Showing papers on "Basis function published in 1973"
TL;DR: In this paper, L2 error estimates for the continuous time and several discrete time Galerkin approxima-tions to solutions of some second order nonlinear parabolic partial differential equations are derived.
Abstract: L2 error estimates for the continuous time and several discrete time Galerkin approxima- tions to solutions of some second order nonlinear parabolic partial differential equations are derived. Both Neumann and Dirichlet boundary conditions are considered. The estimates obtained are the best possible in an L2 sense. These error estimates are derived by relating the error for the nonlinear parabolic problem to known L2 error estimates for a linear elliptic problem. With additional restrictions on basis functions
608 citations
TL;DR: The basis is compared with standard methods in current use and is shown to be superior in terms of energy lowering obtained per additional basis function beyond a minimal number.
Abstract: Generally contracted Gaussian basis functions are defined as those for which each contracted function may have a nonzero contribution from each primitive Gaussian. Alternatives for choice of such bases are tested and guidelines proposed. The basis is compared with standard methods in current use and is shown to be superior in terms of energy lowering obtained per additional basis function beyond a minimal number. A new program for computation of the required multicentered integrals is described.
467 citations
TL;DR: In this paper, the application of matrix methods based on finite differences and on variational (Rayleigh-Ritz-Galerkin) procedures to solution of the radial Schrodinger equation for bound states of the Morse potential was demonstrated.
Abstract: This paper demonstrates the application of matrix methods based on finite differences and on variational (Rayleigh‐Ritz‐Galerkin) procedures to solution of the radial Schrodinger equation for bound states of the Morse potential. It demonstrates sources of numerical inaccuracy: truncation, termination, tolerance, and quadrature. Cubic splines, harmonic oscillators, floating Gaussians, and sines are used as basis functions.
106 citations
TL;DR: In this article, the Galerkin method was used to solve the l = 0 radial Schrodinger equation with piecewise continuous (class C2) polynomial basis functions (B splines or hill functions).
Abstract: This paper demonstrates the use of piecewise continuous (class C2) polynomial basis functions (B splines or hill functions) in solving the l=0 radial Schrodinger equation, with examples of scattering from Eckart, exponential, and static hydrogen potentials, and eigenvalues for Coulomb, harmonic oscillator, and Morse potentials. Simple nonlinear placement of spline centroids can improve accuracy by orders of magnitude. Comparisons demonstrate the greater accuracy of the Galerkin method, compared with collocation, simple finite difference, and Numerov methods.
96 citations
TL;DR: In this paper, it was shown that only for very thin segments that the correct equivalent radius is independent of length when the radius to length ratio (a/L ) is not small, an expansion for the equivalent radius in terms of a/L is given for the self-impedance term.
Abstract: Wire antennas are solved using a moments solution where the method of subsectional basis is applied with both the expansion and testing functions being sinusoidal distributions. This allows not only a simplification of near-field terms but also the far-field expression of the radiated field from each segment, regardless of the length L . Using sinusoidal basis functions, the terms of the impedance matrix obtained become equivalent to the mutual impedances between the subsectional dipoles. These impedances are the familiar impedances found using the induced EMF method. In the induced EMF method an equivalent radius is usually used in the evaluation of the self-impedance term to reduce computation time. However, it is shown that only for very thin segments that the correct equivalent radius is independent of length. When the radius to length ratio ( a/L ) is not small, an expansion for the equivalent radius in terms of a/L is given for the self-impedance term. The use of incorrect self-term, obtained by using a constant equivalent radius term, is shown to be responsible for divergence of numerical solutions as the number of sections is increased. This occurrence is related to the ratio of a/L of the subsections and hence becomes a problem for moderately thick wire antennas even for a reasonably small number of segments per wavelength. Examples are given showing the convergence with the correct self-terms and the divergence when only a length independent equivalent radius is used. The converged solutions are also compared to King's second- and third-order solutions for moderately thick dipoles.
38 citations
TL;DR: The feasibility of many-body perturbation-theory (MBPT) calculations with hole and particle states expanded in a finite set of bound-type functions is investigated in this article.
