Showing papers on "Basis function published in 1997"
TL;DR: These deviations from linearity provide a potential explanation for the weak forms of non-linearity observed in the response properties of cortical simple cells, and they further make predictions about the expected interactions among units in response to naturalistic stimuli.
3,840 citations
TL;DR: In this paper, the auxilliary basis sets for the atoms H to At were optimized for an efficient treatment of molecular electronic Coulomb interactions for molecules with up to 300 atoms and 2500 basis functions.
Abstract: We present auxilliary basis sets for the atoms H to At – excluding the Lanthanides – optimized for an efficient treatment of molecular electronic Coulomb interactions. For atoms beyond Kr our approach is based on effective core potentials to describe core electrons. The approximate representation of the electron density in terms of the auxilliary basis has virtually no effect on computed structures and affects the energy by less than 10−4 a.u. per atom. Efficiency is demonstrated in applications for molecules with up to 300 atoms and 2500 basis functions.
3,035 citations
TL;DR: In this article, a linear scaling, fully self-consistent density-functional method for performing first-principles calculations on systems with a large number of atoms, using standard norm-conserving pseudopotentials and flexible linear combinations of atomic orbitals (LCAO) basis sets, was implemented.
Abstract: We have implemented a linear scaling, fully self-consistent density-functional method for performing first-principles calculations on systems with a large number of atoms, using standard norm-conserving pseudopotentials and flexible linear combinations of atomic orbitals (LCAO) basis sets. Exchange and correlation are treated within the local-spin-density or gradient-corrected approximations. The basis functions and the electron density are projected on a real-space grid in order to calculate the Hartree and exchange–correlation potentials and matrix elements. We substitute the customary diagonalization procedure by the minimization of a modified energy functional, which gives orthogonal wave functions and the same energy and density as the Kohn–Sham energy functional, without the need of an explicit orthogonalization. The additional restriction to a finite range for the electron wave functions allows the computational effort (time and memory) to increase only linearly with the size of the system. Forces and stresses are also calculated efficiently and accurately, allowing structural relaxation and molecular dynamics simulations. We present test calculations beginning with small molecules and ending with a piece of DNA. Using double-z, polarized bases, geometries within 1% of experiments are obtained. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 453–461, 1997
1,383 citations
TL;DR: In this article, a density functional theory-based algorithm for periodic and non-periodic ab initio calculations is presented, which uses pseudopotentials in order to integrate out the core electrons from the problem.
Abstract: A density functional theory-based algorithm for periodic and non-periodic ab initio calculations is presented. This scheme uses pseudopotentials in order to integrate out the core electrons from the problem. The valence pseudo-wavefunctions are expanded in Gaussian-type orbitals and the density is represented in a plane wave auxiliary basis. The Gaussian basis functions make it possible to use the efficient analytical integration schemes and screening algorithms of quantum chemistry. Novel recursion relations are developed for the calculation of the matrix elements of the density-dependent Kohn-Sham self-consistent potential. At the same time the use of a plane wave basis for the electron density permits efficient calculation of the Hartree energy using fast Fourier transforms, thus circumventing one of the major bottlenecks of standard Gaussian based calculations. Furthermore, this algorithm avoids the fitting procedures that go along with intermediate basis sets for the charge density. The performance a...
1,150 citations
TL;DR: This hypothesis is compared with other approaches that are commonly used to model population codes, such as computational maps and vectorial representations, which can fully account for the responses of parietal neurons, and they are computationally less efficient for nonlinear transformations.
