Showing papers on "Basis (linear algebra) published in 1994"

Color constancy - Generalized diagonal transforms suffice

Abstract: This study’s main result is to show that under the conditions imposed by the Maloney–Wandell color constancy algorithm, whereby illuminants are three dimensional and reflectances two dimensional (the 3–2 world), color constancy can be expressed in terms of a simple independent adjustment of the sensor responses (in other words, as a von Kries adaptation type of coefficient rule algorithm) as long as the sensor space is first transformed to a new basis. A consequence of this result is that any color constancy algorithm that makes 3–2 assumptions, such as the Maloney–Wandell subspace algorithm, Forsyth’s MWEXT, and the Funt–Drew lightness algorithm, must effectively calculate a simple von Kries-type scaling of sensor responses, i.e., a diagonal matrix. Our results are strong in the sense that no constraint is placed on the initial spectral sensitivities of the sensors. In addition to purely theoretical arguments, we present results from simulations of von Kries-type color constancy in which the spectra of real illuminants and reflectances along with the human-cone-sensitivity functions are used. The simulations demonstrate that when the cone sensor space is transformed to its new basis in the appropriate manner a diagonal matrix supports nearly optimal color constancy.

Coupled-cluster theory employing approximate integrals : an approach to avoid the input/output and storage bottlenecks

Abstract: By representing orbital products in an expansion basis, certain classes of two‐electron integrals are approximated for use in CCSD(T) calculations (singles and doubles coupled‐cluster plus a perturbational estimate of the effects of connected triple excitations). This leads to a very large reduction in disk storage and input/output requirements, with usually only a modest increase in computational effort. The new procedure will allow very large CCSD(T) calculations to be undertaken, limited only by available processor time. Using the molecular basis as the expansion basis, explicit numerical comparisons of equilibrium geometries, harmonic frequencies, and energy differences indicate that the error due to the use of approximate integrals is less than the error associated with truncation of the molecular basis set.

A nonlinear variational problem for image matching

Abstract: Minimizing a nonlinear functional is presented as a way of obtaining a planar mapping that matches two similar images. A smoothing term is added to the nonlinear functional to penalize discontinuous and irregular solutions. One option for the smoothing term is a quadratic form generated by a linear differential operator. The functional is then minimized using the Fourier representation of the planar mapping. With this representation the quadratic form is diagonalized. Another option is a quadratic form generated via a basis of compactly supported wavelets. In both cases, a natural approximation scheme is described. Both quadratic forms are shown to impose the same smoothing. However, in terms of the finite dimensional approximations, it is easier to accommodate local deformations using the wavelet basis.

On CHF calculations of second-order magnetic properties using the method of continuous transformation of origin of the current density

Abstract: A fully analytical formulation is outlined for computing molecular magnetic susceptibilities and nuclear magnetic shieldings via a continuous change of origin of the electronic current density induced by an external magnetic field. The change of origin is described in terms of a (continuous) arbitrary shift functiond(r). Coupled Hartree-Fock second-order magnetic properties of CH4 and CO2 molecules have been computed, using the special choiced(r)=r as generating function. A detailed analysis of results obtained with a variety of basis sets reveals that such a method is not as good as previously suggested. Large basis sets must be used to obtain accurate magnetic properties. On the other hand, all the components of theoretical nuclear magnetic shielding evaluated via this approach are independent of the origin of the vector potential. In general, theoretical magnetic susceptibilities depend linearly on the distance between different coordinate frames, but are origin independent for centre-symmetric molecules.

Simultaneous Noise Suppression and Signal Compression Using a Library of Orthonormal Bases and the Minimum Description Length Criterion

Abstract: We describe an algorithm to estimate a discrete signal from its noisy observation, using a library of orthonormal bases (consisting of various wavelets, wavelet packets, and local trigonometric bases) and the information-theoretic criterion called minimum description length (MDL). The key to effective random noise suppression is that the signal component in the data may be represented efficiently by one or more of the bases in the library, whereas the noise component cannot be represented efficiently by any basis in the library. The MDL criterion gives the best compromise between the fidelity of the estimation result to the data (noise suppression) and the efficiency of the representation of the estimated signal (signal compression): it selects the “best” basis and the “best” number of terms to be retained out of various bases in the library in an objective manner. Because of the use of the MDL criterion, our algorithm is free from any parameter setting or subjective judgments. This method has been applied usefully to various geophysical datasets containing many transient features.

Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems

Abstract: This paper develops a constructive method for finding a nilpotent basis for a special class of smooth nonholonomic distributions. The main tool is the use of the Goursat normal form theorem which arises in the study of exterior differential systems. The results are applied to the problem of finding a set of nilpotent input vector fields for a nonholonomic control system, which can then used to construct explicit trajectories to drive the system between any two points. A kinematic model of a rolling penny is used to illustrate this approach. The methods presented here extend previous work using "chained form" and cast that work into a coordinate-free setting.

