Distributed Approximating Functional Research Articles

Solving the Generalized Regularized Long Wave Equation Using a Distributed Approximating Functional Method

The generalized regularized long wave (GRLW) equation is solved numerically by using a distributed approximating functional (DAF) method realized by the regularized Hermite local spectral kernel. Test problems including propagation of single solitons, interaction of two and three solitons, and conservation properties of mass, energy, and momentum of the GRLW equation are discussed to test the efficiency and accuracy of the method. Furthermore, using the Maxwellian initial condition, we show that the number of solitons which are generated can be approximately determined. Comparisons are made between the results of the proposed method, analytical solutions, and numerical methods. It is found that the method under consideration is a viable alternative to existing numerical methods.

Lagrange wavelets for signal processing

This paper deals with the design of interpolating wavelets based on a variety of Lagrange functions, combined with novel signal processing techniques for digital imaging. Halfband Lagrange wavelets, B-spline Lagrange wavelets and Gaussian Lagrange (Lagrange distributed approximating functional (DAF)) wavelets are presented as specific examples of the generalized Lagrange wavelets. Our approach combines the perceptually dependent visual group normalization (VGN) technique and a softer logic masking (SLM) method. These are utilized to rescale the wavelet coefficients, remove perceptual redundancy and obtain good visual performance for digital image processing.

Particular and homogeneous solutions of time-independent wavepacket Schrödinger equations: calculations using a subset of eigenstates of undamped or damped Hamiltonians

A variety of causal, particular and homogeneous solutions to the time-independent wavepacket Schrodinger equation have been considered as the basis for calculations using Chebychev expansions, finite-τ expansions obtained from a partial Fourier transform of the time-dependent Schrodinger equation, and the distributed approximating functional (DAF) representation for the spectral density operator (SDO). All the approximations are made computationally robust and reliable by damping the discrete Hamiltonian matrix along the edges of the finite grid to facilitate the use of compact grids. The approximations are found to be completely well behaved at all values of the (continuous) scattering energy. It is found that the DAF–SDO provides a suitable alternative to Chebychev propagation.

NUMERICAL SOLUTION OF THE STATIONARY FPK EQUATION USING SHANNON WAVELETS

The Fokker–Planck–Kolmogorov (FPK) equation governs the probability density function (p.d.f.) of the dynamic response of a particular class of linear or non-linear system to random excitation. This paper proposes a numerical method for calculating the stationary solution of the FPK equation, which is based upon the weighted residual approach using Shannon wavelets as shape functions. The method is developed here for an n -dimensional system and its relationship with the distributed approximating functional (DAF) approach is investigated. For the purposes of validation, numerical results obtained using the proposed method are compared with available exact solutions and numerical solutions for some non-linear oscillators. For the systems considered excellent results over the main body and tails of the marginal distributions are obtained. The accuracy and efficiency of the method are investigated in comparison to the finite element method (FEM).

Iterative and direct methods employing distributed approximating functionals for the reconstruction of a potential energy surface from its sampled values

The reconstruction of a function from knowing only its values on a finite set of grid points, that is the construction of an analytical approximation reproducing the function with good accuracy everywhere within the sampled volume, is an important problem in all branches of sciences. One such problem in chemical physics is the determination of an analytical representation of Born–Oppenheimer potential energy surfaces by ab initio calculations which give the value of the potential at a finite set of grid points in configuration space. This article describes the rudiments of iterative and direct methods of potential surface reconstruction. The major new results are the derivation, numerical demonstration, and interpretation of a reconstruction formula. The reconstruction formula derived approximates the unknown function, say V, by linear combination of functions obtained by discretizing the continuous distributed approximating functional (DAF) approximation of V over the grid of sampling. The simplest of contracted and ordinary Hermite-DAFs are shown to be sufficient for reconstruction. The linear combination coefficients can be obtained either iteratively or directly by finding the minimal norm least-squares solution of a linear system of equations. Several numerical examples of reconstructing functions of one and two variables, and very different shape are given. The examples demonstrate the robustness, high accuracy, as well as the caveats of the proposed method. As to the mathematical foundation of the method, it is shown that the reconstruction formula can be interpreted as, and in fact is, frame expansion. By recognizing the relevance of frames in determining analytical approximation to potential energy surfaces, an extremely rich and beautiful toolbox of mathematics has come to our disposal. Thus, the simple reconstruction method derived in this paper can be refined, extended, and improved in numerous ways.

Distributed approximating functional treatment of noisy signals

Based on their so-called “well-tempered” property, distributed approximating functionals (DAFs) are shown to be very good data filters, and consequently, they have many potential applications in all fields of engineering and the natural sciences. In this paper, a periodic extension of a noisy signal is proposed to generate a “pseudo-signal” in the infinite domain, enabling the use of noncausal, zero-phase window filters that require a knowledge of the signal in the extended domain. The extended signal is also useful for the application of fast Fourier transforms (in which the preferred number of sampled data points is a power of 2). The most attractive feature of the method is that it introduces little aliasing between the original and true signal. The resulting extended signal is then filtered by using DAFs as low pass filters under the assumption that the true signal is bandwidth limited and most of the noise components are in the high frequency region. A “signature” based on computing the root-mean-square value of the filtered signal is introduced to indicate when the high frequency noise has been eliminated with the choice of the DAF parameters. To illustrate the usefulness of the present algorithm, two noisy signal examples are periodically extended and filtered using DAFs.

