Showing papers on "Basis function published in 2003"

Radial Basis Functions: Theory and Implementations

Abstract: Preface 1. Introduction 2. Summary of methods and applications 3. General methods for approximation and interpolation 4. Radial basis function approximation on infinite grids 5. Radial basis functions on scattered data 6. Radial basis functions with compact support 7. Implementations 8. Least squares methods 9. Wavelet methods with radial basis functions 10. Further results and open problems Appendix Bibliography Index.

Optimized Slater-type basis sets for the elements 1-118.

Abstract: Seven different types of Slater type basis sets for the elements H (Z = 1) up to E118 (Z = 118), ranging from a double zeta valence quality up to a quadruple zeta valence quality, are tested in their performance in neutral atomic and diatomic oxide calculations. The exponents of the Slater type functions are optimized for the use in (scalar relativistic) zeroth-order regular approximated (ZORA) equations. Atomic tests reveal that, on average, the absolute basis set error of 0.03 kcal/mol in the density functional calculation of the valence spinor energies of the neutral atoms with the largest all electron basis set of quadruple zeta quality is lower than the average absolute difference of 0.16 kcal/mol in these valence spinor energies if one compares the results of ZORA equation with those of the fully relativistic Dirac equation. This average absolute basis set error increases to about 1 kcal/mol for the all electron basis sets of triple zeta valence quality, and to approximately 4 kcal/mol for the all electron basis sets of double zeta quality. The molecular tests reveal that, on average, the calculated atomization energies of 118 neutral diatomic oxides MO, where the nuclear charge Z of M ranges from Z = 1-118, with the all electron basis sets of triple zeta quality with two polarization functions added are within 1-2 kcal/mol of the benchmark results with the much larger all electron basis sets, which are of quadruple zeta valence quality with four polarization functions added. The accuracy is reduced to about 4-5 kcal/mol if only one polarization function is used in the triple zeta basis sets, and further reduced to approximately 20 kcal/mol if the all electron basis sets of double zeta quality are used. The inclusion of g-type STOs to the large benchmark basis sets had an effect of less than 1 kcal/mol in the calculation of the atomization energies of the group 2 and group 14 diatomic oxides. The basis sets that are optimized for calculations using the frozen core approximation (frozen core basis sets) have a restricted basis set in the core region compared to the all electron basis sets. On average, the use of these frozen core basis sets give atomic basis set errors that are approximately twice as large as the corresponding all electron basis set errors and molecular atomization energies that are close to the corresponding all electron results. Only if spin-orbit coupling is included in the frozen core calculations larger errors are found, especially for the heavier elements, due to the additional approximation that is made that the basis functions are orthogonalized on scalar relativistic core orbitals.

Fast Marginal Likelihood Maximisation for Sparse Bayesian Models

Abstract: The ‘sparse Bayesian’ modelling approach, as exemplified by the ‘relevance vector machine’, enables sparse classification and regression functions to be obtained by linearly-weighting a small number of fixed basis functions from a large dictionary of potential candidates Such a model conveys a number of advantages over the related and very popular ‘support vector machine’, but the necessary ‘training’ procedure — optimisation of the marginal likelihood function — is typically much slower We describe a new and highly accelerated algorithm which exploits recently-elucidated properties of the marginal likelihood function to enable maximisation via a principled and efficient sequential addition and deletion of candidate basis functions

Effectiveness of Diffuse Basis Functions for Calculating Relative Energies by Density Functional Theory

Abstract: The addition of diffuse functions to a double-ζ basis set is shown to be more important than increasing to a triple-ζ basis when calculating reaction energies, reaction barrier heights, and conformational energies with density functional theory, in particular with the modified Perdew−Wang density functional. It is shown that diffuse basis functions are vital to describe the relative energies between reactants, products, and transition states in isogyric reactions, and they provide enormous improvement in accuracy for conformational equilibria, using 1, 2-ethanediol and butadiene as examples. As a byproduct of the present study, we present a one-parameter hybrid density functional method optimized for sugars and sugar-like molecules; this is called MPW1S.