System identification with generalized orthonormal basis functions

Abstract: A least squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function, where the expansion is in terms of generalized basis functions. The basis functions are orthogonal in H/sub 2/ and generalize the pulse, Laguerre and Kautz (1954) bases. The construction of the basis is considered and bias and variance expressions of the identification algorithm are discussed. The basis induces a new transformation (Hambo transform) of signals and systems, for which state space expressions are derived. >

First-order representations of linear systems

Abstract: Part 1 Rational matrices and rational vector spaces: algebraic preliminaries Euclidean domains of rational functions pole/zero structure of a rational matrix Wiener-Hopf structure of a rational matrix minimal basis of a rational vector space preliminary results for matrix pencils. Part 2 Representations of linear time-invariant systems: dynamical systems AR representations ARMA representations first-order representations systems with split external variables. Part 3 Minimality and transformation groups: minimality of a P representation minimality of a D representation minimality of a DZ representation minimality of a DP representation transformation groups. Part 4 Realization in minimal first-order form: realization in pencil form - the abstract procedure the pencil realization in terms of a discrete-time behaviour choosing bases connections with the Fuhrmann realization. Part 5 Structural invariants: observability indices controllability indices the input-output structure.

Adaptive multilevel methods for obstacle problems

Abstract: The authors consider the discretization of obstacle problems for second-order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned conjugate gradient iterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement semilocal and local a posteriors error estimates are derived, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.

Theory of the expansion of wave functions in a gaussian basis

Abstract: The convergence properties of the expansions of (a) the function 1/r and (b) the function exp(-αr) in an even-tempered basis of Gaussians are studied analytically. The starting points are the Gaussian integral representations of 1/r and exp(-αr). One arrives at an expansion in a finite number of Gaussians in three steps: (1) a restriction of the integration domain, (2) a variable transformation, and (3) discretization of the integral. The cutoff error goes in both cases essentially as exp(-ah), and the discretization error, as exp(-b/h). The minimum overall error is reached for the β-parameter of an even-tempered basis β∼exp(c/√n), where n is the dimension of the basis, and the error itself decreases as e∼exp(-d√n)

Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces

Abstract: A problem of enduring interest in connection with the study of frames in Hilbert space is that of characterizing those frames which can essentially be regarded as Riesz bases for computational purposes or which have certain desirable properties of Riesz bases. In this paper we study several aspects of this problem using the notion of a pre-frame operator and a model theory for frames derived from this notion. In particular, we show that the deletion of a finite set of vectors from a frame {x I}'IOl leaves a Riesz basis if and only if the frame is Besselian (i.e., En?Il anXn converges X (an) E 12).

Product-basis method for calculating dielectric matrices

Abstract: We present a method for calculating dielectric matrices of periodic systems. Unlike the conventional method, which uses a plane-wave basis, the present method employs a product basis, which, in the linear-muffin-tin-orbital formalism, consists of products of orbitals. The method can be used for any system, including sp as well as narrow band systems. We demonstrate the applicability of our method by calculating the energy-loss spectra of Ni and Si, including local-field effects that require the full dielectric matrix. Good agreement with experiment is found. The small number of basis functions makes the method suitable for self-energy calculations within the GW approximation, without making the so-called plasmon-pole approximation for the dielectric matrix.

On completeness of certain families of semi-Riemannian manifolds

Abstract: Semi-Riemannian manifolds with a suitable set of conformal symmetries are shown to be complete. Locally warped products are studied and warped-completeness is introduced. In the case of definite and complete basis, several assumptions on the growth of the warping function yield some of the three kinds of completeness. The case of 1-dimensional basis (including a known family of relativistic space-times) is specially studied. Null warped-completeness is related to the completeness of a certain conformal metric on the basis. Several examples and counter-examples explaining the main results are also given.

Basis set effects on the intermolecular interaction of hydrocarbon molecules obtained by an ab initio molecular orbital method : evaluation of dispersion energy

Abstract: The intermolecular interaction potentials of methane, ethane, ethylene and benzene dimers were calculated using several basis sets [up to 6–311G(3d,4p)] with electron correlation correction by the Moller-Plesset perturbation method and basis set superposition error (BSSE) correction. The calculated interaction energies considerably depend on the basis set used. Whereas the interaction energies of the repulsive and Coulombic energy components calculated at the HF level are not affected by the change of the basis set, the dispersion energy component, calculated as the electron correlation energy, greatly depends on the basis set used. A basis set with multiple polarized functions is necessary to calculate the dispersion energy correctly. The use of small basis sets greatly underestimates the dispersion energy.