Abstract: Sensorimotor transformations are nonlinear mappings of sensory inputs to motor responses. We explore here the possibility that the responses of single neurons in the parietal cortex serve as basis functions for these transformations. Basis function decomposition is a general method for approximating nonlinear functions that is computationally efficient and well suited for adaptive modification. In particular, the responses of single parietal neurons can be approximated by the product of a Gaussian function of retinal location and a sigmoid function of eye position, called a gain field. A large set of such functions forms a basis set that can be used to perform an arbitrary motor response through a direct projection. We compare this hypothesis with other approaches that are commonly used to model population codes, such as computational maps and vectorial representations. Neither of these alternatives can fully account for the responses of parietal neurons, and they are computationally less efficient for nonlinear transformations. Basis functions also have the advantage of not depending on any coordinate system or reference frame. As a consequence, the position of an object can be represented in multiple reference frames simultaneously, a property consistent with the behavior of hemineglect patients with lesions in the parietal cortex.
438 citations
TL;DR: In this paper, various time-domain finite-element methods for the simulation of transient electromagnetic wave phenomena are discussed, including nodal and edge/facet element basis functions, along with the numerical stability properties of the different methods.
Abstract: Various time-domain finite-element methods for the simulation of transient electromagnetic wave phenomena are discussed. Detailed descriptions of test/trial spaces, explicit and implicit formulations, nodal and edge/facet element basis functions are given, along with the numerical stability properties of the different methods. The advantages and disadvantages of mass lumping are examined. Finally, the various formulations are compared on the basis of their numerical dispersion performance.
419 citations
TL;DR: A mathematical description of cardiac anatomy is presented for use with finite element models of the electrical activation and mechanical function of the heart, and parameters defined at the nodes of the finite element mesh are fitted.
Abstract: A mathematical description of cardiac anatomy is presented for use with finite element models of the electrical activation and mechanical function of the heart. The geometry of the heart is given in terms of prolate spheroidal coordinates defined at the nodes of a finite element mesh and interpolated within elements by a combination of linear Lagrange and cubic Hermite basis functions. Cardiac microstructure is assumed to have three axes of symmetry: one aligned with the muscle fiber orientation (the fiber axis); a second set orthogonal to the fiber direction and lying in the newly identified myocardial sheet plane (the sheet axis); and a third set orthogonal to the first two, in the sheet-normal direction. The geometry, fiber-axis direction, and sheet-axis direction of a dog heart are fitted with parameters defined at the nodes of the finite element mesh. The fiber and sheet orientation parameters are defined with respect to the ventricular geometry such that 1) they can be applied to any heart of known dimensions, and 2) they can be used for the same heart at various states of deformation, as is needed, for example, in continuum models of ventricular contraction.
296 citations
TL;DR: Simulations and experimental evaluations show that, with little effort, a high-performance motion system can be obtained with this approach, and an emphasis is placed on robustness.
Abstract: The design and realization of an online learning motion controller for a linear motor is presented, and its usefulness is evaluated. The controller consists of two components: (1) a model-based feedback component, and (2) a learning feedforward component. The feedback component is designed on the basis of a simple second-order linear model, which is known to have structural errors. In the design, an emphasis is placed on robustness. The learning feedforward component is a neural-network-based controller, comprised of a one-hidden-layer structure with second-order B-spline basis functions. Simulations and experimental evaluations show that, with little effort, a high-performance motion system can be obtained with this approach.
247 citations
TL;DR: In this article, a new method for large eddy simulations is described and evaluated, where the primary modeled quantity is the unfiltered velocity field appearing in the definition of the subgrid-scale stress tensor.
Abstract: A new method for large eddy simulations is described and evaluated. In the proposed method the primary modeled quantity is the unfiltered velocity field appearing in the definition of the subgrid-scale stress tensor. An estimate of the unfiltered velocity is obtained by expanding the resolved large-scale velocity field to subgrid-scales two times smaller than the grid scale. The estimation procedure consists of two steps. The first step utilizes properties of a filtering operation and the representation of quantities in terms of basis functions such as Fourier polynomials. In the second step, the phases associated with the newly computed smaller scales are adjusted in order to correspond to the small-scale phases generated by nonlinear interactions of the large-scale field. The estimated velocity field is expressed entirely in terms of the known, resolved velocity field without any adjustable constants. The modeling procedure is evaluated in a priori analyses using direct numerical simulation results of c...
