Showing papers on "Basis function published in 1996"

A Reassessment of the Groundwater Inverse Problem

Abstract: This paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms. Unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a (possibly infinite) basis function expansion with random coefficients. The unknown parameter function is related to the measurements used for estimation by a ''forward operator'' which describes the measurement process. In the particular case considered here, the parameter of interest is the large-scale log hydraulic conductivity, the measurements are point values of log conductivity and piezometric head, and the forward operator is derived from an upscaled groundwaterflow equation. The inverse algorithm seeks the ''most probable'' or maximum a posteriori estimate of the unknown parameter function. When the measurement errors and parameter function are Gaussian and independent, the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness-of- fit and prior terms. When the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate. This constraint tends to make the problem well- posed by limiting the range of admissible solutions. The Gaussian maximum a posteriori problem may be solved with variational methods, using functional generalizations of Gauss-Newton or gradient-based search techniques. Several popular groundwater inverse algorithms are either special cases of, or variants on, the functional maximum a posteriori algorithm. These algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use (linear versus nonlinear). The accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a Bayesian extension of the Cramer-Rao lower bound on the estimation error covariance. This bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements.

Simplified support vector decision rules

Abstract: A Support Vector Machine SVM is a uni versal learning machine whose decision sur face is parameterized by a set of support vec tors and by a set of corresponding weights An SVM is also characterized by a kernel function Choice of the kernel determines whether the resulting SVM is a polynomial classi er a two layer neural network a ra dial basis function machine or some other learning machine SVMs are currently considerably slower in test phase than other approaches with sim ilar generalization performance To address this we present a general method to signif icantly decrease the complexity of the deci sion rule obtained using an SVM The pro posed method computes an approximation to the decision rule in terms of a reduced set of vectors These reduced set vectors are not support vectors and can in some cases be computed analytically We give ex perimental results for three pattern recogni tion problems The results show that the method can decrease the computational com plexity of the decision rule by a factor of ten with no loss in generalization perfor mance making the SVM test speed com petitive with that of other methods Fur ther the method allows the generalization performance complexity trade o to be di rectly controlled The proposed method is not speci c to pattern recognition and can be applied to any problem where the Sup port Vector algorithm is used for example regression INTRODUCTION SUPPORT VECTOR MACHINES Consider a two class classi er for which the decision rule takes the form

The MIDI! basis set for quantum mechanical calculations of molecular geometries and partial charges

Abstract: We present a series of calculations designed to identify an economical basis set for geometry optimizations and partial charge calculations on medium-size molecules, including neutrals, cations, and anions, with special emphasis on functional groups that are important for biomolecules and drug design. A new combination of valence basis functions and polarization functions, called the MIDI! basis set, is identified as a good compromise of speed and accuracy, yielding excellent geometries and charge balances at a cost that is as affordable as possible for large molecules. The basis set is optimized for molecules containing H, C, N, O, F, P, S, and Cl. Although much smaller than the popular 6-31G* basis set, in direct comparisons it yields more accurate geometries and charges as judged by comparison to MP2/cc-pVDZ calculations.

A cellular texture basis function

Abstract: Solid texturing is a powerful way to add detail to the surface of rendered objects. Perlin’s “noise” is a 3D basis function used in some of the most dramatic and useful surface texture algorithms. We present a new basis function which complements Perlin noise, based on a partitioning of space into a random array of cells. We have used this new basis function to produce textured surfaces resembling flagstone-like tiled areas, organic crusty skin, crumpled paper, ice, rock, mountain ranges, and craters. The new basis function can be computed efficiently without the need for precalculation or table storage.

Semiempirical molecular orbital calculations with linear system size scaling

Abstract: Details are provided for the implementation of a density matrix divide‐and‐conquer approximation into the framework of molecular orbital theory on nonperiodic systems. Originally developed for density functional theory, the divide‐and‐conquer procedure is one of the most promising in a growing list of techniques that exhibit linear scaling with respect to the number of basis functions in the system. The key to linear scaling is the division of the electronic structure calculation into a series of calculations over a set of small, overlapping subsystems. A semiempirical molecular orbital program designed around the divide‐and‐conquer approach has been written and a number of tests are carried out on polyglycine structures in order to evaluate its performance. For the systems examined, linear scaling is indeed observed, and the accuracy of the calculations can be controlled quite readily by the manner in which the system is divided into its component subsystems. For very large structures, the expense associated with the computation of two‐center interactions will ultimately dominate the calculation, and quadratic scaling will become apparent. Techniques to linearize this aspect of the calculation are investigated and discussed.

