helioseismic forward and inverse problem for solar differential rotation is presented. For the forward problem, differential rotation is represented as the axisymmetric component of a general toroidal flow field using velocity spherical harmonics. This approach allows each degree of differential rotation to be estimated independently from all other degrees. In the inverse problem, the splitting caused by differential rotation is expressed as an expansion in a set of orthonormal polynomials that are intimately related to the solution of the forward problem. The combined use of vector spherical harmonics as basis functions for differential ratio and the Clebsch-Gordon coefficients to represent splitting provides a unified approach to the forward and inverse problems of differential rotation which greatly simplify inversion.
TL;DR: The author proposes a technique based on the idea that for most of the data, only a few dimensions of the input may be necessary to compute the desired output function, and it can be used to reduce the number of required measurements in situations where there is a cost associated with sensing.
Abstract: Nonlinear function approximation is often solved by finding a set of coefficients for a finite number of fixed nonlinear basis functions. However, if the input data are drawn from a high-dimensional space, the number of required basis functions grows exponentially with dimension, leading many to suggest the use of adaptive nonlinear basis functions whose parameters can be determined by iterative methods. The author proposes a technique based on the idea that for most of the data, only a few dimensions of the input may be necessary to compute the desired output function. Additional input dimensions are incorporated only where needed. The learning procedure grows a tree whose structure depends upon the input data and the function to be approximated. This technique has a fast learning algorithm with no local minima once the network shape is fixed, and it can be used to reduce the number of required measurements in situations where there is a cost associated with sensing. Three examples are given: controlling the dynamics of a simulated planar two-joint robot arm, predicting the dynamics of the chaotic Mackey-Glass equation, and predicting pixel values in real images from pixel values above and to the left. >
TL;DR: A parameter optimization method is suggested based on the non-linear least-squares optimization procedure due to Levenberg-Marquardt, which has been compared with a number of published methods using cubic B-spline basis functions.
TL;DR: In this paper, a new approach to the evaluation of two-electron repulsion integrals over contracted Gaussian basis functions is developed, which encompasses 20 distinct, but interrelated, paths from simple shell-quartet parameters to the target integrals, and, for any given integral class, the path requiring the fewest floating-point operations (FLOPS) is used.
Abstract: A new approach to the evaluation of two-electron repulsion integrals over contracted Gaussian basis functions is developed. The new scheme encompasses 20 distinct, but interrelated, paths from simple shell-quartet parameters to the target integrals, and, for any given integral class, the path requiring the fewest floating-point operations (FLOPS) is that used. Both theoretical (FLOP counting) and practical (CPU timing) measures indicate that the method represents a substantial improvement over the HGP algorithm.
TL;DR: In this article, the dipole moment of CO has been calculated with manybody perturbation theory (MBPT) and coupled cluster (CC) methods using basis sets which have been optimized at the MBPT•2 level.
Abstract: The dipole moment of CO has been calculated with many‐body perturbation theory (MBPT) and coupled cluster (CC) methods using basis sets which have been optimized at the MBPT‐2 level. It is demonstrated that triple excitations as well as g‐type functions in the basis set are crucial to obtain satisfactory agreement with experiment. The most reliable prediction (0.125 D) is obtained at the CCSD(T) (coupled cluster including all single, double, and connected triple excitations, perturbatively) level of theory using a 10s9p4d2f1g basis set (160 basis functions). This result is in excellent agreement with the experimental value of 0.122 D.
TL;DR: In this paper, the authors introduce an approach to basis set design that uses physically motivated atomic orbitals as basis functions for molecular calculations, and demonstrate that the natural orbitals derived from correlated atomic wave functions prove to be a compact, computationally efficient, and physically meaningful set of basis functions.
Abstract: Publisher Summary This chapter introduces an approach to basis set design— that is, to use physically motivated atomic orbitals as basis functions for molecular calculations. The fact that the AOs are expanded as fixed contractions of Gaussian functions is a physically irrelevant computational device. For correlated calculations, the natural orbitals derived from correlated atomic wave functions prove to be a compact, computationally efficient, and physically meaningful set of basis functions. The efficient evaluation of AO integrals over such basis sets imposes certain computational requirements on an integral code. The chapter discusses how these requirements are implemented in evaluation of two-electron integrals over a generally contracted Gaussian basis set in the molecule integral program. The chapter also discusses how ANO basis sets are effectively able to exhaust the capabilities of the underlying Gaussian expansion basis— that is, to minimize the contraction error. Finally, the chapter describes several applications of quantum chemistry to problems in which the use of ANO basis sets is significant in obtaining the required high accuracy.
TL;DR: In this article, an adaptive refinement algorithm is presented and interpreted as the selective enrichment of a finite-element space through the hierarchical basis, where each elemental division corresponds exactly to the inclusion of a small number of new basis functions, while existing basis functions remain unchanged.