231 citations
TL;DR: A new technique is developed that allows identified objects in the image to move as rigid bodies, while the remainder smoothly deforms, valid in any dimension, subject to the choice of the basis function.
215 citations
TL;DR: In this paper, a hierarchical multipole method was developed for fast computation of the Coulomb matrix, and a linear scaling algorithm for calculation of the Fock matrix was demonstrated for a sequence of water clusters at the restricted Hartree-Fock/3-21G level of theory.
Abstract: Computation of the Fock matrix is currently the limiting factor in the application of Hartree-Fock and hybrid Hartree-Fock/density functional theories to larger systems. Computation of the Fock matrix is dominated by calculation of the Coulomb and exchange matrices. With conventional Gaussian-based methods, computation of the Fock matrix typically scales as ∼N2.7, where N is the number of basis functions. A hierarchical multipole method is developed for fast computation of the Coulomb matrix. This method, together with a recently described approach to computing the Hartree-Fock exchange matrix of insulators [J. Chem. Phys. 105, 2726 (1900)], leads to a linear scaling algorithm for calculation of the Fock matrix. Linear scaling computation the Fock matrix is demonstrated for a sequence of water clusters at the restricted Hartree-Fock/3-21G level of theory, and corresponding accuracies in converged total energies are shown to be comparable with those obtained from standard quantum chemistry programs. Restri...
TL;DR: In this paper, a new version of the filter diagonalization method of diagonalizing large real symmetric Hamiltonian matrices is presented, which produces a small set of adapted basis functions by applying the Chebyshev polynomial expansion of the Green's function on a generic initial vector χ. The small Hamiltonian, H and overlap, S, matrices would then be evaluated in this adapted basis and the corresponding generalized eigenvalue problem would be solved yielding the desired spectral information.
Abstract: A new version of the filter diagonalization method of diagonalizing large real symmetric Hamiltonian matrices is presented. Our previous version would first produce a small set of adapted basis functions by applying the Chebyshev polynomial expansion of the Green’s function on a generic initial vector χ. The small Hamiltonian, H, and overlap, S, matrices would then be evaluated in this adapted basis and the corresponding generalized eigenvalue problem would be solved yielding the desired spectral information. Here in analogy to a recent work by Wall and Neuhauser [J. Chem. Phys. 102, 8011 (1995)] H and S are computed directly using only the Chebyshev coefficients cn=〈χ|Tn(Ĥ)|χ〉, calculation of which requires a minimal storage if the Ĥ matrix is sparse. The expressions for H and S are analytically simple, computationally very inexpensive and stable. The method can be used to obtain all the eigenvalues of Ĥ using the same sequence {cn}. We present an application of the method to a realistic quantum dynamics...
TL;DR: In this paper, a new linear scaling method for Hartree-Fock exchange matrix is described, which employs a method numerically equivalent to standard direct SCF, and which does not enforce locality of the density matrix.
Abstract: A new linear scaling method for computation of the Cartesian Gaussian-based Hartree-Fock exchange matrix is described, which employs a method numerically equivalent to standard direct SCF, and which does not enforce locality of the density matrix. With a previously described method for computing the Coulomb matrix [J. Chem. Phys. 106, 5526 (1997)], linear scaling incremental Fock builds are demonstrated for the first time. Microhartree accuracy and linear scaling are achieved for restricted Hartree-Fock calculations on sequences of water clusters and polyglycine α-helices with the 3-21G and 6-31G basis sets. Eightfold speedups are found relative to our previous method. For systems with a small ionization potential, such as graphitic sheets, the method naturally reverts to the expected quadratic behavior. Also, benchmark 3-21G calculations attaining microhartree accuracy are reported for the P53 tetramerization monomer involving 698 atoms and 3836 basis functions.
