Showing papers on "Basis (linear algebra) published in 1997"

Density-functional method for very large systems with LCAO basis sets

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

Signal-space projection method for separating MEG or EEG into components.

Abstract: CURRENTS INSIDE a conducting body can be estimated by measuring the magnetic and/or the electric field at multiple locations outside and then constructing a solution to the inverse problem, i.e. determining a current configuration that could have produced the measured field. Unfortunately, there is no unique solution to this problem (HELMHOLTZ, 1853) unless restricting assumptions are made. The minimum-norm estimate (HAM/~.L,~INEN and ILMONIEMI, 1994) provides a solution with the smallest expected overall error when minimum a priori information about the source distribution is available. Other methods to estimate a continuous current distribution producing the measured signals have been studied (PASCUAL-MARQUI et al., 1994; WANG et aL, 1995; GORODNITSKY, et al., 1995). A different approach is to divide the brain activity into discrete components such as current dipoles (ScHERG, 1990; MOSHER et al., 1992). Here we widen this approach into arbitrary current configurations. In our signal-space projection (SSP) method, the signals measured by d sensors are considered to form a time-varying vector in a d-dimensional signal space. The component vectors,, i.e. the signals caused by the different neuronal sources, have different and fixed orientations in the signal space. In other words, each source has a distinct and stable field pattern. All the current eonfi~marations producing the same measured field pattern are indistinguishable on the basis of the field: they have the same vector direction in the signal space and thus belong to the same equivalence class of current configurations (TESCHE et al., 1995a). The angle in the signal space between vectors representing different equivalence classes, e.g. between component vectors, is a measure of similarity of the equivalence classes in signal space and a way to characterise the separability of sources. The cosine of this angle has previously been used as a numerical charaeterisation of the difference between topographical distributions (DESMEDT and CHALK[.IN, 1989). If the direction of at least one of the component vectors forming the measured multi-channel signal can be determined from the data, or is known otherwise, SSP can be used to simplify subsequent analysis. For example, if an early deflection in an evoked response is produced by one source, and the rest of the response is a mixture of signals from this and other sources, SSP can separate the data into two parts so that the early source contributes only to one part. In general, the signals are divided into two orthogonal parts: s~, including the time-varying contribution from sources with known signalspace directions; and s~_, including the rest of the signals. Both sl~ and s j_ can then be analysed separately in more detail. By analysing s t , we can detect activity originally masked by s~. On the other hand, the sources included in stl are seen with an enhanced signal-to-noise ratio. By forward modelling of sources in selected patches of cortex, it is possible to form a spatial filter that selectively passes only the signals that may have been generated by currents in the given patches. If the subspace defined by artefacts can be determined, the artefactflee S L can be analysed. In SSP, in contrast to PCA (HARRIS, 1975; MAIER et al., 1987) and other analysis methods (GRUMMICH et al., 1991; KOLES et aL, 1990; KOLES, 1991; SOONG and KOLES, 1995; BESA*), the source decomposition does not depend on the orthogonality of source components or the availability of source or conductivity models. No conductivity or source models are needed if the component vectors are estimated directly from the measured signals. This is useful when no source estimation is needed, e.g. when artefacts or somatomotor activity in a cogrritive study must be filtered out. The angles between the components provide an easy and illustrative way to characterise the linear dependence between the components and thus the separability of sources. The concept of signal space in MEG was introduced previously ([LMONIEMI, 1981; [LMONIEMI and WILLIAMSON,

A class of neural networks for independent component analysis

Abstract: Independent component analysis (ICA) is a recently developed, useful extension of standard principal component analysis (PCA). The ICA model is utilized mainly in blind separation of unknown source signals from their linear mixtures. In this application only the source signals which correspond to the coefficients of the ICA expansion are of interest. In this paper, we propose neural structures related to multilayer feedforward networks for performing complete ICA. The basic ICA network consists of whitening, separation, and basis vector estimation layers. It can be used for both blind source separation and estimation of the basis vectors of ICA. We consider learning algorithms for each layer, and modify our previous nonlinear PCA type algorithms so that their separation capabilities are greatly improved. The proposed class of networks yields good results in test examples with both artificial and real-world data.