A chemical potential equalization method for molecular simulations

Abstract: A formulation of the chemical potential (electronegativity) equalization principle is presented from the perspective of density‐functional theory. The resulting equations provide a linear‐response framework for describing the redistribution of electrons upon perturbation by an applied field. The method has two main advantages over existing electronegativity equalization and charge equilibration methods that allow extension to accurate molecular dynamics simulations. Firstly, the expansion of the energy is taken about the molecular ground state instead of the neutral atom ground states; hence, in the absence of an external field, the molecular charge distribution can be represented by static point charges and dipoles obtained from fitting to high‐level ab initio calculations without modification. Secondly, in the presence of applied fields or interactions with other molecules, the density response can be modeled accurately using basis functions. Inclusion of basis functions with dipolar or higher order mul...

Dynamic structure neural networks for stable adaptive control of nonlinear systems

Abstract: An adaptive control technique, using dynamic structure Gaussian radial basis function neural networks, that grow in time according to the location of the system's state in space is presented for the affine class of nonlinear systems having unknown or partially known dynamics. The method results in a network that is "economic" in terms of network size, for cases where the state spans only a small subset of state space, by utilizing less basis functions than would have been the case if basis functions were centered on discrete locations covering the whole, relevant region of state space. Additionally, the system is augmented with sliding control so as to ensure global stability if and when the state moves outside the region of state space spanned by the basis functions, and to ensure robustness to disturbances that arise due to the network inherent approximation errors and to the fact that for limiting the network size, a minimal number of basis functions are actually being used. Adaptation laws and sliding control gains that ensure system stability in a Lyapunov sense are presented, together with techniques for determining which basis functions are to form part of the network structure. The effectiveness of the method is demonstrated by experiment simulations.

On the Relationship between Generalization Error, Hypothesis Complexity, and Sample Complexity for Radial Basis Functions

Abstract: Feedforward networks together with their training algorithms are a class of regression techniques that can be used to learn to perform some task from a set of examples. The question of generalization of network performance from a finite training set to unseen data is clearly of crucial importance. In this article we first show that the generalization error can be decomposed into two terms: the approximation error, due to the insufficient representational capacity of a finite sized network, and the estimation error, due to insufficient information about the target function because of the finite number of samples. We then consider the problem of learning functions belonging to certain Sobolev spaces with gaussian radial basis functions. Using the above-mentioned decomposition we bound the generalization error in terms of the number of basis functions and number of examples. While the bound that we derive is specific for radial basis functions, a number of observations deriving from it apply to any approximation technique. Our result also sheds light on ways to choose an appropriate network architecture for a particular problem and the kinds of problems that can be effectively solved with finite resources, i.e., with a finite number of parameters and finite amounts of data.

Basis function networks for interpolation of local linear models

Abstract: In this paper, a new algorithm (LOLIMOT) for nonlinear dynamic system identification with local linear models is proposed. The input space is partitioned by a tree-construction algorithm. The local models are interpolated by overlapping local basis functions. The resulting structure is equivalent to a Sugeno-Takagi fuzzy system and a local model network and can therefore be interpreted correspondingly. The LOLIMOT algorithm is very simple, easy to implement, and fast. Moreover, this approach has the following appealing properties: it does not underlie the "curse of dimensionality", it reveals irrelevant inputs, it detects inputs that influence the output mainly in a linear way, and it applies robust local linear estimation schemes. The drawbacks are that only orthogonal cuts are performed and that the local estimation approach may lead to interpolation errors.

Numerical steady state analysis of electronic circuits driven by multi-tone signals

Abstract: Characteristics of analogue circuits such as intermodulation distortion and transfer characteristics can often be received from the steady state behavior. This paper presents a unified approach for the simulation of non-autonomous circuits with multi-tone excitation. The steady state is here regarded as the solution of a partial differential-algebraic equation. A suitable numerical method for its solution is a variational method with trigonometric basis functions. The Harmonic Balance technique based either on the multi-dimensional Fourier transformation or the Artificial Frequency Map technique can be interpreted as a special variant of this method.