TL;DR: In this paper, an adaptive spline method for smoothing is proposed that combines features from both regression spline and smoothing spline approaches, which can be applied to many multivariate function estimation problems.
Abstract: An adaptive spline method for smoothing is proposed that combines features from both regression spline and smoothing spline approaches. One of its advantages is the ability to vary the amount of smoothing in response to the inhomogeneous “curvature” of true functions at different locations. This method can be applied to many multivariate function estimation problems, which is illustrated by an application to smoothing temperature data on the globe. The method's performance in a simulation study is found to be comparable to the wavelet shrinkage methods proposed by Donoho and Johnstone. The problem of how to count the degrees of freedom for an adaptively chosen set of basis functions is addressed. This issue arises also in the MARS procedure proposed by Friedman and other adaptive regression spline procedures.
TL;DR: In this paper, fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the form = Lu+ Nf(u), where L and N are linear differential operators and f(u) is a nonlinear function are developed.
TL;DR: In this article, the magnetic field integral equation (MFIE) with the use of vector triangular basis functions has been developed, and the 1/R2 singular behavior is extracted from the integral, and closed-form expressions are given for the singular terms.
Abstract: Formulas that provide an efficient and reliable numerical evaluation of the magnetic field integral equation (MFIE) with the use of vector triangular basis functions are developed. The MFIE integrals for the three basis functions on a triangular facet are converted from three vector integrals to three scalar integrals, which are evaluated simultaneously. The 1/R2 singular behavior is extracted from the integral, and closed-form expressions are given for the singular terms. Consequently, the formulas are valid for all observation points and are suitable for general-purpose modeling. © 1997 John Wiley & Sons, Inc.
TL;DR: An algorithm is presented for calculating a suitable normalized B-spline representation for Powell-Sabin splines in which the basis functions are all positive, have local support and form a partition of unity.
TL;DR: In this paper, the authors investigated the relation between blip function basis sets and the plane-wave basis used in standard pseudopotential methods, derive criteria for the approximate equivalence of the two, and describe practical tests of these criteria.
Abstract: In the framework of a recently reported linear-scaling method for density-functional-pseudopotential calculations, we investigate the use of localized basis functions for such work. These basis functions (referred to as ``blip functions'') are centered on the points of a grid, and vanish exactly outside a limited domain surrounding each grid point. We analyze the relation between blip-function basis sets and the plane-wave basis used in standard pseudopotential methods, derive criteria for the approximate equivalence of the two, and describe practical tests of these criteria. Techniques are presented for using blip-function basis sets in linear-scaling calculations, and numerical tests of these techniques are reported for Si crystals using both local and nonlocal pseudopotentials. We find rapid convergence of the total energy to the values given by standard plane-wave calculations as the radius of the linear-scaling localized orbitals is increased.
TL;DR: In this paper, a generalization of Simon's exterior complex scaling technique for long-range potentials is proposed, based on the method of complex basis functions, which can be used for computing scattering amplitudes and cross sections.
Abstract: We examine finite basis set implementations of complex scaling procedures for computing scattering amplitudes and cross sections. While ordinary complex scaling, i.e., the technique of multiplying all interparticle distances in the Hamiltonian by a complex phase factor, is known to provide convergent cross-section expressions only for exponentially bounded potentials, we propose a generalization, based on Simon's exterior complex scaling technique, that works for long-range potentials as well. We establish an equivalence between a class of complex scaling transformations carried out on the time-independent Schr\"odinger equation and a procedure commonly referred to as the method of complex basis functions. The procedure is illustrated with a numerical example.
TL;DR: This paper details the general construction principle of the WEM to the 1D, 2D and 3D cases and addresses additional features such as symmetry, vanishing moments and minimal support of the wavelet functions in each particular dimension.
TL;DR: A procedure is developed for identification of probabilistic system uncertainty regions for a linear time-invariant system with unknown dynamics, on the basis of time sequences of input and output data, using an instrumental variable method for identification.