Time-domain finite-element 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.

An optimization approach to reconstructing force-free fields from boundary data: I. Theoretical basis.

Abstract: A new method for reconstructing force-free magnetic fields from their boundary values, based on minimizing the global departure of an initial field from a force-free and solenoidal state, is presented. The method is tested by application to a known nonlinear solution. We discuss the obstacles to be overcome in the application of this method to the solar case: the reconstruction of force-free fields in the corona from measurements of the vector magnetic field in the low atmosphere.

Weyl-Heisenberg Frames and Riesz Bases in L2(Rd).

Abstract: : We study Weyl-Heisenberg (= Gabor) expansions for either L2(Rd) or a subspace of it. These are expansions in terms of the spanning set, involving K and L are some discrete lattices in Rd, P, in L2(Rd), is finite, E is the translation operator, and M is a modulation operator. Such sets X are known as WH systems. The analysis of the 'basis' properties of WH systems (e.g. being a frame or a Riesz basis) is our central topic, with the fiberization-decomposition techniques of shift-invariant systems, developed in a previous paper of us, being the main tool. Of particular interest is the notion of the adjoint of a WH set, and the duality principle which characterizes a WH (tight) frame in term of the stability (orthonormality) of its adjoint. The actions of passing to the adjoint and passing to the dual system commute, hence the dual WH frame can be computed via the dual basis of the adjoint. Estimates for the underlying frame/basis bounds are obtained by two different methods. The Gramian analysis applies to all WH systems, albeit provides estimates that might be quite crude. This approach is invoked to show how, under only mild conditions on X, a frame can be obtained by oversampling a Bessel sequence. Finally, finer estimates of the frame bounds, based on the Zak transform, are obtained for a large collection of WH systems.

Linear scaling computation of the Fock matrix. II. Rigorous bounds on exchange integrals and incremental Fock build

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.

Learning parameterized models of image motion

Abstract: A framework for learning parameterized models of optical flow from image sequences is presented. A class of motions is represented by a set of orthogonal basis flow fields that are computed from a training set using principal component analysis. Many complex image motions can be represented by a linear combination of a small number of these basis flows. The learned motion models may be used for optical flow estimation and for model-based recognition. For optical flow estimation we describe a robust, multi-resolution scheme for directly computing the parameters of the learned flow models from image derivatives. As examples we consider learning motion discontinuities, non-rigid motion of human mouths, and articulated human motion.

Linear bases for representation of natural and artificial illuminants

Abstract: By the principal-value decomposition process, we have obtained two linear bases for representing the spectral power distributions of illuminants, applicable for algorithms of color synthesis and analysis in artificial vision: one from experimental measurements of daylight and another combining both natural and artificial illuminants. The first basis adequately represents daylight with dimension 3, in accordance with the previous results of Judd [J. Opt. Soc. Am.54, 1031 (1964)]; however, it does not adequately represent artificial illuminants, even with a higher dimension. In the case of the second basis, many good results are obtained in the reconstruction of the spectral power distribution both of daylight and of artificial illuminants, including some fluorescent lights, with dimension 7 or even less. In consequence, we show the possibility of obtaining linear bases of a low dimension, even when the set of illuminants that we try to represent presents a certain variability in shape.

Human conceptions of spaces: Implications for GIS

Abstract: The way people conceptualize space is an important consideration for the design of GIS, because a better match with people's thinking is expected to lead to easier-to-use information systems. Everyday space, the basis to GIS, has been characterized in the literature as being either small-scale (from table-top to room-size spaces) or large-scale (inside-of-building spaces to city-size spaces). While this dichotomy of space is grounded in the view from psychology that people's perception of space, spatial cognition, and spatial behaviour are experience-based, it is in contrast to current GIS, which enable us to interact with large-scale spaces as though they were small-scale or manipulable. We analyse different approaches to characterizing spaces and propose a unified view in which space is based on the physical properties of manipulability, locomotion, and size of space. Within the structure of our framework, we distinguish six types of spaces: manipulable object space (smaller than the human body), non-manipulable object space (greater than the human body, but less than the size of a building), environmental space (from inside-of-building spaces to city-size spaces), geographic space (state, country, and continent-size spaces), panoramic space (spaces perceived via scanning the landscape), and map space. Such a categorization is an important part of Naive Geography, a set of theories on how people intuitively or spontaneously conceptualize geographic space and time, because it has implications for various theoretical and methodological questions concerning the design and use of spatial information tools. Of particular concern is the design of effective spatial information tools that lead to better communication.