TL;DR: In this article, a method to truncate and recouple basis functions in general variational calculations based on a direct-product representation of multidimensional wave functions is presented for molecular vibrations; however, the procedure is quite general and can be used in any basis set expansion method.
Abstract: We describe a new method to truncate and recouple basis functions in general variational calculations based on a direct‐product representation of multidimensional wave functions. The method is presented for molecular vibrations; however, the procedure is quite general and can be used in any basis set expansion method. The direct‐product Hamiltonian matrix H is decomposed into a block diagonal matrix H0 plus a remainder H1 . A new subset of basis functions is obtained by diagonalizing H0 . This subset of basis functions is shown to be eigenfunctions of a Hamiltonian in a reduced dimensionality space, ‘‘dressed’’ by the remaining degrees of freedom. These dressed eigenfunctions are then augmented by the component of the original direct‐product basis in which H0 is diagonal. The new basis is recoupled using an energy selection criterion, yielding a substantial reduction in the size of the final full Hamiltonian matrix. The method also suggests a generalization of the vibrational self‐consistent field method,...
TL;DR: In this paper, the Muntz-Szatzatz series expansion is used to generate global spans within large classes of functions, where the relevant constraints are inequality restrictions, and the approach using Bayesian methods to avoid the problems of sampling distribution truncation.
TL;DR: In this paper, the use of spin-space groups (SSG) in the analysis of the electronic states of spiral magnetic order was studied, and the conditions on the symmetry operations belonging to the group of a given Bloch vector and the formula describing the symmetry of eigenvalues in reciprocal space were obtained.
Abstract: This paper is devoted to the systematic study of questions concerning the use of spin-space groups (SSG) in calculation and qualitative analysis of the electronic states for crystals with spiral magnetic order. The types of operators that may enter into the symmetry group of a spiral are investigated. Introduction of the wavevector of an electronic state on the basis of the generalized Bloch theorem is discussed, and it is shown that the possibility of the choice of the wavevector is not unique. The condition imposed on the symmetry operations belonging to the group of a given Bloch vector and the formula describing the symmetry of eigenvalues in reciprocal space are obtained and appear to be substantially different from counterparts used in traditional cases of collinear and non-magnetic crystals. It is shown that, contrary to the traditional cases, there are spiral structures whose spectral symmetry is described by non-symmorphic space groups. The irreducible domain of reciprocal space is found for a number of concrete spiral structures. Methods of construction of the SSG double-valued irreducible representations and also of their basis functions are suggested. Special attention is devoted to the possibility of using the corresponding tables of ordinary space groups. In particular, it is shown that for spirals with hexagonal close-packed crystal structure the traditional tables may be used after minor corrections. The peculiarities of allowance for the operation of time inversion are discussed.
TL;DR: In this paper, the spectral-domain approach was combined with the perturbation method to analyze lossy coplanar-type transmission lines, where a finite thickness of metallization was introduced to prevent the integrals used for calculating the conductor losses from becoming singular when evaluated at the conductor edge.
Abstract: Lossy coplanar-type transmission lines are analyzed based on the hybrid-mode formulation by combining the spectral-domain approach with the perturbation method. Introducing a finite thickness of metallization and choosing the proper basis functions for the thick conductor model prevent the integrals used for calculating the conductor losses from becoming singular when evaluated at the conductor edge. An orthogonality relation is used to reduce the double infinite or semi-infinite integral to a single integral, thus reducing the computation effort drastically. Numerical computations by new basis functions for the thick conductor show convergence rates as fast as those for the zero-thickness cases. Numerical results include the effective dielectric constants, characteristic impedances, and total losses (conductor and dielectric losses) for slot lines and symmetrical and asymmetrical coplanar waveguides. >
TL;DR: In this paper, the potential energy function of the N2 molecule is calculated using the internally contracted multireference CI method (CMRCI) and complete active space SCF (CASSCF) reference wave functions.
Abstract: The potential energy function of the N2 molecule is calculated using the internally contracted multireference CI method (CMRCI) and complete active space SCF (CASSCF) reference wave functions. A full CI calculation in a DZP basis set is used to estimate the errors associated with the CMRCI wave function. The dependence of the computed spectroscopic constants and the dissociation energy on the basis set is also investigated. Uncontracted and segmented basis sets are compared with ANO (atomic natural orbital) and other generally contracted basis sets. It is found that the energy optimized ‘‘correlation consistent’’ basis sets of Dunning yield substantially better results than ANO basis sets of the same size. In the largest calculations, which included up to h type basis functions and also accounted for core–core and core–valence correlation effects, the remaining errors are 0. 0003 A, 8 cm−1, and 0.7 kcal/mol for re, ωe, and De, respectively. The inclusion of an i type basis function reduces the error in th...
TL;DR: In this article, the radial orbit instability for a family of anisotropic isochrone spheres is studied by a numerical linear stability analysis, and the results are compared with N-body simulations.