The Linear Programming Approach to Approximate Dynamic Programming

Abstract: The curse of dimensionality gives rise to prohibitive computational requirements that render infeasible the exact solution of large-scale stochastic control problems. We study an efficient method based on linear programming for approximating solutions to such problems. The approach "fits" a linear combination of pre-selected basis functions to the dynamic programming cost-to-go function. We develop error bounds that offer performance guarantees and also guide the selection of both basis functions and "state-relevance weights" that influence quality of the approximation. Experimental results in the domain of queueing network control provide empirical support for the methodology.

Characteristic basis function method: A new technique for efficient solution of method of moments matrix equations

Abstract: In this paper, we introduce a novel approach for an efficient solution of matrix equations arising in the method of moments (MoM) formulation of electromagnetic scattering problems. This approach is based on the characteristic basis functions (CBFs), which are used to substantially reduce the matrix size because these bases are not bound by the conventional λ/20 domain discretization. As a result, it is possible to electrically solve large problems with much fewer unknowns than those needed when using conventional RWG basis functions. The accuracy and efficiency of the CBFs are demonstrated in a variety of scattering problems to illustrate the versatility of the approach. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 36: 95–100, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10685

Efficient solution algorithms for factored MDPs

Abstract: This paper addresses the problem of planning under uncertainty in large Markov Decision Processes (MDPs). Factored MDPs represent a complex state space using state variables and the transition model using a dynamic Bayesian network. This representation often allows an exponential reduction in the representation size of structured MDPs, but the complexity of exact solution algorithms for such MDPs can grow exponentially in the representation size. In this paper, we present two approximate solution algorithms that exploit structure in factored MDPs. Both use an approximate value function represented as a linear combination of basis functions, where each basis function involves only a small subset of the domain variables. A key contribution of this paper is that it shows how the basic operations of both algorithms can be performed efficiently in closed form, by exploiting both additive and context-specific structure in a factored MDP. A central element of our algorithms is a novel linear program decomposition technique, analogous to variable elimination in Bayesian networks, which reduces an exponentially large LP to a provably equivalent, polynomial-sized one. One algorithm uses approximate linear programming, and the second approximate dynamic programming. Our dynamic programming algorithm is novel in that it uses an approximation based on max-norm, a technique that more directly minimizes the terms that appear in error bounds for approximate MDP algorithms. We provide experimental results on problems with over 1040 states, demonstrating a promising indication of the scalability of our approach, and compare our algorithm to an existing state-of-the-art approach, showing, in some problems, exponential gains in computation time.

An efficient orbital transformation method for electronic structure calculations

Abstract: An efficient method for optimizing single-determinant wave functions of medium and large systems is presented. It is based on a minimization of the energy functional using a new set of variables to perform orbital transformations. With this method convergence of the wave function is guaranteed. Preconditioners with different computational cost and efficiency have been constructed. Depending on the preconditioner, the method needs a number of iterations that is very similar to the established diagonalization–DIIS approach, in cases where the latter converges well. Diagonalization of the Kohn–Sham matrix can be avoided and the sparsity of the overlap and Kohn–Sham matrix can be exploited. If sparsity is taken into account, the method scales as O(MN2), where M is the total number of basis functions and N is the number of occupied orbitals. The relative performance of the method is optimal for large systems that are described with high quality basis sets, and for which the density matrices are not yet sparse....

The adaptive bases algorithm for intensity-based nonrigid image registration

Abstract: Nonrigid registration of medical images is important for a number of applications such as the creation of population averages, atlas-based segmentation, or geometric correction of functional magnetic resonance imaging (IMRI) images to name a few. In recent years, a number of methods have been proposed to solve this problem, one class of which involves maximizing a mutual information (Ml)-based objective function over a regular grid of splines. This approach has produced good results but its computational complexity is proportional to the compliance of the transformation required to register the smallest structures in the image. Here, we propose a method that permits the spatial adaptation of the transformation's compliance. This spatial adaptation allows us to reduce the number of degrees of freedom in the overall transformation, thus speeding up the process and improving its convergence properties. To develop this method, we introduce several novelties: 1) we rely on radially symmetric basis functions rather than B-splines traditionally used to model the deformation field; 2) we propose a metric to identify regions that are poorly registered and over which the transformation needs to be improved; 3) we partition the global registration problem into several smaller ones; and 4) we introduce a new constraint scheme that allows us to produce transformations that are topologically correct. We compare the approach we propose to more traditional ones and show that our new algorithm compares favorably to those in current use.

