Showing papers on "Basis function published in 1999"

Nonlinear spatial normalization using basis functions

Abstract: We describe a comprehensive framework for performing rapid and automatic nonlabel-based nonlinear spatial normalizations. The approach adopted minimizes the residual squared difference between an image and a template of the same modality. In order to reduce the number of parameters to be fitted, the nonlinear warps are described by a linear combination of low spatial frequency basis functions. The objective is to determine the optimum coefficients fur each of the bases by minimizing the sum of squared differences between the image and template, while simultaneously maximizing the smoothness of the transformation using a maximum a posteriori (MAP) approach. Most MAT approaches assume that the variance associated with each voxel is already known and that there is no covariance between neighboring voxels. The approach described here attempts to estimate this variance from the data, and also corrects fur the correlations between neighboring voxels. This makes the same approach suitable for the spatial normalization of both high-quality magnetic resonance images, and low-resolution noisy positron emission tomography images. A fast algorithm has been developed that utilizes Taylor's theorem and the separable nature of the basis functions, meaning that most of the nonlinear spatial variability between images can be automatically corrected within a few minutes. Hum. Brain Mapping 7:254-266, 1999. (C) 1999 Wiley-Liss, Inc.

Low-order scaling local electron correlation methods. I. Linear scaling local MP2

Abstract: A new implementation of local second-order Mo/ller-Plesset perturbation theory (LMP2) is presented for which asymptotically all computational resources (CPU, memory, and disk) scale only linearly with the molecular size. This is achieved by (i) using orbital domains for each electron pair that are independent of molecular size; (ii) classifying the pairs according to a distance criterion and neglecting very distant pairs; (iii) treating distant pairs by a multipole approximation, and (iv) using efficient prescreening algorithms in the integral transformation. The errors caused by the various approximations are negligible. LMP2 calculations on molecules including up to 500 correlated electrons and over 1500 basis functions in C1 symmetry are reported, all carried out on a single low-cost personal computer.

Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements

Abstract: New vector finite elements are proposed for electromagnetics. The new elements are triangular or tetrahedral edge elements (tangential vector elements) of arbitrary polynomial order. They are hierarchal, so that different orders can be used together in the same mesh and p-adaption is possible. They provide separate representation of the gradient and rotational parts of the vector field. Explicit formulas are presented for generating the basis functions to arbitrary order. The basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning. Matrix assembly for the frequency-domain curl-curl equation is conveniently carried out by means of universal matrices. Application of the new elements to the solution of a parallel-plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient, for tetrahedral elements to order 4. In particular, the full-order elements have only the same asymptotic convergence rate as elements with a reduced gradient space (such as the Whitney element). However, further tests reveal that the optimum balance of the gradient and rotational components is problem-dependent.

Linear scaling second-order Moller–Plesset theory in the atomic orbital basis for large molecular systems

Abstract: We have used Almlof and Haser’s Laplace transform idea to eliminate the energy denominator in second-order perturbation theory (MP2) and obtain an energy expression in the atomic orbital basis. We show that the asymptotic computational cost of this method scales quadratically with molecular size. We then define atomic orbital domains such that selective pairwise interactions can be neglected using well-defined thresholding criteria based on the power law decay properties of the long-range contributions. For large molecules, our scheme yields linear scaling computational cost as a function of molecular size. The errors can be controlled in a precise manner and our method reproduces canonical MP2 energies. We present benchmark calculations of polyglycine chains and water clusters containing up to 3040 basis functions.

Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives

Abstract: The authors develop a theory characterizing optimal stopping times for discrete-time ergodic Markov processes with discounted rewards. The theory differs from prior work by its view of per-stage and terminal reward functions as elements of a certain Hilbert space. In addition to a streamlined analysis establishing existence and uniqueness of a solution to Bellman's equation, this approach provides an elegant framework for the study of approximate solutions. In particular, the authors propose a stochastic approximation algorithm that tunes weights of a linear combination of basis functions in order to approximate a value function. They prove that this algorithm converges (almost surely) and that the limit of convergence has some desirable properties. The utility of the approximation method is illustrated via a computational case study involving the pricing of a path dependent financial derivative security that gives rise to an optimal stopping problem with a 100-dimensional state space.