Singularity analysis of mechanisms and robots via a motion-space model of the instantaneous kinematics

Abstract: This paper investigates the kinematic singularities of a general mechanism (arbitrary kinematic chain), considered as a non-redundant input-output device with equal number of inputs and outputs. The instantaneous kinematics of a mechanism is described by the orientation of a linear subspace, the motion space, inside the velocity space of all potential instantaneous motions. The definition of singularity for a general mechanism is provided. On the basis of the six types of singular configurations introduced in the paper a comprehensive singularity classification is developed. >

The basis of nonlocal curvature invariants in quantum gravity theory. Third order.

Abstract: A complete basis of nonlocal invariants in quantum gravity theory is built to third order in space–time curvature and matter‐field strengths. The nonlocal identities are obtained which reduce this basis for manifolds with dimensionality 2ω<6. The present results are used in heat‐kernel theory, theory of gauge fields and serve as a basis for the model‐independent approach to quantum gravity and, in particular, for the study of nonlocal vacuum effects in the gravitational collapse problem.

Highly accurate relativistic universal Gaussian basis set: Dirac-Fock-Coulomb calculations for atomic systems up to nobelium

Abstract: A universal Gaussian basis set is developed that leads to relativistic Dirac–Fock SCF energies of comparable accuracy as that obtained by the accurate numerical finite‐difference method (GRASP2 package) [J. Phys. B 25, 1 (1992)]. The Gaussian‐type functions of our universal basis set satisfy the relativistic boundary conditions associated with the finite nuclear model for a finite speed of light and conform to the so‐called kinetic balance at the nonrelativistic limit. We attribute the exceptionally high accuracy obtained in our calculations to the fact that the representation of the relativistic dynamics of an electron in a spherical ball finite nucleus near the origin in terms of our universal Gaussian basis set is as accurate as that provided by the numerical finite‐difference method. Results of the Dirac–Fock–Coulomb energies for a number of atoms up to No (Z=102) and some negative ions are presented and compared with the recent results obtained with the numerical finite‐difference method and geometrical Gaussian basis sets by Parpia, Mohanty, and Clementi [J. Phys. B 25, 1 (1992)]. The accuracy of our calculations is estimated to be within a few parts in 109 for all the atomic systems studied.

The basis of nonlocal curvature invariants in quantum gravity theory

Abstract: A complete basis of nonlocal invariants in quantum gravity theory is built to third order in spacetime curvature and matter-field strengths. The nonlocal identities are obtained which reduce this basis for manifolds with dimensionality $2\omega<6$. The present results are used in heat-kernel theory, theory of gauge fields and serve as a basis for the model-independent approach to quantum gravity and, in particular, for the study of nonlocal vacuum effects in the gravitational collapse problem.

High dimensional feature reduction via projection pursuit

Abstract: The recent development of more sophisticated remote sensing systems enables the measurement of radiation in many more spectral intervals than previously possible. An example of that technology is the AVIRIS system, which collects image data in 220 bands. As a result of this, new algorithms must be developed in order to analyze the more complex data effectively. Data in a high dimensional space presents a substantial challenge, since intuitive concepts valid in a 2-3 dimensional space do not necessarily apply in higher dimensional spaces. For example, high dimensional space is mostly empty. This results from the concentration of data in the corners of hypercubes. Other examples may be cited. Such observations suggest the need to project data to a subspace of a much lower dimension on a problem specific basis in such a manner that information is not lost. Projection pursuit is a technique that will accomplish such a goal. Since it processes data in lower dimensions, it should avoid many of the difficulties of high dimensional spaces. The authors investigate some of the properties of projection pursuit. >

Discrete variable representations of complicated kinetic energy operators

Abstract: Probably the most important advantage of the discrete variable representation (DVR) is its simplicity. The DVR potential energy matrix is constructed directly from the potential function without evaluating integrals. For simple kinetic energy operators the DVR kinetic energy matrix is determined from transformation matrices and exact matrix representations of one‐dimensional kinetic energy operators in the original delocalized polynomial basis set. For complicated kinetic energy operators, for which matrix elements of terms or factors with derivatives must be calculated numerically, defining a DVR is harder. A DVR may be defined from a finite basis representation (FBR) where matrix elements of terms or factors in the kinetic energy operator are computed by quadrature but implicating quadrature undermines the simplicity and convenience of the DVR. One may bypass quadrature by replacing the matrix representation of each kinetic energy operator term with a product of matrix representations. This product appr...