Abstract: The feasibility of many-body perturbation-theory (MBPT) calculations with hole and particle states expanded in a finite set of bound-type functions is investigated. The correlation energy of atomic beryllium is used as a test case. Particle states with negative and positive energy values are treated uniformly. We find that convergence of individual diagrams to better than ${10}^{\ensuremath{-}4}$ a.u., and similar agreement with Kelly's numerical results can be obtained using basis sets composed of $9s$, $7p$, $5d$, and $4f$ Slater orbitals for intrashell correlation and a $10s8p6d4f$ set for the intershell effects. The total correlation energy calculated with these basis functions is in good agreement with experiment. These results indicate that MBPT calculations by the expansion method are indeed feasible. While this method may not be more convenient than the numerical approach for atomic systems, it should be useful for molecular calculations, where finding a suitable complete set of numerical orbitals presents difficult problems.
38 citations
TL;DR: In this article, a solution procedure for discrete stochastic programs with recourse linear programs under uncertainty is presented, in which the m-dimensional space in which each combination of the discrete values is a lattice point is used to delete infeasible points from the space.
Abstract: This paper presents a solution procedure for discrete stochastic programs with recourse linear programs under uncertainty. It views the m stochastic elements of the requirements vector as an m-dimensional space in which each combination of the discrete values is a lattice point. For a given second-stage basis, certain of the lattice points are feasible. A procedure is presented to delete infeasible points from the space. Thus, the aggregate probability associated with points feasible for this basis can be enumerated, and used to weight the vector of dual variables defined by the basis. Finally, the paper presents a systematic procedure for changing optimal bases so that a feasible and optimal basis is found for every lattice point.
33 citations
TL;DR: In this paper, a device simulation method derived from a numerical method for solving the Boltzmann transport equation is described, which is essentially exact, in particular free from the assumption that electrons respond instantaneously to changes of electric field.
Abstract: A device simulation method derived from a numerical method for solving the Boltzmann transport equation is described. It is essentially exact, in particular free from the assumption that electrons respond instantaneously to changes of electric field. A key feature is the representation of the k dependence of the free carrier distribution by an expansion in a set of basis functions. Reasonable accuracy can be obtained for a fairly small number of basis functions, leading to high computational efficiency compared with alternative exact simulation methods.
32 citations
TL;DR: In this paper, a simple scheme of basis functions piecewise-continuous cubic polynomials, together with a simple quadrature approximation for matrix elements, is presented for the hydrogen atom.
Abstract: The author presents a simple scheme of basis functions piecewise-continuous cubic polynomials, together with a simple quadrature approximation for matrix elements. Applied to the hydrogen atom, the method allows a dozen basis functions to approximate the 14 bound states 1s ... 5f with an average error of 0.03% in energies and 0.2% in the oscillator strengths of the 28 allowed transitions.
28 citations
TL;DR: In this article, a method for perturbation of quantum systems with a symmetry is presented. But the method is not suitable for the case of the center-of-mass motion of a nucleus, the symmetry being the translational invariance.
Abstract: For quantum systems with a symmetry it is often convenient to start from a simple soluble model lacking the symmetry, and restore the symmetry by projection. If this is done for all eigenstates of the model system, one obtains an overcomplete set of basis functions, which is not suitable for standard perturbation theory. The paper develops a method by which one can do perturbation theory in this situation. The method is illustrated by application to the problem of the centre-of-mass motion of a nucleus, the symmetry being the translational and Galileo invariance.
28 citations
TL;DR: Using basis functions with complex coordinates, it is possible to construct discrete approximations to the Fredholm determinant directly at real energies using only square-integrable functions as discussed by the authors, which is equivalent to an analytic continuation of the coordinate dependence of the Hamiltonian.
Abstract: Using basis functions with complex coordinates, it is possible to construct discrete approximations to the Fredholm determinant directly at real energies using only square-integrable functions. For the case of purely elastic scattering, the procedure is equivalent to an analytic continuation of the coordinate dependence of the Hamiltonian. It is shown that improved convergence is obtained by an application of the "dispersion-correction" method. The method is generalized to allow calculation of the "substituted" Fredholm determinants needed to construct the $S$ matrix for many-channel potential-scattering problems. This generalization is not equivalent to a simple continuation of the coordinate dependence of the many-channel Hamiltonian. Results of calculations on several model problems are presented.
