Showing papers on "Basis (linear algebra) published in 1990"

Application of the Karhunen-Loeve procedure for the characterization of human faces

Abstract: The use of natural symmetries (mirror images) in a well-defined family of patterns (human faces) is discussed within the framework of the Karhunen-Loeve expansion This results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix, without increasing the complexity of the calculation The resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average >

Finite basis set corrections to total energy pseudopotential calculations

Abstract: A means of correcting total energy pseudopotential calculations performed using a fixed cut-off energy for the plane waves in the basis set is presented. The use of a finite set of special k-points in such a calculation will introduce errors in the total energies which decrease only slowly with increasing cut-off energy. In particular, total energy differences are not accurate unless the cut-off energy used is sufficiently large that the total energies are themselves converged. This would not be the case if a truly constant cut-off energy could be used. Unfortunately this can only be achieved by using an infinite k-point set. We have derived a correction which will explicitly eliminate these errors to give total energies which can correspond to a strictly constant cut-off energy. In this way, total energy differences and hence many physical properties can be accurately calculated using cut-off energies significantly lower than otherwise possible, with substantial savings in computational time. Total energy pseudopotential calculations can be used to determine a wide variety of physical properties of materials. Calculations are performed on periodic supercells thereby allowing the electronic wavefunctions to be expanded in terms of a discrete set of plane waves at each of an infinite set of k-points in the Brillouin zone. This in turn allows the application of the following two approximations. Firstly, a small number of carefully chosen k-points can be used to accurately represent the wavefunction at all k- points (Chadi and Cohen 1973, Monkhorst and Pack 1976), and secondly, by truncating the basis set the wavefunctions at each k-point can be expanded in terms of a finite basis set. In principle by increasing the number of k-points and the size of the basis set it is possible to achieve absolute energy convergence. However, even in the case of very small systems, this proves to be extremely computationally expensive. In order to perform calculations on larger, more complex systems it is necessary to be able to use smaller plane wave basis sets at each k-point without reducing the accuracy of the calculation. It is known that differences in the total energies of systems of the same size can be accurately calculated for numbers of plane waves and of k-points very much smaller than those required to ensure convergence of the absolute energies provided that identical basis sets are used for each calculation (Cheng et a1 1988). However, when computing energy differences between systems of varying size it is impossible to use identical plane wave basis sets unless an infinite number of k-points are used in the calculation. We must choose instead either to use a constant number of plane waves in

Vector-subspace model for color representation

Abstract: In multispectral imaging it is advantageous to compress spectral information with a minimum loss of information in a way that permits accurate recovery of the spectrum. By use of the simple vector-subspace model that we propose, spectral information can be stored and recovered by the use of a few inner products, which are easy to measure optically. Two large data sets, the first consisting of 1257 Munsell colors and the other of 218 naturally occurring spectral reflectances, were analyzed to form two bases for the model. The Munsell basis can be used to represent the natural colors, and the basis derived from the natural data can be used to represent the Munsell data. The proposed vector-subspace model includes a simple relation between the inner products and conventional color coordinates. It also provides a way to estimate the spectrum of an object that has known chromaticity coordinates.

New method for calculating wave packet dynamics: Strongly coupled surfaces and the adiabatic basis

Abstract: The nuclear dynamics on potential energy surfaces with a conical intersection is investigated on the basis of exact (numerical) integration of the time‐dependent Schrodinger equation. The ethylene cation is chosen as a typical realistic model system. Complementing earlier work we study the dynamics also in the adiabatic basis, which will be seen to allow for a more profound understanding of the decay and dephasing processes occurring in the system. The computational effort exceeds considerably that of propagation in the diabatic basis, to which previous related studies have been confined. To solve the resulting computational problems we develop and present a special multidimensional adaptation of the finite basis set method utilizing the product structure of the basis. It allows us to calculate propagation in a general potential including three vibrational modes. For the time integration a fourth order differencing scheme is introduced which is faster than the second order differencing‐scheme and predictor–corrector approaches.

New approximate optimization method for distribution system planning

Abstract: An algorithm to obtain an approximate optimal solution to the problem of large-scale radial distribution system planning is proposed. The distribution planning problem is formulated as a MIP (mixed integer programming) problem. The set of constraints is reduced to a set of continuous variable linear equations by using the fact that the basis of the simplex tableau consists of the power flow variables of radial branch. This linear problem is solved by pivot operations which correspond to a branch-exchange of the radial network. Numerical examples are presented to demonstrate the validity and effectiveness of the algorithm. >