Abstract: The radial orbit instability for a family of anisotropic isochrone spheres is studied by a numerical linear stability analysis, and the results are compared with N-body simulations. A new way of choosing basis functions for the modes is introduced that allows spatially infinite systems to be handled without truncation. Previous studies of the same models, by adiabatic deformation and N-body integrations, found instability for b = / ≥ 2; the present computation indicate that very slowly growing modes persist for anisotropies as small as b = 1.4.
TL;DR: In this paper, a weak form of this integral equation is obtained by testing it with subdomain basis functions defined over the plate domain only, and then the vector potential is expanded in a sequence of sub domain basis functions.
Abstract: A number of electromagnetic field problems for planar structures can be formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a convolution of the free-space Green's function and some surface current density over the domain of interest. A weak form of this integral equation is obtained by testing it with subdomain basis functions defined over the plate domain only. As a next step, the vector potential is expanded in a sequence of subdomain basis functions and the grad-div operator is integrated analytically over the plate domain only. For the problem of electromagnetic scattering by a plate, the method shows excellent numerical performance. The numerical difficulties encountered in some previous conjugate gradient fast Fourier transform (CGFFT) methods have been eliminated. >
03 Jun 1991
TL;DR: Experimental results that demonstrate the applicability of the technique to generating multi-orientation multiscale 2-D edge-detection kernels are presented and the implementation issues are discussed.
Abstract: A technique is presented that allows (1) computing the best approximation of a given family using linear combinations of a small number of basis functions; and (2) describing all finite-dimensional families, i.e. the families of filters for which a finite-dimensional representation is possible with no error. The technique is general and can be applied to generating filters in arbitrary dimensions. Experimental results that demonstrate the applicability of the technique to generating multi-orientation multiscale 2-D edge-detection kernels are presented. The implementation issues are also discussed. >
TL;DR: In this article, a general formalism for the application of explicitly correlated Gaussian-type basis functions for nonadiabatic calculations on many-body systems is presented, where the motions of all particles are correlated in the same time.
Abstract: General formalism for the application of explicitly correlated Gaussian‐type basis functions for nonadiabatic calculations on many‐body systems is presented. In this approach the motions of all particles are correlated in the same time. The energy associated with the external degrees of freedom, i.e., the motion of the center of mass, is eliminated in an effective way from the total energy of the system. In order to achieve this, methodology for construction of the many‐body nonadiabatic wave function and algorithms for evaluation of the multicenter and multiparticle integrals involving explicitly correlated Gaussian cluster functions are derived. Next the computational implementation of the method is discussed. Finally, variational calculations for a model three‐body system are presented.
TL;DR: In this article, a numerical solution to the integral equation is obtained using the method of weighted residuals with linear B splines as basis functions and a regularization technique is employed to stabilize the solution.
Abstract: The bubble population near the ocean surface is of considerable interest. This population affects surface scattering strength, propagation near the surface, and the exchange of gases between the atmosphere and the sea. Both optical and acoustical means have been used to measure the bubble population with varying degrees of success. The acoustic method requires measurements at multiple frequencies and their subsequent conversion to bubble densities through either the resonance theory approximation or numerical solution of the resulting integral equation. In this paper, a numerical solution to the integral equation is obtained using the method of weighted residuals with linear B splines as basis functions. A regularization technique is employed to stabilize the solution. A number of plausible bubble distribution functions are generated along with their acoustic properties to test the robustness of the technique. This approach is shown to yield very accurate bubble distributions from high‐quality attenuation...
TL;DR: In this paper, the authors choose the Hermite-Gauss functions as the set of orthogonal basis functions to solve the eigenvalue problem based on the two-dimensional scalar-wave equation subject to the radiation boundary conditions at infinity.
Abstract: In Galerkin's method, an orthogonal set of functions is used to convert a differential equation into a set of simultaneous linear equations. The authors choose the Hermite-Gauss functions as the set of orthogonal basis functions to solve the eigenvalue problem based on the two-dimensional scalar-wave equation subject to the radiation boundary conditions at infinity. The method gives an accurate prediction of modal propagation constant and of the field distribution. The method is tested by using the step-index optical fiber, which has a known exact solution, and the truncated parabolic profile fiber, which has a known exact solution. The authors also test the method using square and elliptic core fibers. The method is found to agree with known results. >
TL;DR: A new algorithm for approximating continuous functions in high-dimensional input spaces and an example that predicts future values of the Mackey-Glass differential delay equation are presented.
Abstract: I describe a new algorithm for approximating continuous functions in high-dimensional input spaces. The algorithm builds a tree-structured network of variable size, which is determined both by the distribution of the input data and by the function to be approximated. Unlike other tree-structured algorithms, learning occurs through completely local mechanisms and the weights and structure are modified incrementally as data arrives. Efficient computation in the tree structure takes advantage of the potential for low-order dependencies between the output and the individual dimensions of the input. This algorithm is related to the ideas behind k-d trees (Bentley 1975), CART (Breiman et al. 1984), and MARS (Friedman 1988). I present an example that predicts future values of the Mackey-Glass differential delay equation.