TL;DR: In this paper, it was shown that the application of a projection operator from a given group to a function is equivalent to the successive application of projection operators from factor groups of the starting group to that function.
Abstract: It is shown that the application of a projection operator from a given group to a function is equivalent to the successive application of projection operators from factor groups of the starting group to that function. When used with the factor groups representing the site symmetry of a position and the simplest group of interchanges of positions, this concept provides a very simple method for obtaining symmetry adapted linear combinations of basis functions.
TL;DR: In this article, a moment method with mixed basis functions is introduced, where modal basis functions are used for the expansion of the currents corresponding to the scattered propagating modes, while pulse basis functions were used to expand the current corresponding to scattered envancescent waves, together with the Dirac /spl delta/ weighting functions, reducing the number of total basis functions needed.
Abstract: A moment method with mixed basis functions is introduced. In this formulation, modal basis functions are used for the expansion of the currents corresponding to the scattered propagating modes, while pulse basis functions are used for the expansion of the current corresponding to the scattered envancescent waves. This, together with the Dirac /spl delta/ weighting functions, reduces the number of total basis functions needed while retaining the simplicity and versatility of the method to cover junctions of an arbitrary shape. This method is applied to study examples of homogeneous and inhomogeneous waveguide junctions of parallel-plate waveguide propagating TE waves. It is found that for junctions that are not electrically large the convergence of the solutions is good. An appendix is included to transform and quicken the numerical integration of the modal basis functions.
01 Jan 1973
TL;DR: This chapter discusses the mathematical theory of finite element methods, and the Sobolev spaces and variational formulation of elliptic boundary value problems, nodal finite element method, abstract finiteelement method, and nonlinear elliptic problems and time dependent problems.
Abstract: Publisher Summary This chapter discusses the mathematical theory of finite element methods. It also discusses the Sobolev spaces and variational formulation of elliptic boundary value problems, nodal finite element method, abstract finite element method, and nonlinear elliptic problems and time dependent problems. It is well known that the finite element method is a special case of the Ritz–Galerkin method. The classical Ritz approach has two great shortcomings: (1) in practice, construction of the basis functions is only possible for some special domains: (2) the corresponding Ritz matrices are full matrices, and are very often, for simple problems, catastrophically ill-conditioned. The crucial difference between the finite element method and the classical Ritz–Galerkin technique lie in the construction of the basis functions. In the finite element method, the basis functions for general domains can easily be computed. The main feature of these basis functions is that they vanish over all but a fixed number of the elements into which the given domain is divided. This property causes the Ritz matrices to be sparse band matrices, and the resulting Ritz process is stable. There are in addition to the Ritz–Galerkin method many other direct variational methods, and the differences among these stem from the variational principles used.
01 Jan 1973
TL;DR: The objective is to show the relationship of the finite element method to a variety of classical methods of approximate analysis, to demonstrate techniques for formulating various types of finite element models of boundary and initial value problems, and to establish the generality and flexibility of the method.
Abstract: Publisher Summary This chapter discusses finite element applications in mathematical physics. The objective is to show the relationship of the finite element method to a variety of classical methods of approximate analysis, to demonstrate techniques for formulating various types of finite element models of boundary and initial value problems, and then, to establish the generality and flexibility of the method, to discuss a broad range of problems in mathematical physics in which the method has been applied successfully. The finite-element method is a systematic technique for constructing the basis functions ωi(x) for Ritz–Galerkin approximations for irregular domains. Apart from a number of other advantages, the method overcomes all of the traditional disadvantages of Ritz–Galerkin procedures. The basis functions ωi(x) are generated in a straightforward and systematic manner, irregular domains and mixed boundary conditions are easily accommodated, the resulting equations describing the discrete model are generally well-conditioned, and the method is exceptionally well suited for implementation via electronic computers.