Function approximation and time series prediction with neural networks

Abstract: Neural networks are examined in the context of function approximation and the related field of time series prediction. A natural extension of radial basis nets is introduced. It is found that use of an adaptable gradient and normalized basis functions can significantly reduce the amount of data necessary to train the net while maintaining the speed advantage of a net that is linear in the weights. The local nature of the network permits the use of simple learning algorithms with short memories of earlier training data. In particular, it is shown that a one-dimensional Newton method is quite fast and reasonably accurate

Reconstructing convex sets from support line measurements

Abstract: Algorithms are proposed for reconstructing convex sets given noisy support line measurements. It is observed that a set of measured support lines may not be consistent with any set in the plane. A theory of consistent support lines which serves as a basis for reconstruction algorithms that take the form of constrained optimization algorithms is developed. The formal statement of the problem and constraints reveals a rich geometry that makes it possible to include prior information about object position and boundary smoothness. The algorithms, which use explicit noise models and prior knowledge, are based on maximum-likelihood and maximum a posteriori estimation principles and are implemented using efficient linear and quadratic programming codes. Experimental results are presented. This research sets the stage for a more general approach to the incorporation of prior information concerning the estimation of object shape. >

Signal processing via least squares error modeling

Abstract: The signal model presently considered is composed of a linear combination of basis signals chosen to reflect the basic nature believed to characterize the data being modeled. The basis signals are dependent on a set of real parameters selected to ensure that the signal model best approximates the data in a least-square-error (LSE) sense. In the nonlinear programming algorithms presented for computing the optimum parameter selection, the emphasis is placed on computational efficiency considerations. The development is formulated in a vector-space setting and uses such fundamental vector-space concepts as inner products, the range- and null-space matrices, orthogonal vectors, and the generalized Gramm-Schmidt orthogonalization procedure. A running set of representative signal-processing examples are presented to illustrate the theoretical concepts as well as point out the utility of LSE modeling. These examples include the modeling of empirical data as a sum of complex exponentials and sinusoids, linear prediction, linear recursive identification, and direction finding. >

On the analysis of frequency-selective surfaces using subdomain basis functions

Abstract: The problem of scattering from frequency-selective surfaces (FSSs) has been investigated by expanding the unknown current distribution with three different sets of basis functions, namely the roof top, surface patch, and triangular patch. The boundary condition on the total electric field on the FSS due to this current distribution is tested either by a line integral or by the Galerkin procedure. This results in an operator equation that can be solved either by a direct matrix inversion method or by an iterative procedure, namely the conjugate gradient method (CGM). The performance of each of these basis and testing functions is evaluated. It is found that the roof-top and the surface-patch basis functions in conjunction with the Galerkin testing are superior in computational efficiency to other combinations of basis and testing functions that have been studied. Comparison of the CPU times on a Cray X-MP/48 supercomputer in solving the operator equation by the direct matrix inversion method and the CGM is provided. Frequency responses of free-standing, periodic arrays of conducting and resistive plates are also presented. >

Representations of integers as the sum of k terms

Abstract: A set of natural numbers is called an asymptotic basis of order k if every number (sufficiently large) can be expressed as a sum of k distinct numbers from the set. in this paper we prove that, for every fixed k, there exists an asymptotic basis of order k such that the number of representations of n is Θ(log n). © 1990 Wiley Periodicals, Inc.

Preference-based deontic logic (PDL)

Abstract: A new possible world semantics for deontic logic is proposed. Its intuitive basis is that prohibitive predicates (such as “wrong” and “prohibited”) have the property of negativity, i.e. that what is worse than something wrong is itself wrong. The logic of prohibitive predicates is built on this property and on preference logic. Prescriptive predicates are defined in terms of prohibitive predicates, according to the well-known formula “ought” = “wrong that not”. In this preference-based deontic logic (PDL), those theorems that give rise to the paradoxes of standard deontic logic (SDL) are not obtained. (E.g., O(p & q) → Op & Oq and Op → O(p v q)) are theorems of SDL but not of PDL. The more plausible theorems of SDL, however, can be derived in PDL.

Uniqueness of unconditional bases in quasi-banach spaces with applications to hardy spaces, II

Abstract: We prove that a wide class of quasi-Banach spaces has a unique up to a permutation unconditional basis. This applies in particular to Hardy spacesHp forp<1. We also investigate the structure of complemented subspaces ofHp(D). The proofs use in essential way matching theory.

A build-down scheme for linear programming

Abstract: We propose a “build-down” scheme for Karmarkar's algorithm and the simplex method for linear programming. The scheme starts with an optimal basis “candidate” setΞ including all columns of the constraint matrix, then constructs a dual ellipsoid containing all optimal dual solutions. A pricing rule is developed for checking whether or not a dual hyperplane corresponding to a column intersects the containing ellipsoid. If the dual hyperplane has no intersection with the ellipsoid, its corresponding column will not appear in any of the optimal bases, and can be eliminated fromΞ. As these methods iterate,Ξ is eventually built-down to a set that contains only the optimal basic columns.