Quantifying Generalization from Trial-by-Trial Behavior of Adaptive Systems that Learn with Basis Functions: Theory and Experiments in Human Motor Control

Abstract: During reaching movements, the brain's internal models map desired limb motion into predicted forces. When the forces in the task change, these models adapt. Adaptation is guided by generalization: errors in one movement influence prediction in other types of movement. If the mapping is accomplished with population coding, combining basis elements that encode different regions of movement space, then generalization can reveal the encoding of the basis elements. We present a theory that relates encoding to generalization using trial-by-trial changes in behavior during adaptation. We consider adaptation during reaching movements in various velocity-dependent force fields and quantify how errors generalize across direction. We find that the measurement of error is critical to the theory. A typical assumption in motor control is that error is the difference between a current trajectory and a desired trajectory (DJ) that does not change during adaptation. Under this assumption, in all force fields that we examined, including one in which force randomly changes from trial to trial, we found a bimodal generalization pattern, perhaps reflecting basis elements that encode direction bimodally. If the DJ was allowed to vary, bimodality was reduced or eliminated, but the generalization function accounted for nearly twice as much variance. We suggest, therefore, that basis elements representing the internal model of dynamics are sensitive to limb velocity with bimodal tuning; however, it is also possible that during adaptation the error metric itself adapts, which affects the implied shape of the basis elements.

New Cartesian grid methods for interface problems using the finite element formulation

Abstract: New finite element methods based on Cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients. The triangulations in these methods do not need to fit the interfaces. The basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately. Both non-conforming and conforming finite element spaces are considered. Corresponding interpolation functions are proved to be second order accurate in the maximum norm. The conforming finite element method has been shown to be convergent. With Cartesian triangulations, these new methods can be used as finite difference methods. Numerical examples are provided to support the methods and the theoretical analysis.

Shapelets: I. a method for image analysis

Abstract: We present a new method for the analysis of images, a fundamental task in observational astronomy. It is based on the linear decomposition of each object in the image into a series of localized basis functions of different shapes, which we call ‘shapelets’. A particularly useful set of complete and orthonormal shapelets is that consisting of weighted Hermite polynomials, which correspond to perturbations around a circular Gaussian. They are also the eigenstates of the two-dimensional quantum harmonic oscillator, and thus allow us to use the powerful formalism developed for this problem. One of their special properties is their invariance under Fourier transforms (up to a rescaling), leading to an analytic form for convolutions. The generator of linear transformations such as translations, rotations, shears and dilatations can be written as simple combinations of raising and lowering operators. We derive analytic expressions for practical quantities, such as the centroid (astrometry), flux (photometry) and radius of the object, in terms of its shapelet coefficients. We also construct polar basis functions which are eigenstates of the angular momentum operator, and thus have simple properties under rotations. As an example, we apply the method to Hubble Space Telescope images, and show that the small number of shapelet coefficients required to represent galaxy images lead to compression factors of about 40 to 90. We discuss applications of shapelets for the archival of large photometric surveys, for weak and strong gravitational lensing and for image deprojection.

A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions

Abstract: We propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed along a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate the sets starting from the coarsest level. We interpolate a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. An original point set and its coarse-to-fine hierarchy of interpolated sets is presented. According to our numerical experiments, the method is essentially faster than the state-of-the-art scattered data approximation with globally supported RBFs (Carr et al., 2001) and much simpler to implement.