TL;DR: In this article, it was shown that in addition to the usual extended basis functions (normal modes), a set of localized basis functions are available for lattices with several atoms per unit cell.
Abstract: : For the case of electrons in a periodic lattice there exist extended basis functions (Bloch waves) and localized basis functions (Wannier functions). It is shown that also for lattice vibrations there exists, in addition to the usual extended basis functions (normal modes), a set of localized basis functions. These are non-trivial only for lattices with several atoms per unit cell. Examples are worked out and discussed. These functions appear to be of interest for the treatment of defects and other problems calling for a localized description. (Author)
TL;DR: In this article, the Schmidt orthogonalization method is applied to perturbative expansions for the energies and transition amplitudes in systems described by nonorthogonal basis functions.
Abstract: A new procedure is presented which enables the development of perturbative expansions for the energies and transition amplitudes in systems described by nonorthogonal basis functions The procedure is based upon defining a new metric matrix, that is, by redefining the quantum-mechanical scalar product The results justify certain previous applications of the Schmidt orthogonalization method It is shown that the same results can be obtained using Lowdin orthogonalization While the procedure was developed for application with the correlated basis function (CBF) approach to dilute3He-4He solutions, it can be applied in any nonorthogonality situation
TL;DR: In this paper, the second-order perturbation energy expression is derived from a Gaussian quadrature, and the bounds obtained from the larger basis set are of comparable quality to those reported using Gaussian Quadrature or the [2, 1] Pade approximants to bound the polarizabilities in the Casimir-Polder formula.
Abstract: Starting from the second‐order perturbation energy expression and utilizing inner projection and operator inequalities techniques, easy to evaluate expressions for the bounds to dispersion energy coefficients are obtained in terms of ground state sum rule values of the separated atoms for two sets of basis functions. The resulting bounds are narrower than those obtained starting from the Casimir‐Polder integral formula and bounding each of the polarizabilities in that expression by using either the present technique and basis set or the [1,0] Pade approximants. The bounds obtained here from the larger basis set are of comparable quality to those reported using Gaussian quadrature or the [2,1] Pade approximants to bound the polarizabilities in the Casimir‐Polder formula. A derivation of the Kramer‐Herschbach combination rule from one of the bounds is also presented.
TL;DR: In this article, the binomial theorem on trace N−n2 (⩾ p for all positive and negative integral values of n except n = −1 and ⩽ p for n = 0.
01 Jan 1973
TL;DR: In this paper, generalized matrix inverse techniques for local approximations of operator equations are presented, which are based on the Galerkin approach and the Rayleigh-Ritz analysis.
Abstract: Publisher Summary This chapter presents the generalized matrix inverse techniques for local approximations of operator equations. Considerable advance has been made in the practical implementation of the Galerkin and the Rayleigh–Ritz methods to equations of importance to engineering and physics. The prime features of this advance have been the use of localized basis functions and the resulting discretization processes that are amenable to computer instruction. In terms of the Galerkin approach, the discretization process has resulted in approximate general solutions of differential equations, while with Rayleigh–Ritz analysis it has fostered the conventional finite element method. As with any discretization process, both of these methods consist of two separate steps. First, an approximation of the functions comprising the domain space of the operator must be established and, second, a numerical equivalent of the operator acting on this space must be devised. Unfortunately, all existing developments of these methods emphasize the second of these steps. Accordingly, while methods furnishing good solutions of differential equations are available, uncertainties and difficulties in the development and the use of legitimate approximation functions for the domain space of an operator are often encountered.
TL;DR: In this paper, the authors studied the unitary irreducible representations of the group SL(2C) by taking the coupled states of two angular momenta J1 and J2 as basis functions.
Abstract: The unitary irreducible representations of the group SL(2C) are studied by taking the coupled states of two angular momenta J1 and J2 as basis functions. The unitarity of the representations leads to unphysical values of j1 and j2 such that one of the quantities sigma =j1+j2+1 and j12=j1-j2 is integral or half-integral and the other is purely imaginary or real. The formulae derived in this paper exhibit a beautiful symmetry with respect to the interchange of these two quantities. The basis functions are expressible in terms of terminating hypergeometric series and, by using the properties of the latter, the matrices of the generators and of finite transformations are easily determined. The matrix element of the 'boost operator' corresponding to a pure Lorentz transformation in the x3-x4 plane is found to take the form of a finite linear combination of 3F2 functions.