Generalized symmetric interpolating wavelets

A new class of biorthogonal wavelets-interpolating distributed approximating functional (DAF) wavelets are proposed as a powerful basis for scale-space functional analysis and approximation. The important advantage is that these wavelets can be designed with infinite smoothness in both time and frequency spaces, and have as well symmetric interpolating characteristics. Boundary adaptive wavelets can be implemented conveniently by simply shifting the window envelope. As examples, generalized Lagrange wavelets and generalized Sinc wavelets are presented and discussed in detail. Efficient applications in computational science and engineering are explored.

Distributed approximating functional approach to Burgers' equation in one and two space dimensions

The method of distributed approximating functionals (DAFs) is applied to Burgers' equation, as a typical nonlinear PDE, in one and two space dimensions. This equation is similar to, but simpler than, the Navier-Stokes equation in fluid dynamics. The present approach uses DAFs for spatial discretization and a Taylor expansion for time discretization. Several moderately large values of the Reynolds number are considered in the present application. The results obtained, which are in excellent agreement with the formally exact series solutions, are compared with those obtained by other authors using various different methods. It is found that a simple numerical propagation scheme based on DAFs provides highly accurate numerical solutions for Burgers' equation, while requiring very few grid points.

Lagrange distributed approximating functional method for the solution of the Schrödinger equation

The utility and accuracy of a new distributed approximating functional (DAF), combining the Gaussian weighted DAF concept with Lagrange interpolation, is explored for the discrete spectrum solution of the Schrödinger equation. Two instructive examples, an I 2 Morse oscillator and the 2-dimensional Henon–Heiles potential, are considered in the present study. The present “Lagrange DAF” (LDAF) approach achieves extremely high accuracy for I 2 while using fewer grid points than previous approaches. The present results for the Henon–Heiles system are in excellent agreement with those of earlier established methods, such as that of Shizgal.

Distributed approximating functional fit of the H3 ab initio potential-energy data of Liu and Siegbahn

We report a distributed approximating functional (DAF) fit of the ab initio potential-energy data of Liu [J. Chem. Phys. 58, 1925 (1973)] and Siegbahn and Liu [ibid. 68, 2457 (1978)]. The DAF-fit procedure is based on a variational principle, and is systematic and general. Only two adjustable parameters occur in the DAF leading to a fit which is both accurate (to the level inherent in the input data; RMS error of 0.2765 kcal/mol) and smooth (“well-tempered,” in DAF terminology). In addition, the LSTH surface of Truhlar and Horowitz based on this same data [J. Chem. Phys. 68, 2466 (1978)] is itself approximated using only the values of the LSTH surface on the same grid coordinate points as the ab initio data, and the same DAF parameters. The purpose of this exercise is to demonstrate that the DAF delivers a well-tempered approximation to a known function that closely mimics the true potential-energy surface. As is to be expected, since there is only roundoff error present in the LSTH input data, even more significant figures of fitting accuracy are obtained. The RMS error of the DAF fit, of the LSTH surface at the input points, is 0.0274 kcal/mol, and a smooth fit, accurate to better than 1 cm−1, can be obtained using more than 287 input data points.

Distributed approximating functional approach to fitting and predicting potential surfaces. 1. Atom-atom potentials

A procedure is presented for obtaining an analytical representation of the potential energy for a diatomic system, knowing the potential only at finite number of internuclear separations. The approach is based on the distributed approximating functional (DAF), which has the property of delivering comparable accuracy for approximating a function and its derivatives, given such limited input data. The method is illustrated by applications to several atom-atom potentials, including those for the Li2 and H2 systems.

Distributed approximating functional approach to fitting and predicting potential surfaces. 1. Atom-atom potentials

A procedure is presented for obtaining an analytical representation of the potential energy for a diatomic system, knowing the potential only at finite number of internuclear separations. The approach is based on the distributed approximating functional (DAF), which has the property of delivering comparable accuracy for approximating a function and its derivatives, given such limited input data. The method is illustrated by applications to several atom-atom potentials, including those for the Li 2 and H 2 systems.

Direct approach to density functional theory: iterative treatment using a polynomial representation of the Heaviside step function operator

A new approach to density functional theory is presented. The ground state electronic density and energy of a many-electron system are obtained directly using a polynomial representation of the Heaviside step operator acting on a trial wavefunction. A radial coordinate extension of the distributed approximating functional (DAF) is developed to treat Coulomb singularities and cusps accurately. Examples of electronic structure for the He and Ne atomic systems are presented.