Probabilistic framework for the adaptation and comparison of image codes

Abstract: We apply a Bayesian method for inferring an optimal basis to the problem of finding efficient image codes for natural scenes. The basis functions learned by the algorithm are oriented and localized in both space and frequency, bearing a resemblance to two-dimensional Gabor functions, and increasing the number of basis functions results in a greater sampling density in position, orientation, and scale. These properties also resemble the spatial receptive fields of neurons in the primary visual cortex of mammals, suggesting that the receptive-field structure of these neurons can be accounted for by a general efficient coding principle. The probabilistic framework provides a method for comparing the coding efficiency of different bases objectively by calculating their probability given the observed data or by measuring the entropy of the basis function coefficients. The learned bases are shown to have better coding efficiency than traditional Fourier and wavelet bases. This framework also provides a Bayesian solution to the problems of image denoising and filling in of missing pixels. We demonstrate that the results obtained by applying the learned bases to these problems are improved over those obtained with traditional techniques.

Loop-star decomposition of basis functions in the discretization of the EFIE

Abstract: A general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder. The proposed approach brings into correspondence these two parts with two scalar functions and generates the known loop and star basis functions. The completeness of the loop-star basis is discussed, employing the presented scheme; the issue of the irrotational property of the nonsolenoidal functions is addressed.

Generating efficient dynamical models for microelectromechanical systems from a few finite-element simulation runs

Abstract: In this paper, we demonstrate how efficient low-order dynamical models for micromechanical devices can be constructed using data from a few runs of fully meshed but slow numerical models such as those created by the finite-element method (FEM). These reduced-order macromodels are generated by extracting global basis functions from the fully meshed model runs in order to parameterize solutions with far fewer degrees of freedom. The macromodels may be used for subsequent simulations of the time-dependent behavior of nonlinear devices in order to rapidly explore the design space of the device. As an example, the method is used to capture the behavior of a pressure sensor based on the pull-in time of an electrostatically actuated microbeam, including the effects of squeeze-film damping due to ambient air under the beam. Results show that the reduced-order model decreases simulation time by at least a factor of 37 with less than 2% error. More complicated simulation problems show significantly higher speedup factors. The simulations also show good agreement with experimental data.

Optimization of Gaussian basis sets for density-functional calculations

Abstract: We introduce a scheme for the optimization of Gaussian basis sets for use in density-functional calculations. It is applicable to both all-electron and pseudopotential methodologies. In contrast to earlier approaches, the number of primitive Gaussians (exponents) used to define the basis functions is not fixed but adjusted, based on a total-energy criterion. Furthermore, all basis functions share the same set of exponents. The numerical results for the scaling of the shortest-range Gaussian exponent as a function of the nuclear charge are explained by analytical derivations. We have generated all-electron basis sets for H, B through F, Al, Si, Mn, and Cu. Our results show that they efficiently and accurately reproduce structural properties and binding energies for a variety of clusters and molecules for both local and gradient-corrected density functionals.

High-level canonical piecewise linear representation using a simplicial partition

Abstract: In this work, we propose a set of high-level canonical piecewise linear (HL-CPWL) functions to form a representation basis for the set of piecewise linear functions f: D/spl rarr/R/sup 1/ defined over a simplicial partition of a rectangular compact set D in R/sup n/. In consequence, the representation proposed uses the minimum number of parameters. The basis functions are obtained recursively by multiple compositions of a unique generating function /spl gamma/, resulting in several types of nested absolute-value functions. It is shown that the representation in a domain in R/sup n/ requires functions up to nesting level n. As a consequence of the choice of the basis functions, an efficient numerical method for the resolution of the parameters of the high-level (HL) canonical representation results. Finally, an application to the approximation of continuous functions is shown.

Robust radial basis function neural networks

Abstract: Function approximation has been found in many applications. The radial basis function (RBF) network is one approach which has shown a great promise in this sort of problems because of its faster learning capacity. A traditional RBF network takes Gaussian functions as its basis functions and adopts the least-squares criterion as the objective function, However, it still suffers from two major problems. First, it is difficult to use Gaussian functions to approximate constant values. If a function has nearly constant values in some intervals, the RBF network will be found inefficient in approximating these values. Second, when the training patterns incur a large error, the network will interpolate these training patterns incorrectly. In order to cope with these problems, an RBF network is proposed in this paper which is based on sequences of sigmoidal functions and a robust objective function. The former replaces the Gaussian functions as the basis function of the network so that constant-valued functions can be approximated accurately by an RBF network, while the latter is used to restrain the influence of large errors. Compared with traditional RBF networks, the proposed network demonstrates the following advantages: (1) better capability of approximation to underlying functions; (2) faster learning speed; (3) better size of network; (4) high robustness to outliers.