Fast surface interpolation using multiresolution wavelet transform

Abstract: Discrete formulation of the surface interpolation problem usually leads to a large sparse linear equation system. Due to the poor convergence condition of the equation system, the convergence rate of solving this problem with iterative method is very slow. To improve this condition, a multiresolution basis transfer scheme based on the wavelet transform is proposed. By applying the wavelet transform, the original interpolation basis is transformed into two sets of bases with larger supports while the admissible solution space remains unchanged. With this basis transfer, a new set of nodal variables results and an equivalent equation system with better convergence condition can be solved. The basis transfer can be easily implemented by using an QMF matrix pair associated with the chosen interpolation basis. The consequence of the basis transfer scheme can be regarded as a preconditioner to the subsequent iterative computation method. The effect of the transfer is that the interpolated surface is decomposed into its low-frequency and high-frequency portions in the frequency domain. It has been indicated that the convergence rate of the interpolated surface is dominated by the low-frequency portion. With this frequency domain decomposition, the low-frequency portion of the interpolated surface can be emphasized. As compared with other acceleration methods, this basis transfer scheme provides a more systematical approach for fast surface interpolation. The easy implementation and high flexibility of the proposed algorithm also make it applicable to various regularization problems. >

Transformations of real-time finite-temperature Feynman rules.

Abstract: We consider transformations of the 2[times]2 propagator matrix in real-time finite-temperature field theory, resulting in transformed [ital n]-point functions. As special cases of such a transformation we examine the Keldysh basis, the retarded/advanced [ital RA] basis, and a Feynman-like [ital F[bar F]] basis, which differ in this context as to how economically'' certain constraints on the original propagator matrix elements are implemented. We also obtain the relation between some of these real-time functions and certain analytic continuations of the imaginary-time functions. Finally, we compare some aspects of these bases which arise in practical calculations.

Intrinsic Statistical Weakness of Keystream Generators

Abstract: It is shown that an arbitrary binary keystream generator with M bits of memory can be linearly modelled as a non-autonomous linear feedback shift register of length at most M with an additive input sequence of nonbalanced identically distributed binary random variables. An effective method for the linear model determination based on the linear sequential circuit approximation of autonomous finite-state machines is developed. Linear models for clock-controlled shift registers and arbitrary shift register based keystream generators are derived. Several examples including the time-variant memoryless combiner, the basic summation generator, the stop-and-go cascade, and the shrinking generator are presented. Linear models are the basis for a general structure-dependent and initial-state-independent statistical test and they may also be used for correlation attacks on the initial-state. Theoretical security against the introduced statistical attack appears hard to control in practice and hard to achieve with simple schemes.

Finite element approximation

Abstract: In this section we study certain linear classes V of approximating functions resp. sequences of them {V j , j = 0, 1, ...} which are particularly useful for efficient numerical methods in different fields. They share some properties which can be described in short as follows: existence of a well-localized and stable (with respect to L p -norms) algebraic basis, simple recursions (prolongation and restriction operators) for exchanging information between different V j , good approximation properties for smooth functions, e.g., from Sobolev spaces.

A comparison of finite basis set and finite difference methods for the ground state of the CS molecule

Abstract: A comparison is made of the accuracy with which the total electronic energy can be calculated by using either the finite basis set approach (the algebraic approximation) or finite difference methods in calculations using the Hartree-Fock model for the ground (X1 Sigma +) state of the carbon monosulphide molecule. The CS molecule is considered as a prototype for systems containing atoms from different rows of the periodic table. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The dependence of the finite difference calculations on the numerical grid employed is studied.

Strictly positive definite kernels on the Hilbert sphere

Abstract: We completely characterize the strictly positive definite and the strictly conditionally negative definite radial continuous kernels on the real Hilbert sphere. Any functions generating such kernels, can be used in radial basis interpolation of arbitrary data on a set of points in any finite–dimensional sphere.

An accurate ab initio potential energy surface of the he-h2 interaction

Abstract: An accurate potential energy surface of the He–H2 interaction is calculated with a large basis set at the complete fourth‐order Mo/ller–Plesset approximation. The basis set—a combination of a nucleus‐centered set 6s4p2d and a bond function set 3s3p2d centered at the midpoint between He and the H2 center of mass—is designed to give the optimal description of both the intra‐ and intersystem correlation effects. The validity of the basis set is confirmed by extensive preliminary calculations on the linear (orientation angle θ=0°), bent (45°), and T‐shaped (90°) structures at a fixed separation (R=6.5a0) with a series of large basis sets containing different polarization functions and/or bond functions. Bond functions are found more effective than polarization functions in recovering the intersystem correlation energy and they are particularly useful in removing the geometric bias of a basis to give an accurate description for the potential anisotropy and the relative energies of different structures. The eff...