Quantum similarity measures under atomic shell approximation: First order density fitting using elementary Jacobi rotations

Abstract: The elementary Jacobi rotations technique is proposed as a useful tool to obtain fitted electronic density functions expressed as linear combinations of atomic spherical shells, with the additional constraint that all coefficients are kept positive. Moreover, a Newton algorithm has been implemented to optimize atomic shell exponents, minimizing the quadratic error integral function between ab initio and fitted electronic density functions. Although the procedure is completely general, as an application example both techniques have been used to compute a 1S-type Gaussian basis for atoms H through Kr, fitted from a 3-21G basis set. Subsequently, molecular electronic densities are modeled in a promolecular approximation, as a simple sum of parameterized atomic contributions. This simple molecular approximation has been employed to show, in practice, its usefulness to some computational examples in the field of molecular quantum similarity measures. © 1997 John Wiley & Sons, Inc. J Comput Chem18: 2023–2039, 1997

Wavelets, Vibrations and Scalings

Abstract: Introduction Scaling exponents at small scales Infrared divergences and Hadamard's finite parts The 2-microlocal spaces C^{s,s^{\prime}}_{x_0}$ New characterizations of the two-microlocal spaces An adapted wavelet basis Combining a Wilson basis with a wavelet basis Bibliography Index Greek symbols Roman symbols.

Rates of convex approximation in non-hilbert spaces

Abstract: This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subsetS of a function space. Specifically, the focus is on therate at which the lowest achievable error can be reduced as larger subsets ofS are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including, in particular, a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, forL p spaces with 1

Basis functions for linear-scaling first-principles calculations

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.

Multiwavelets for Second-Kind Integral Equations

Abstract: We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a two-dimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N2 to O(N log N) nonzero entries and still obtain (up to log N terms) the same convergence rates as for the exact Galerkin method.

A new class of unbalanced haar wavelets that form an unconditional basis for Lp on general measure spaces

Abstract: Given a complete separable σ-finite measure space (X,Σ, μ) and nested partitions of X, we construct unbalanced Haar-like wavelets on X that form an unconditional basis for Lp (X,Σ, μ) where1

A Systematization of Convexity Concepts for Sets and Functions

Abstract: The paper surveys several convexity concepts, referring to sets and to functions respectively, with the purpose to put them in some kind of order according to the same principle. First it examines the connections between six convexity concepts regarding sets in topological linear spaces and points out the most general concept among these concepts. On the basis of this analysis it is then revealed that twelve convexity concepts concerning functions, that take values in topological linear spaces, can be naturally dened by reduction to the investigated convexities for sets. The most general convexity concept for functions is also found. It is applied to establish an alternative theorem as well as necessary optimality conditions for weak multiobjective optimization problems.

Motion compensated interframe prediction method based on adaptive motion vector interpolation

Abstract: In a method of interpolating the motion vectors at all pixel positions in a frame on the basis of the motion vectors at plural predetermined representative points to perform motion compensated prediction of moving pictures, there is adaptively selected one of a method for dividing the frame into plural areas and interpolating the motion vector every area is selected from a conventional method of linearly interpolating a motion vector by using all the surrounding representative points, a method of separately detecting motion vectors for an area, a method of further dividing an area into plural subareas, a method of temporarily converting the values of the motion vectors at the representative points into values to be used for the motion vector interpolation, and a method of setting a fixed value as a prediction value.