Fixed point analysis of frequency to instantaneous frequency mapping for accurate estimation of F0 and periodicity

Abstract: An accurate fundamental frequency (F0) estimation method for non-stationary, speech-like sounds is proposed based on the differential properties of the instantaneous frequencies of two sets of filter outputs. A specific type of fixed points of mapping from the filter center frequency to the output instantaneous frequency provides frequencies of the constituent sinusoidal components of the input signal. When the filter is made from an isometric Gabor function convoluted with a cardinal B-spline basis function, the differential properties at the fixed points provide practical estimates of the carrier-to-noise ratio of the corresponding components. These estimates are used to select the fundamental component and to integrate the F0 information distributed among the other harmonic components.

A simplified density matrix minimization for linear scaling self-consistent field theory

Abstract: A simplified version of the Li, Nunes and Vanderbilt [Phys Rev B 47, 10891 (1993)] and Daw [Phys Rev B 47, 10895 (1993)] density matrix minimization is introduced that requires four fewer matrix multiplies per minimization step relative to previous formulations The simplified method also exhibits superior convergence properties, such that the bulk of the work may be shifted to the quadratically convergent McWeeny purification, which brings the density matrix to idempotency Both orthogonal and nonorthogonal versions are derived The AINV algorithm of Benzi, Meyer, and Tůma [SIAM J Sci Comp 17, 1135 (1996)] is introduced to linear scaling electronic structure theory, and found to be essential in transformations between orthogonal and nonorthogonal representations These methods have been developed with an atom-blocked sparse matrix algebra that achieves sustained megafloating point operations per second rates as high as 50% of theoretical, and implemented in the MondoSCF suite of linear scaling SCF programs For the first time, linear scaling Hartree–Fock theory is demonstrated with three-dimensional systems, including water clusters and estane polymers The nonorthogonal minimization is shown to be uncompetitive with minimization in an orthonormal representation An early onset of linear scaling is found for both minimal and double zeta basis sets, and crossovers with a highly optimized eigensolver are achieved Calculations with up to 6000 basis functions are reported The scaling of errors with system size is investigated for various levels of approximation

Fast computation, rotation, and comparison of low resolution spherical harmonic molecular surfaces

Abstract: A procedure that rapidly generates an approximate parametric representation of macromolecular surface shapes is described. The parametrization is expressed as an expansion of real spherical harmonic basis functions. The advantage of using a parametric representation is that a pair of surfaces can be matched by using a quasi-Newton algorithm to minimize a suitably chosen objective function. Spherical harmonics are a natural and convenient choice of basis function when the task is one of search in a rotational search space. In particular, rotations of a molecular surface can be simulated by rotating only the harmonic expansion coefficients. This rotational property is applied for the first time to the 3-dimensional molecular similarity problem in which a pair of similar macromolecular surfaces are to be maximally superposed. The method is demonstrated with the superposition of antibody heavy chain variable domains. Special attention is given to computational efficiency. The spherical harmonic expansion coefficients are determined using fast surface sampling and integration schemes based on the tessellation of a regular icosahedron. Low resolution surfaces can be generated and displayed in under 10 s and a pair of surfaces can be maximally superposed in under 3 s on a contemporary workstation. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 383–395, 1999

Signals induced in semiconductor gamma-ray imaging detectors

Abstract: The signal induced in a readout circuit connected to a pixel electrode in a semiconductor gamma-ray imaging array is calculated by solving the Laplace equation. Two approaches are presented that use Green functions in solving the boundary value problem: decomposition into basis functions, and construction of an infinite series of image charges. Another approach is developed based on the Ramo–Shockley theorem, which makes use of weighting potentials. These potentials may be readily calculated in three dimensions using a Fourier-transform propagation technique. An analytic solution is found for the special two-dimensional case of a strip detector. Experiments on CdZnTe square-pixel test structures using alpha radiation confirm the expected trends in pulse shape as a function of pixel size. Signals observed simultaneously on adjacent pixels also follow the predicted division of currents. Trends with pixel size are also confirmed in the shape of pulse-height spectra taken using a 99mTc source.