Abstract: A procedure is developed for identification of probabilistic system uncertainty regions for a linear time-invariant system with unknown dynamics, on the basis of time sequences of input and output data. The classical framework is handled in which the system output is contaminated by a realization of a stationary stochastic process. Given minor and verifiable prior information on the system and the noise process, frequency response, pulse response, and step response confidence regions are constructed by explicitly evaluating the bias and variance errors of a linear regression estimate. In the model parametrizations, use is made of general forms of basis functions. Conservatism of the uncertainty regions is limited by focusing on direct computational solutions rather than on closed-form expressions. Using an instrumental variable method for identification, the procedure is suitable also for input-output data obtained from closed-loop experiments.
TL;DR: In this paper, a localised spherical-wave basis set suitable for O(N ) total energy pseudopotential calculations is considered. But the basis set is conveniently truncated using a single parameter, the kinetic energy cutoff used with the plane-wave based basis.
TL;DR: In this paper, a generalized and more complete methodology for treating boundary conditions in the Differential Quadrature Method (DQM) is presented, which eliminates the deficiencies of the δ-type grid arrangement, which represents an approximation, by applying the boundary conditions exactly.
Abstract: A generalized and more complete methodology for treating boundary conditions in the Differential Quadrature Method (DQM) is presented. This improved approach eliminates the deficiencies of the δ-type grid arrangement, which represents an approximation, by applying the boundary conditions exactly. Two kinds of basis functions, Chebyshev and Lagrange, are used for concept demonstration. It is found that the new approach cures most deficiencies of the current DQM. © 1997 by John Wiley & Sons, Ltd.
Proceedings Article•
01 Dec 1997TL;DR: The utility of overcomplete representations on natural speech is demonstrated and it is shown that compared to the traditional Fourier basis the inferred representations potentially have much greater coding efficiency.
Abstract: We derive a learning algorithm for inferring an overcomplete basis by viewing it as probabilistic model of the observed data. Overcomplete bases allow for better approximation of the underlying statistical density. Using a Laplacian prior on the basis coefficients removes redundancy and leads to representations that are sparse and are a nonlinear function of the data. This can be viewed as a generalization of the technique of independent component analysis and provides a method for blind source separation of fewer mixtures than sources. We demonstrate the utility of overcomplete representations on natural speech and show that compared to the traditional Fourier basis the inferred representations potentially have much greater coding efficiency.
TL;DR: It is shown that the estimation of the expansion parameters is equivalent to estimating the second-order parameters of an unobservable FIR single-input-many-output (SIMO) process, which are directly computed (under some assumptions) from the observation data.
Abstract: Novel linear algorithms are proposed in this paper for estimating time-varying FIR systems, without resorting to higher order statistics. The proposed methods are applicable to systems where each time-varying tap coefficient can be described (with respect to time) as a linear combination of a finite number of basis functions. Examples of such channels include almost periodically varying ones (Fourier series description) or channels locally modeled by a truncated Taylor series or by a wavelet expansion. It is shown that the estimation of the expansion parameters is equivalent to estimating the second-order parameters of an unobservable FIR single-input-many-output (SIMO) process, which are directly computed (under some assumptions) from the observation data. By exploiting this equivalence, a number of different blind subspace methods are applicable, which have been originally developed in the context of time-invariant SIMO systems. Identifiability issues are investigated, and some illustrative simulations are presented.
16 Jul 1997
TL;DR: A new neural network structure is given for approximation of piecewise continuous functions of the sort that appear in friction, deadzone, backlash and other motion control actuator nonlinearities, which is ideal for motion control applications in robotics and other industrial systems.