A maximum likelihood approach to single-channel source separation

Abstract: This paper presents a new technique for achieving blind signal separation when given only a single channel recording. The main concept is based on exploiting a priori sets of time-domain basis functions learned by independent component analysis (ICA) to the separation of mixed source signals observed in a single channel. The inherent time structure of sound sources is reflected in the ICA basis functions, which encode the sources in a statistically efficient manner. We derive a learning algorithm using a maximum likelihood approach given the observed single channel data and sets of basis functions. For each time point we infer the source parameters and their contribution factors. This inference is possible due to prior knowledge of the basis functions and the associated coefficient densities. A flexible model for density estimation allows accurate modeling of the observation and our experimental results exhibit a high level of separation performance for simulated mixtures as well as real environment recordings employing mixtures of two different sources.

Fresnelets: new multiresolution wavelet bases for digital holography

Abstract: We propose a construction of new wavelet-like bases that are well suited for the reconstruction and processing of optically generated Fresnel holograms recorded on CCD-arrays. The starting point is a wavelet basis of L/sub 2/ to which we apply a unitary Fresnel transform. The transformed basis functions are shift-invariant on a level-by-level basis but their multiresolution properties are governed by the special form that the dilation operator takes in the Fresnel domain. We derive a Heisenberg-like uncertainty relation that relates the localization of Fresnelets with that of their associated wavelet basis. According to this criterion, the optimal functions for digital hologram processing turn out to be Gabor (1948) functions, bringing together two separate aspects of the holography inventor's work. We give the explicit expression of orthogonal and semi-orthogonal Fresnelet bases corresponding to polynomial spline wavelets. This special choice of Fresnelets is motivated by their near-optimal localization properties and their approximation characteristics. We then present an efficient multiresolution Fresnel transform algorithm, the Fresnelet transform. This algorithm allows for the reconstruction (backpropagation) of complex scalar waves at several user-defined, wavelength-independent resolutions. Furthermore, when reconstructing numerical holograms, the subband decomposition of the Fresnelet transform naturally separates the image to reconstruct from the unwanted zero-order and twin image terms. This greatly facilitates their suppression. We show results of experiments carried out on both synthetic (simulated) data sets as well as on digitally acquired holograms.

A contracted basis-Lanczos calculation of vibrational levels of methane: Solving the Schrödinger equation in nine dimensions

Abstract: We present a contracted basis-iterative method for calculating numerically exact vibrational energy levels of methane (a 9D calculation). The basis functions we use are products of eigenfunctions of bend and stretch Hamiltonians obtained by freezing coordinates at equilibrium. The basis functions represent the desired wavefunctions well, yet are simple enough that matrix-vector products may be evaluated efficiently. We use Radau polyspherical coordinates. The bend functions are computed in a nondirect product finite basis representation [J. Chem. Phys. 118, 6956 (2003)] and the stretch functions are computed in a product potential optimized discrete variable (PODVR) basis. The memory required to store the bend basis is reduced by a factor of ten by storing it on a compacted grid. The stretch basis is optimized by discarding PODVR functions with high potential energies. The size of the primitive basis is 33 billion. The size of the product contracted basis is six orders of magnitude smaller. Parity symmetr...

Wavelet theory demystified

Abstract: We revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory - including some new extensions for fractional orders n a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in the L/sub p/-sense and a sharper theorem stating that smoothness implies order.

Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDT

Abstract: The equation-of-motion (EOM) coupled cluster (CC) method with full inclusion of the connected triple excitations for ionization energies has been formulated and implemented. Using proper factorization of the three- and four-body parts of the effective Hamiltonian, an efficient computational procedure has been proposed for IP-EOM-CCSDT which at the EOM level requires no-higher-than nocc3nvir4 scaling. The method is calibrated by the evaluation of the valence vertical ionization potentials for CO, N2, and F2 molecules for several basis sets up to 160 basis functions. At the basis set limit, errors vary from 0.0 to 0.2 eV, compared to “experimental” vertical ionization potentials.

Loop star basis functions and a robust preconditioner for EFIE scattering problems

Abstract: An electric field integral equation (EFIE) formulation using the loop-star basis functions has been developed for modeling plane wave scattering from perfect conducting objects. A stability analysis at the DC limit shows that the use of the Rao-Wilton-Glisson (RWG) basis functions results in a singular matrix operator. However, the use of the loop-star basis functions results in a well-conditioned matrix. Moreover, a preconditioner constructed from a two-step process, based on near interactions and an incomplete factorization with a heuristic drop strategy, has been proposed in conjunction with the conjugate gradient method to solve the resulting matrix equation. The approach is shown to be effective for resolving both the low frequency instability and the bad conditioning of the EFIE method. The computational complexity of the proposed approach is shown to be O(N/sup 2/).