Resonant modes of circular microstrip patches in multilayered substrates

Abstract: Galerkin's method in the Hankel transform domain (HTD) is used for computing the resonant frequencies, quality factors, and radiation patterns of the resonant modes of circular microstrip patches. The patches are assumed to be embedded in multilayered dielectric substrates. In this paper, the dyadic Green's function of the problem in the HTD is determined in terms of the two-dimensional Fourier transform of a related Green's function. New basis functions for the current density on the patches are introduced. It is shown that these new basis functions ensure a very quick convergence of the numerical results obtained via the Galerkin's method with respect to the number of basis functions. Also, a very efficient technique is presented, which makes possible the fast computation of the infinite integrals arising from the application of Galerkin's method in the HTD. At the end of this paper, the numerical results obtained are compared with previously published numerical results, with numerical results computed by means of the electromagnetic simulator "Ensemble", and with measurements carried out by the authors. Good agreement is found in all cases among all sets of results.

The slantlet transform

Abstract: The discrete wavelet transform (DWT) is usually carried out by filterbank iteration; however, for a fixed number of zero moments, this does not yield a discrete-time basis that is optimal with respect to time localization. This paper discusses the implementation and properties of an orthogonal DWT, with two zero moments and with improved time localization. The basis is not based on filterbank iteration; instead, different filters are used for each scale. For coarse scales, the support of the discrete-time basis functions approaches two thirds that of the corresponding functions obtained by filterbank iteration. This basis, which is a special case of a class of bases described by Alpert (1992, 1993), retains the octave-band characteristic and is piecewise linear (but discontinuous). Closed-form expressions for the filters are given, an efficient implementation of the transform is described, and improvement in a denoising example is shown. This basis, being piecewise linear, is reminiscent of the slant transform, to which it is compared.

Variable neural networks for adaptive control of nonlinear systems

Abstract: This paper is concerned with the adaptive control of continuous-time nonlinear dynamical systems using neural networks. A novel neural network architecture, referred to as a variable neural network, is proposed and shown to be useful in approximating the unknown nonlinearities of dynamical systems. In the variable neural networks, the number of basis functions can be either increased or decreased with time, according to specified design strategies, so that the network will not overfit or underfit the data set. Based on the Gaussian radial basis function (GRBF) variable neural network, an adaptive control scheme is presented. The location of the centers and the determination of the widths of the GRBFs in the variable neural network are analyzed to make a compromise between orthogonality and smoothness. The weight-adaptive laws developed using the Lyapunov synthesis approach guarantee the stability of the overall control scheme, even in the presence of modeling error(s). The tracking errors converge to the required accuracy through the adaptive control algorithm derived by combining the variable neural network and Lyapunov synthesis techniques. The operation of an adaptive control scheme using the variable neural network is demonstrated using two simulated examples.

Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach

Abstract: We present an approach to solid-state electronic-structure calculations based on the finite-element method. In this method, the basis functions are strictly local, piecewise polynomials. Because the basis is composed of polynomials, the method is completely general and its convergence can be controlled systematically. Because the basis functions are strictly local in real space, the method allows for variable resolution in real space; produces sparse, structured matrices, enabling the effective use of iterative solution methods; and is well suited to parallel implementation. The method thus combines the significant advantages of both real-space-grid and basis-oriented approaches and so promises to be particularly well suited for large, accurate {ital ab initio} calculations. We develop the theory of our approach in detail, discuss advantages and disadvantages, and report initial results, including electronic band structures and details of the convergence of the method. {copyright} {ital 1999} {ital The American Physical Society}

Learning Sparse Codes with a Mixture-of-Gaussians Prior

Abstract: We describe a method for learning an overcomplete set of basis functions for the purpose of modeling sparse structure in images. The sparsity of the basis function coefficients is modeled with a mixture-of-Gaussians distribution. One Gaussian captures nonactive coefficients with a small-variance distribution centered at zero, while one or more other Gaussians capture active coefficients with a large-variance distribution. We show that when the prior is in such a form, there exist efficient methods for learning the basis functions as well as the parameters of the prior. The performance of the algorithm is demonstrated on a number of test cases and also on natural images. The basis functions learned on natural images are similar to those obtained with other methods, but the sparse form of the coefficient distribution is much better described. Also, since the parameters of the prior are adapted to the data, no assumption about sparse structure in the images need be made a priori, rather it is learned from the data.