Web bases for sl(3) are not dual canonical

Abstract: We compare two natural bases for the invariant space of a tensor product of irreducible representations of A_2, or sl(3). One basis is the web basis, defined from a skein theory called the combinatorial A_2 spider. The other basis is the dual canonical basis, the dual of the basis defined by Lusztig and Kashiwara. For sl(2) or A_1, the web bases have been discovered many times and were recently shown to be dual canonical by Frenkel and Khovanov. We prove that for sl(3), the two bases eventually diverge even though they agree in many small cases. The first disagreement comes in the invariant space Inv((V^+ tensor V^+ tensor V^- tensor V^-)^{tensor 3}), where V^+ and V^- are the two 3-dimensional representations of sl(3). If the tensor factors are listed in the indicated order, only 511 of the 512 invariant basis vectors coincide.

Towards optimal measurement of power spectra - II. A basis of positive, compact, statistically orthogonal kernels

Abstract: This is the second of two papers which address the problem of measuring the unredshifted power spectrum of fluctuations from a galaxy survey in optimal fashion. A key quantity is the Fisher matrix, which is the inverse of the covariance matrix of minimum variance estimators of the power spectrum of the survey. It is shown that bases of kernels which give rise to complete sets of statistically orthogonal windowed power spectra are obtained in general from the eigenfunctions of the Fisher matrix scaled by some arbitrary positive definite scaling matrix. Among the many possible bases of kernels, there is a basis, obtained by applying an infinitely steep scaling function, which leads to kernels which are positive and compact in Fourier space. This basis of kernels, along with the associated minimum variance pair weighting derived in the previous paper, would appear to offer a solution to the problem of how to measure the unredshifted power spectrum optimally. Illustrative kernels are presented for the case of the PSCz survey.

Nonlinear approximation of random functions

Abstract: Given an orthonormal basis and a certain class X of vectors in a Hilbert space H, consider the following nonlinear approximation process: approach a vector $x\in X$ by keeping only its N largest coordinates, and let N go to infinity. In this paper, we study the accuracy of this process in the case where $H=L^2(I)$, and we use either the trigonometric system or a wavelet basis to expand this space. The class of function that we are interested in is described by a stochastic process. We focus on the case of "piecewise stationary processes" that describe functions which are smooth except at isolated points. We show that the nonlinear wavelet approximation is optimal in terms of mean square error and that this optimality is lost either by using the trigonometric system or by using any type of linear approximation method, i.e., keeping the N first coordinates. The main motivation of this work is the search for a suitable mathematical model to study the compression of images and of certain types of signals.

Exact analytical solution of the n-level landau-zener-type bow-tie model

Abstract: The non-stationary Schrodinger equation in a finite basis of states is considered for the Hamiltonian matrix linearly depending on time. Exact analytical solutions of asymptotic transition probabilities are obtained for a bow-tie model, in which an arbitrary number of linear time-dependent diabatic potential curves cross at one point and only a particular horizontal curve has interactions with the others. Based on the contour integral method used, some mathematical aspects such as a possible generalization of the Whittaker functions are also briefly discussed.

Molecular integrals over Gaussian-type geminal basis functions

Abstract: We present formulas for the evaluation of molecular integrals over basis functions with an explicit Gaussian dependence on interelectronic coordinates. These formulas use expansions in Hermite Gaussian functions and represent an extension to the work of McMurchie and Davidson to two-electron basis functions. Integrals that depend on the coordinates of up to four electrons are discussed explicitly. A key feature of this approach is that it allows full exploitation of the shell structure of the orbital part of the basis.

Stabilizing the Hierarchical Basis by Approximate Wavelets, I: Theory

Abstract: This paper proposes a stabilization of the classical hierarchical basis (HB) method by modifying the HB functions using some computationally feasible approximate L2-projections onto finite element spaces of relatively coarse levels. The corresponding multilevel additive and multiplicative algorithms give spectrally equivalent preconditioners, and one action of such a preconditioner is of optimal order computationally. The results are regularity-free for the continuous problem (second order elliptic) and can be applied to problems with rough coefficients and local refinement. © 1997 by John Wiley & Sons, Ltd.