Abstract: A new neural network (NN) structure is given for approximation of piecewise continuous (PC) functions of the sort that appear in friction, deadzone, backlash and other motion control actuator nonlinearities. The NN consists of neurons having a special class of nonsmooth activation functions termed 'jump approximation basis functions'. This 'jump approximation' NN plus a NN based on standard smooth sigmoidal activation functions can approximate any piecewise continuous function with discontinuities at a finite number of known points. Industrial motion device actuator nonlinearities are in this class of functions, therefore, the new NN structure is ideal for motion control applications in robotics and other industrial systems.
TL;DR: A relative-error statistic is developed which is only sensitive to calculation errors that could affect the inverse solution for source position, and insensitive to errors that only affect the solution for sources strength.
Abstract: A comprehensive review of factors affecting the accuracy of the boundary element method (BEM) for calculating surface potentials is presented. A relative-error statistic is developed which is only sensitive to calculation errors that could affect the inverse solution for source position, and insensitive to errors that only affect the solution for source strength. The factors considered in this paper are numerical approximations intrinsic to the BEM, such as constant-potential versus linear-potential basis functions and sharp-edged versus smooth-surfaced volumes; aspects of the volume conductor including the volume shape, density of surface elements, and element shape; source position and orientation; and effects of "refinements" in the numerical methods. The effects of these factors are considered in both smooth-shaped (spheres and spheroids) and sharp-edged (cubes) volume conductors. This represents the first attempt to assess the effects of many of these factors pertaining to the numerical methods commonly used in fields such as electrocardiography (ECG) and electroencephalography (EEG). Strategies for obtaining the most accurate solutions are presented.
TL;DR: In this paper, a multireference perturbation algorithm for ab initio electronic structure calculations, based on a generalized valence bond (GVB) reference system, a local version of second-order Mo/ller-Plesset perturbations theory (LMP2), and pseudospectral (PS) numerical methods, is presented.
Abstract: We describe a new multireference perturbation algorithm for ab initio electronic structure calculations, based on a generalized valence bond (GVB) reference system, a local version of second-order Mo/ller–Plesset perturbation theory (LMP2), and pseudospectral (PS) numerical methods. This PS-GVB-LMP2 algorithm is shown to have a computational scaling of approximately N3 with basis set size N, and is readily applicable to medium to large size molecules using workstations with relatively modest memory and disk storage. Furthermore, the PS-GVB-LMP2 method is applicable to an arbitrary molecule in an automated fashion (although specific protocols for resonance interactions must be incorporated) and hence constitutes a well-defined model chemistry, in contrast to some alternative multireference methodologies. A calculation on the alanine dipeptide using the cc-pVTZ(−f) basis set (338 basis functions total) is presented as an example. We then apply the method to the calculation of 36 conformational energy differ...
01 Nov 1997
TL;DR: In this paper, a hybrid learning approach for local model (LM) networks consisting of ARX local models and normalised gaussian basis functions is described. And a new nonlinear internal model control (IMC) scheme is proposed, based on this LM network model of the nonlinear plant, which has the important property that the controller can be derived analytically.
Abstract: Local model (LM) networks represent a nonlinear dynamical system by a set of locally valid submodels across the operating range. Training such feedforward structures involves the combined estimation of the submodel parameters and those of the interpolation or validity functions. The paper describes a new hybrid learning approach for LM networks comprising ARX local models and normalised gaussian basis functions. Singular value decomposition (SVD) is used to identify the local linear models in conjunction with quasiNewton optimisation for determining the centres and widths of the interpolation functions. A new nonlinear internal model control (IMC) scheme is proposed, based on this LM network model of the nonlinear plant, which has the important property that the controller can be derived analytically. Accuracy and stability issues for the nonlinear feedback control scheme are discussed. Simulation studies of a pH neutralisation process confirm the excellent nonlinear modelling properties of LM networks and illustrate the potential for setpoint tracking and disturbance rejection within an IMC framework.
TL;DR: The analysis, modeling and simulation of measured full-scale wind pressure and velocity data is addressed, followed by the simulation of pressure data through new static transformation techniques.