Adding configuration interaction to the time‐dependent Hartree grid approximation

Abstract: The time‐dependent Hartree grid (TDHG) method is extended into an ab initio algorithm for obtaining exact quantum wave packet dynamics. The new algorithm employs a superposition of orthogonal zeroth order time‐dependent basis functions generated from a single TDHG wave packet trajectory. The superposition coefficients are themselves time‐dependent, and are responsible for mixing the basis functions in such a way as to represent exact solutions of the time‐dependent Schrodinger equation. Evolution of the superposition coefficients is governed by a set of first‐order linearly coupled ordinary differential equations. The couplings between coefficients are given by matrix elements of a naturally identified interaction potential taken between members of the zeroth order basis. In numerical tests involving computation of S‐matrix elements for collinear inelastic atom–Morse oscillator scattering the method proves accurate, flexible and efficient, and appears to be easily extendable to more complicated systems.

Approximate system identification using system based orthonormal functions

Abstract: The problem of approximate system identification is addressed. The use of prefilters that are based on a special class of system based orthonormal functions is proposed. It is shown that every linear finite dimensional time invariant discrete time system gives rise to two sets of orthonormal functions and that both can be used as a basis of the space l/sub 2/. The derivation of these functions, to be considered as a generalization of the Laguerre polynomials, is based on the properties of discrete time all-pass functions. Transformation of the input/output signals of a linear system in terms of these functions leads to new system descriptions, and new possibilities arise for the construction of approximate identification methods, with favorable properties allowing the use of simple estimation techniques and a systematic choice of prefilters. >

Hopf-algebraic structure of combinatorial objects and differential operators

Abstract: A Hopf-algebraic structure on a vector space which has as basis a family of trees is described. Some applications of this structure to combinatorics and to differential operators are surveyed. Some possible future directions for this work are indicated.

Ritz method for dynamic analysis of large discrete linear systems with non-proportional damping

Abstract: Real and complex Ritz vector bases for dynamic analysis of large linear systems with non-proportional damping are presented and compared. Both vector bases are generated utilizing load dependent vector algorithms that employ recurrence equations analogous to the Lanczos algorithm. The choice of static response to fixed spatial loading distribution, as a starting vector in recurrence equations, is motivated by the static correction concept. Different phases of dynamic response analysis are compared with respect to computational efficiency and accuracy. It is concluded that the real vector basis approach is approximately eight times more efficient than the complex vector basis approach. The complex vector basis has some advantages with respect to accuracy, if the excitation is of piecewise linear form, since the exact solution can be utilized. In addition, it is demonstrated that both Ritz vector bases, real and complex, possess superior accuracy over the adequate eigenvector bases.

A method for calculating vibrational bound states: Iterative solution of the collocation equations constructed from localized basis sets

Abstract: We propose a simple and efficient method for calculating vibrational bound states of molecular systems. The technique is based upon iterative solution of the collocation equations. A localized basis set is used which is very efficient for strongly coupled modes and also leads to a diagonally dominant set of collocation equations. The iterative scheme developed is based upon Davidson’s method and takes advantage of this diagonal dominance. The approach is capable of exploiting the efficiency with which the matrix elements are calculated in the collocation method by evaluating the matrix elements as they are required. This combination of techniques should allow the method to be used for systems which have more degrees of freedom than have been treated by conventional methods.

Finite-dimensional compensators for infinite-dimensional systems via Galerkin-type approximation

Abstract: In this paper existence and construction of stabilizing compensators for linear time-invariant systems defined on Hilbert spaces are discussed. An existence result is established using Galerkin-type approximations in which independent basis elements are used instead of the complete set of eigenvectors. A design procedure based on approximate solutions of the optimal regulator and optimal observer via Galerkin-type approximation is given and the Schumacher approach is used to reduce the dimension of compensators. A detailed discussion for parabolic and hereditary differential systems is included.

Optimized Gaussian basis sets for representation of continuum wavefunctions

Abstract: Exponents of Gaussian-type basis functions have been optimized for electron-molecule scattering purposes. The criterion for optimization was to obtain the best least-squares fit to six Bessel functions jl(kh(l)*r) representing the continuum functions. The values for the radial momentum kh(l) are defined by the boundary conditions for the Bessel functions to have vanishing radial derivatives at r=20 au. For each l=0, 1 and 2, Gaussian basis sets of eight functions have been optimized. The results are of excellent quality. It is therefore concluded that usual atomic Gaussian basis sets, augmented by these functions, can be sufficient in electron-molecule scattering calculations, such as R-matrix calculations, for example.