Model Estimations Biased by Truncated Expansions: Possible Artifacts in Seismic Tomography

Abstract: In most linear imaging problems, where the model to be sought is expanded in a set of basis functions, it is common practice to truncate the set at a certain (arbitrary) level. The solution then depends on the chosen parameterization, and neglected basis functions may leak into the solution to produce artifacts in the retrieved model. An unbiased estimate of the coefficients of the true model may be obtained in the chosen finite basis set; here, a method to suppress leakage is illustrated on an example of global seismic tomography.

Accurate quantum‐chemical calculations: The use of Gaussian‐type geminal functions in the treatment of electron correlation

Abstract: We investigate augmenting conventional Gaussian‐type one‐electron orbital basis sets with two‐electron functions that have a Gaussian dependence on the interelectronic distance. We observe substantial improvements in calculated correlation energies for helium and neon atoms and for the water molecule. A feature of our approach is that there is no nonlinear optimization of the two‐electron basis function parameters at all.

On the efficiency of the orthogonal least squares training method for radial basis function networks

Abstract: The efficiency of the orthogonal least squares (OLS) method for training approximation networks is examined using the criterion of energy compaction. We show that the selection of basis vectors produced by the procedure is not the most compact when the approximation is performed using a nonorthogonal basis. Hence, the algorithm does not produce the smallest possible networks for a given approximation error. Specific examples are given using the Gaussian radial basis functions type of approximation networks.

Variational Calculations of Rovibrational Energy Levels and Transition Intensities for Tetratomic Molecules

Abstract: A description is given of an algorithm for computing rovibrational energy levels for tetratomic molecules. The expressions required for evaluating transition intensities are also given. The variational principle is used to determine the energy levels, and the kinetic energy operator is simple and evaluated exactly. The computational procedure is split up into the determination of one-dimensional radial basis functions, the computation of a contracted rotational−bending basis, followed by a final variational step coupling all degrees of freedom. An angular basis is proposed whereby the rotational−bending contraction takes place in three steps. Angular matrix elements of the potential are evaluated by expansion in terms of a suitable basis, and the angular integrals are given in a factorized form which simplifies their evaluation. The basis functions in the final variational step have the full permutation symmetries of the identical particles. Sample results are given for HCCH and BH3.

An algorithm for coarsening unstructured meshes

Abstract: We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order \(J^2\), where \(J\) is the number of hierarchical basis levels.

Higher-order vector finite elements for tetrahedral cells

Abstract: Edge-based vector finite elements are widely used for two-dimensional (2-D) and three-dimensional (3-D) electromagnetic modeling. This paper seeks to extend these low-order elements to higher orders to improve the accuracy of numerical solutions. These elements have relaxed normal-component continuity to prohibit spurious modes, and also satisfy Nedelec's constraints to eliminate unnecessary degrees of freedom while remaining entirely local in character. Element matrix derivations are given for the first two vector finite element sets. Also, results of the application of these basis functions to cavity resonators demonstrate the superiority of the higher-order elements.

Scale-based clustering using the radial basis function network

Abstract: This paper shows how scale-based clustering can be done using the radial basis function network (RBFN), with the RBF width as the scale parameter and a dummy target as the desired output. The technique suggests the "right" scale at which the given data set should be clustered, thereby providing a solution to the problem of determining the number of RBF units and the widths required to get a good network solution. The network compares favorably with other standard techniques on benchmark clustering examples. Properties that are required of non-Gaussian basis functions, if they are to serve in alternative clustering networks, are identified. This work, on the whole, points out an important role played by the width parameter in RBFN, when observed over several scales, and provides a fundamental link to the scale space theory developed in computational vision.

Nonparametric estimation and classification using radial basis function nets and empirical risk minimization

Abstract: Studies convergence properties of radial basis function (RBF) networks for a large class of basis functions, and reviews the methods and results related to this topic. The authors obtain the network parameters through empirical risk minimization. The authors show the optimal nets to be consistent in the problem of nonlinear function approximation and in nonparametric classification. For the classification problem the authors consider two approaches: the selection of the RBF classifier via nonlinear function estimation and the direct method of minimizing the empirical error probability. The tools used in the analysis include distribution-free nonasymptotic probability inequalities and covering numbers for classes of functions.