Non-Born–Oppenheimer calculations of atoms and molecules

Abstract: We review a recent development in high-accuracy non-Born–Oppenheimer calculations of atomic and molecular systems in a basis of explicitly correlated Gaussian functions. Much of the recent progress in this area is due to the derivation and implementation of analytical gradients of the energy functional with respect to variational linear and non-linear parameters of the basis functions. This method has been used to obtain atomic and molecular ground and excited state energies and the corresponding wave functions with accuracy that exceeds previous calculations. Further, we have performed the first calculations of non-linear electrical properties of molecules without the Born–Oppenheimer approximation for systems with more than one electron. The results for the dipole moments of such systems as HD and LiH agree very well with experiment. After reviewing our non-Born–Oppenheimer results we will discuss ways this method can be extended to deal with larger molecular systems with and without an external perturbation.

Evaluation of two-loop self-energy basis integrals using differential equations

Abstract: I study the Feynman integrals needed to compute two-loop self-energy functions for general masses and external momenta. A convenient basis for these functions consists of the four integrals obtained at the end of Tarasov's recurrence relation algorithm. The basis functions are modified here to include one-loop and two-loop counterterms to render them finite; this simplifies the presentation of results in practical applications. I find the derivatives of these basis functions with respect to all squared-mass arguments, the renormalization scale, and the external momentum invariant, and express the results algebraically in terms of the basis. This allows all necessary two-loop self-energy integrals to be efficiently computed numerically using the differential equation in the external momentum invariant. I also use the differential equations method to derive analytic forms for various special cases, including a four-propagator integral with three distinct nonzero masses.

Efficient analysis of a class of microstrip antennas using the characteristic basis function method (CBFM)

Abstract: This paper presents a novel approach for the efficient solution of a class of microstrip antennas using the newly introduced characteristic basis functions (CBFs) in conjunction with the method of moments (MoM). The CBFs are special types of high-level basis functions, defined over domains that encompass a relatively large number of conventional subdomain basis functions, for example, triangular patches or rooftops. The advantages of applying the CBF method (CBFM) are illustrated by several representative examples, and the accuracy as well as the computation time are compared to those of conventional direct computation. It is demonstrated that the use of CBFs can result in significant savings in computation time, with little or no compromise in the accuracy of the solution. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 39: 456–464, 2003

On the construction and training of reformulated radial basis function neural networks

Abstract: Presents a systematic approach for constructing reformulated radial basis function (RBF) neural networks, which was developed to facilitate their training by supervised learning algorithms based on gradient descent. This approach reduces the construction of radial basis function models to the selection of admissible generator functions. The selection of generator functions relies on the concept of the blind spot, which is introduced in the paper. The paper also introduces a new family of reformulated radial basis function neural networks, which are referred to as cosine radial basis functions. Cosine radial basis functions are constructed by linear generator functions of a special form and their use as similarity measures in radial basis function models is justified by their geometric interpretation. A set of experiments on a variety of datasets indicate that cosine radial basis functions outperform considerably conventional radial basis function neural networks with Gaussian radial basis functions. Cosine radial basis functions are also strong competitors to existing reformulated radial basis function models trained by gradient descent and feedforward neural networks with sigmoid hidden units.

Numerical solution of elliptic partial differential equation by growing radial basis function neural networks

Abstract: In this paper a neural network for solving partial differential equations (PDE) is described The activation functions of the hidden nodes are the radial basis functions (RBF) whose parameters are learnt by a two-stage gradient descent strategy A new growing radial basis functions-node insertion strategy with different radial basis functions is used in order to improve the net performances The learning strategy is able to save computational time and memory space because of the selective growing of nodes whose activation functions consist of different radial basis functions An analysis of the learning capabilities and a comparison of the net performances with other approaches have been performed It is shown that the resulting network improves the approximation results