Set Membership Approach to Identification and Prediction of Lake Eutrophication

Abstract: Generally, ecosystems modeling is obstructed by the problem of sparse and unreliable data, and lack of knowledge about processes dominating the system. Under these circumstances, set theoretic uncertainty models are an appropriate alternative to probabilistic models. The only requirement is that the uncertainty is pointwise bounded. A newly developed set membership identification procedure is presented and demonstrated by an application to the modeling of shallow lake eutrophication. First, a set of parameter vectors is identified. Analysis of the set reveals a dominant direction spanned by four algal growth and death parameters. Second, on the basis of additional fuzzy set theoretic assumptions, a formal min-max estimation is performed to obtain information about the model validity. If the model appears to be (partially) invalid, the degree of invalidity, affecting the model prediction uncertainty, can be represented by an estimate of the model structure error in addition to the uncertainty contained in the identified set of parameter vectors.

Method of estimating motion in a picture signal

Abstract: In a method of estimating motion per picture portion in an image of a picture signal, wherein, starting from a first (x0=x1, y0=y2) and a second (x0=x3, y0=y4) starting vector a motion vector is determined, a first and a second candidate motion vector, respectively, are determined on the basis of the first and second starting vector in accordance with a predetermined criterion, a motion vector is selected from both candidate motion vectors, and as components of the starting vectors, respective corresponding components of candidate motion vectors (x1, y1), (x2, y2), (x3, y3) and (x4, y4) already previously determined are taken, which candidate motion vectors correspond respectively to the starting vectors.

Dynamical basis sets for algebraic variational calculations in quantum-mechanical scattering theory

Abstract: New basis sets are proposed for linear algebraic variational calculations of transition amplitudes in quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained, even for basis sets with the majority of the members independent of energy.

Basis of Vector Space

Abstract: Summary. We prove the existence of a basis of a vector space, i.e., a set of vectors that generates the vector space and is linearly independent. We also introduce the notion of a subspace generated by a set of vectors and linear independence of set of vectors.

Localisation of field energy

Abstract: The authors define conservation laws in general relativity with respect to a flat reference spacetime, via Noether's theorem and the standard Lagrangian function, quadratic in first-order derivatives. The covariant superpotential obtained in that way fulfils all standard global requirements at spatial and at null infinity and has no anomalous factor of two for the ratio of the mass to angular momentum. Next they attempt to localise the conservation laws and obtain linear and angular momentum densities. They put forward local mapping equations in which a key role is played by a family of artificial, short living, closed shells of matter whose interior is flat. The equations are derived on the basis that the linear momentum densities at each point of the flat interior must be equal to zero. They gain some insight in the mapping equations by considering static spacetimes and spaces with spherically symmetric static shells.

A domain decomposition algorithm using a hierarchical basis

Abstract: Over the last five years, several new fast algorithms have been developed for the solution of elliptic finite element problems with many degrees of freedom. Among them are Yserentant's hierarchical basis multigrid method and a number of domain decomposition algorithms for problems divided into many subproblems. The condition number of the relevant operators, for the best of these preconditioned conjugate gradient methods, grows only slowly with the size of the problems; the growth typically is quadratic in the logarithm of the number of degrees of freedom associated with a subregion or an element of the coarsest mesh.Important components of many domain decomposition preconditioners are subproblems related to the sets of variables on the interfaces between neighboring subregions. In this paper, it is demonstrated that, for problems in two dimensions, a very simple change of basis for these one-dimensional problems leads to a preconditioner which is as effective as those previously considered. In the proof, only tools of linear algebra and a result of Yserentant's are used. The numerical experiments confirm that the new algorithm is better conditioned than the original hierarchical basis multigrid method.

Nonlocal integral‐equation approximations. I. The zeroth order (hydrostatic) approximation with applications

Abstract: A formally exact nonlocal density‐functional expansion procedure for direct correlation functions developed earlier by Stell for a homogeneous system, and extended by Blum and Stell, Sullivan and Stell, and ourselves to various inhomogeneous systems, is used here to derive nonlocal integral‐equation approximations. Two of the simplest of these approximations (zeroth order), which we shall characterize here as the hydrostatic Percus–Yevick (HPY) approximation and the hydrostatic hypernetted‐chain (HHNC) approximation, respectively, are shown to be capable of accounting for wetting transitions on the basis of general theoretical considerations. Before turning to such transitions, we investigate in this first paper of a series the case of homogeneous hard‐sphere fluids and hard spheres near a hard wall as well as the case of hard spheres inside a slit pore. Numerical results show that the HHNC approximation is better than the HNC approximation for both the homogeneous and inhomogeneous systems considered here while the HPY approximation appears to overcorrect the PY approximation.