Some Studies on Dual Reciprocity BEM for Elastodynamic Analysis

Abstract: The accuracy of the dual reciprocity boundary element method for two-dimensional elastodynamic interior problems is investigated. A general analytical method is described for the closed form determination of the displacement and traction tensor corresponding to radial basis functions and explicit expressions of these tensors are provided for a number of specific basis functions. For all these basis functions the accuracy of the dual reciprocity boundary element method is numerically assessed for three interior plane stress elastodynamic problems. The influence of internal points on the accuracy of the solution is also considered. Useful results concerning the suitability of the various basis functions for solving plane elastodynamic problems are obtained.

Local basis function networks for identification of a turbocharger

Abstract: This paper deals with nonlinear dynamic system identification by local basis function networks A special kind of local basis function network generated by a tree construction algorithm is proposed This local linear model tree (LOLIMOT) is applied for identification of a truck diesel engine exhaust turbocharger The charging pressure is modelled as the output of a nonlinear second order multiple input system with engine speed and injection rate as inputs The LOLIMOT approach was capable to identify the turbocharger with measured signals during road driving and with ten local linear models in less than one minute on a Pentium PC

Further analysis of solutions to the time‐independent wave packet equations of quantum dynamics. II. Scattering as a continuous function of energy using finite, discrete approximate Hamiltonians

Abstract: We consider further how scattering information (the S‐matrix) can be obtained, as a continuous function of energy, by studying wave packet dynamics on a finite grid of restricted size. Solutions are expanded using recursively generated basis functions for calculating Green’s functions and the spectral density operator. These basis functions allow one to construct a general solution to both the standard homogeneous Schrodinger’s equation and the time‐independent wave packet, inhomogeneous Schrodinger equation, in the non‐interacting region (away from the boundaries and the interaction region) from which the scattering solution obeying the desired boundary conditions can be constructed. In addition, we derive new expressions for a ‘‘remainder or error term,’’ which can hopefully be used to optimize the choice of grid points at which the scattering information is evaluated. Problems with reflections at finite boundaries are dealt with using a Hamiltonian which is damped in the boundary region as was done by Mandelshtam and Taylor [J. Chem. Phys. 103, 2903 (1995)]. This enables smaller Hamiltonian matrices to be used. The analysis and numerical methods are illustrated by application to collinear H+H2 reactive scattering.

Photonic Band Using Vector Spherical Waves : I. Various Properties of Bloch Electric Fields and Heavy Photons

Abstract: The calculation is presented for the photonic bands for the fcc array of dielectric spheres, using the vector spherical-waves as basis functions. It is shown that the dimension of the secular determinant is less than 100, or less than one tenth of that of the plane-wave basis functions. Based upon the calculated eigenvectors, the symmetry properties of the electric-field vectors, the angle between the electric- and displcement-field vectors and the orthonormality of the Bloch states are investigated. Also the convergence of the plane-wave expansion is examined. Furthermore, we have demonstrated the existence of the modes appropriately called heavy photons, whose origin is exactly identical to the electronic analogues, heavy fermions.

Local Linear Model Trees for On-Line Identification of Time-Variant Nonlinear Dynamic Systems

Abstract: This paper discusses on-line identification of time-variant nonlinear dynamic systems. A neural network (LOLIMOT, [1]) based on local linear models weighted by basis functions and constructed by a tree algorithm is introduced. Training of this network can be divided into a structure and a parameter optimization part. Since the network is linear in its parameters a recursive least-squares algorithm can be applied for on-line identification. Other advantages of the proposed local approach are robustness and high training and generalisation speed. The simplest recursive version of the algorithm requires only slightly more computations than a recursive linear model identification. The locality of LOLIMOT enables on-line learning in one operating region without forgetting in the others. A drawback of this approach is that systems with large structural changes over time cannot be properly identified, since the model structure is fixed.

A Comparison of Six Deconvolution Techniques

Abstract: We present results for the comparison of six deconvolution techniques. The methods we consider are based on Fourier transforms, system identification, constrained optimization, the use of cubic spline basis functions, maximum entropy, and a genetic algorithm. We compare the performance of these techniques by applying them to simulated noisy data, in order to extract an input function when the unit impulse response is known. The simulated data are generated by convolving the known impulse response with each of five different input functions, and then adding noise of constant coefficient of variation. Each algorithm was tested on 500 data sets, and we define error measures in order to compare the performance of the different methods.