Linear programs for automatic accuracy control in regression.

Abstract: We have recently proposed a new approach to control the number of basis functions and the accuracy in support vector machines. The latter is transferred to a linear programming setting, which inherently enforces sparseness of the solution. The algorithm computes a nonlinear estimate in terms of kernel functions and an /spl epsiv/>0 with the property that at most a fraction /spl nu/ of the training set has an error exceeding /spl epsiv/. The algorithm is robust to local perturbations of these points' target values. We give an explicit formulation of the optimization equations needed to solve the linear program and point out which modifications of the standard optimization setting are necessary to take advantage of the particular structure of the equations in the regression case.

Wavelet-Galerkin scheme of time-dependent inhomogeneous electromagnetic problems

Abstract: A wavelet-Galerkin scheme based on the time-dependent Maxwell's equations is presented. Daubechies' wavelet with two vanishing wavelet moments is expanded for basis function in spatial domain, and Yee's leap-frog approach is applied. The shifted interpolation property of Daubechies' wavelet family leads to the simplified formulations for inhomogeneous media without the additional matrices for the integral or material operator. The storage effectiveness, execution time reduction, and accuracy of this scheme are demonstrated by calculating the resonant frequencies of the homogeneous and inhomogeneous cavities.

Accurate density-functional calculation of core-electron binding energies by a total-energy difference approach

Abstract: A procedure for calculating core-electron binding energies (CEBEs), based on a total-energy difference approach within Kohn–Sham density functional theory, was investigated. Ten functional combinations and several basis sets (including unscaled, scaled, and core-valence correlated functions) were studied using a database of reliable observed CEBEs. The functionals designed by Perdew and Wang (1986 exchange and 1991 correlation) were found to give the best performance with an average absolute deviation from experiment of 0.15 eV. The scaled basis sets did not perform satisfactorily, but it was found that the core-valence correlated cc-pCVTZ basis functions were an excellent alternative to the cc-pV5Z set as they provided equally accurate results and could be applied to larger molecules.

The basis set convergence of the Hartree–Fock energy for H2

Abstract: By using completely optimized basis functions, it is shown that the Hartree–Fock energy for the H2 molecule converges exponentially, both with respect to the number of basis functions of a given type and with respect to the highest angular momentum functions included. With a basis set including up to h-functions, a subnanohartree accuracy can be obtained. These results clearly suggest that estimation of the complete basis-set limit should employ different extrapolation schemes for the Hartree–Fock and correlation energies.

Elastic registration of medical images using radial basis functions with compact support

Abstract: We introduce radial basis functions with compact support for elastic registration of medical images. With these basis functions the influence of a landmark on the registration result is limited to a circle in 2D and, respectively, to a sphere in 3D. Therefore, the registration can be locally constrained which especially allows to deal with rather local changes in medical images due to, e.g., tumor resection. An important property of the used RBFs is that they are positive definite. Thus, the solvability of the resulting system of equations is always guaranteed. We demonstrate our approach for synthetic as well as for 2D and 3D tomographic images.

Proper Orthogonal Decomposition in Optimal Control of Fluids

Abstract: In this article, we present a reduced order modeling approach suitable for active control of fluid dynamical systems based on proper orthogonal decomposition (POD). The rationale behind the reduced order modeling is that numerical simulation of Navier-Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. We examine the possibility of obtaining reduced order models that reduce computational complexity associated with the Navier-Stokes equations while capturing the essential dynamics by using the POD. The POD allows extraction of certain optimal set of basis functions, perhaps few, from a computational or experimental data-base through an eigenvalue analysis. The solution is then obtained as a linear combination of these optimal set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations. We here use it in active control of fluid flows governed by the Navier-Stokes equations. We show that the resulting reduced order model can be very efficient for the computations of optimization and control problems in unsteady flows. Finally, implementational issues and numerical experiments are presented for simulations and optimal control of fluid flow through channels.