TL;DR: In this paper, it is shown that, in the absence of computational error, the iterative procedure would converge to the minimum-residue approximate solution of minimum norm, when appropriate ellipsoidal norms are employed which are defined by the noise and error processes present.
01 Jan 1973
TL;DR: This chapter presents two iterative solution techniques that have been observed to effect substantial reductions in both storage and time requirements for a number of practical heat conduction applications and investigates the relative efficiency of SOR and direct solution procedures as a function of the mesh spacings on rectangular grids and the order of the elliptic operator.
Abstract: Publisher Summary This chapter reviews the dependence of computing storage and time requirements on the size and structure of finite element problems and presents two iterative solution techniques that have been observed to effect substantial reductions in both these quantities for a number of practical heat conduction applications. The chapter discusses the remarks informally presented by L. H. Seitelman at the September, 1971 Symposium on Sparse Matrices and their Applications at the IBM Thomas J. Watson Research Center. Since that time, related work by Fix and Larsen has appeared. Their study of SOR convergence for several different families of basis functions for elliptic problems on rectangular grids shows that convergence is fastest for spline functions. In addition, they investigate the relative efficiency of SOR and direct solution procedures as a function of the mesh spacings on rectangular grids and the order of the elliptic operator. While they recognize the efficiency of SOR for three-dimensional work, their numerical experiments are confined to one-and two-dimensional problems.
TL;DR: In this article, the authors formulated constrained minimization problems from a quasilinear parabolic boundary value problem (possibly with nonlinear boundary conditions), making use of the latter's (conditional) inverse-positive property.
Abstract: Constrained minimization problems are formulated from a quasilinear parabolic boundary value problem (possibly with nonlinear boundary conditions), making use of the latter’s (conditional) inverse-positive property. Approximate solutions and three error bounds can be obtained by solving these minimization problems by linear programming and discretization techniques. Numerical results are obtained using splines as basis functions.
TL;DR: The Gauss elimination technique with full pivoting as generalized to nonsquare, complex matrices is used throughout to determine basis vectors and dependent vector relationships.
TL;DR: In this article, a spinor-scalar Bethe-Salpeter equation with a scalar ladder-like interaction with a relativistic parton model of baryons is considered.
TL;DR: In this article, the excitation spectrum of a charged Bose gas in the high density limit is calculated to two orders in the expansion parameter r s 3/4 by the method of correlated basis functions (CBF).
Abstract: The excitation spectrum of a charged Bose gas in the high-density limit is calculated to two orders in the expansion parameter r s 3/4 by the method of correlated basis functions (CBF). To O(r s 3/4 ), the result differs from that obtained variationally, but agrees with that obtained by Ma and Woo via diagrammatic perturbation theory. The choice of the basis functions is, however, not unique. By choosing an alternate set of CBF which employs the ground-state eigenfunction as a correlating factor, we find it possible to extract from the excitation spectrum an expression for the liquid-structure function, also to two orders in r s 3/4 , which represents a new result. This new facet of CBF helps to demonstrate the flexibility of the method and the ease with which it can be applied to calculating properties of many-particle systems, model as well as realistic.
TL;DR: In this paper, the energy spectrum of elementary excitations and the liquid-structure function of liquid4He are calculated by means of the method of correlated basis functions in the approximation of second-order perturbation theory.
Abstract: The energy spectrum of elementary excitations and the liquid-structure function of liquid4He are calculated by means of the method of correlated basis functions in the approximation of second-order perturbation theory. The procedure is based on (i) the construction of phonon functions in terms of collective coordinates and the optimum Bijl-Dingle-Jastrow type of ground-state wave function and (ii) the evaluation of leading correction terms to the Bijl-Feynman excitation spectrum, which are generated by two types of three-phonon vertices. Numerical results are obtained using the optimum liquid-structure function computed by Campbell and Feenberg in the paired-phonon analysis.