Natural Basis Functions Research Articles

Using ensemble Monte Carlo methods to evaluate non-equilibrium Green’s functions

The use of ensemble Monte Carlo methods for the simulation of transport in semiconductor devices has become extensive over the past few decades. This method allows for simulation utilizing particles while addressing the full physics within the device, leaving the computational difficulties to the computer. More recently, the study of quantum mechanical effects within the devices, effects which also strongly affect the carrier transport itself, have become important. While particles have continued to be useful in quantum simulations using Wigner functions, interest in analytical solutions based upon the non-equilibrium Green’s functions (NEGF) have become of greater interest in device simulation. While NEGF has been adopted by many commercial semiconductor, there remains considerable computational difficulty in this approach. Here, a particle approach to NEGF is discussed, and preliminary results presented illustrating the computational efficiency that remains with the use of particles. This approach adopts the natural basis functions for use in a high electric field and the preliminary results are obtained for quantum transport in Si at 300 K. This approach appears to offer significant advantages for the use of NEGF.

A Basis Approach to Surface Clustering

This paper presents a novel method for clustering surfaces. The proposal involves first using natural splines basis functions in a tensor product to smooth the data and thus reduce the dimension to a finite number of coefficients, and then using these estimated coefficients to cluster the surfaces via k-means or spectral clustering. An extension of the algorithm to clustering higher-dimensional tensors is also discussed. We show that the proposed algorithm exhibits the property of strong consistency, with or without measurement errors, in correctly clustering the data as the sample size increases. Simulation studies suggest that the proposed method outperforms the benchmark k-means and spectral algorithm which use the original data. In addition, an EGG real data example is considered to illustrate the practical application of the proposal.

Supersymmetric generalized power functions

Complex-valued functions defined on a finite interval [a, b] generalizing power functions of the type (x−x0)n for n ≥ 0 are studied. These functions called Φ-generalized powers, Φ being a given nonzero complex-valued function on the interval, were considered to construct a general solution representation of the Sturm–Liouville equation in terms of the spectral parameter [V. V. Kravchenko and R. M. Porter, Math. Methods Appl. Sci. 33(4), 459–468 (2010)]. The Φ-generalized powers can be considered as natural basis functions for the one-dimensional supersymmetric quantum mechanics systems taking Φ=ψ02, where the function ψ0(x) is the ground state wave function of one of the supersymmetric scalar Hamiltonians. Several properties are obtained such as Φ-symmetric conjugate and antisymmetry of the Φ-generalized powers, a supersymmetric binomial identity for these functions, a supersymmetric Pythagorean elliptic (hyperbolic) identity involving four Φ-trigonometric (Φ-hyperbolic) functions, as well as a supersymmetric Taylor series expressed in terms of the Φ-derivatives. We show that the first n Φ-generalized powers are a fundamental set of solutions associated with nonconstant coefficient homogeneous linear ordinary differential equations of order n + 1. Finally, we present a general solution representation of the stationary Schrödinger equation in terms of geometric series where the Volterra compositions of the first type are considered.

Closed-form expression for cross-channel conformal blocks near the lightcone

In the study of conformal field theories, conformal blocks in the lightcone limit are fundamental to the analytic conformal bootstrap method. Here we consider the lightcone limit of 4-point functions of generic scalar primaries. Based on the nonperturbative pole structure in spin of Lorentzian inversion, we propose the natural basis functions for cross-channel conformal blocks. In this new basis, we find a closed-form expression for crossed conformal blocks in terms of the Kampé de Fériet function, which applies to intermediate operators of arbitrary spin in general dimensions. We derive the general Lorentzian inversion for the case of identical external scaling dimensions. Our results for the lightcone limit also shed light on the complete analytic structure of conformal blocks in the lightcone expansion.

Generation of Coulomb matrix elements for the 2D quantum harmonic oscillator

The Fock-Darwin states are the natural basis functions for a system of interacting electrons trapped inside a 2D quantum dot. Interaction effects at the mean field level or more elaborate quantum many body descriptions rely on an accurate evaluation of Coulomb matrix elements. In this work, we derive a highly efficient recurrence scheme to compute these elements in Fock-Darwin basis. The algorithm is best implemented on symbolic calculus platforms, preventing the appearance of rounding-off errors. The high speed achieved allows us to calculate all matrix elements in a basis set of several hundred states in very reasonable times (a few hours on a standard computer). We also make use of symmetry to reduce the number of elements to be computed. Finally, we check the reliability of floating-point evaluation for existing analytical expressions. We confirm the stability of a two-folded nonalternating sum for elements involving only the lowest energy levels and put on display the daunting limitations of the expressions commonly used to compute the elements in the general case.

Spectrally Accurate and Well-Posed Jacobi Galerkin Method for Multiple Delay Pantograph Integral Equations

Operational Jacobi Galerkin method is developed and analyzed to solve multiple delay pantograph integral equations. This algorithm is an application of the matrix vector product approach in Galerkin formulation of the problem when the classical Jacobi polynomials are employed as natural basis functions. Our scheme has two main advantages. First, it produces a well-posed approximate method in the sense that the algebraic form of the Galerkin discretization of the problem has an upper triangular structure with a bounded condition number which can be solved directly by the forward substitution method and second, it has a potential convergence property such that its numerical error decay with the familiar spectral rate of convergence. Finally, some numerical examples are used to illustrate the efficiency and accuracy of the method.

Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integro-differential equations

The main concern of this paper is to develop and analyze an operational Tau method for obtaining the numerical solution of fractional weakly singular integro-differential equations when the Jacobi polynomials are used as natural basis functions. This strategy is an application of the matrix–vector–product approach in Tau formulation of the problem. We first study the regularity of the exact solution and show that some derivatives of the exact solution have a singularity at origin dependence on both order of fractional derivative and weakly singular kernel function which makes poor convergence results for the Tau discretization of the problem. In order to recover high-order of convergence, we propose a new variable transformation to regularize the given functions and then to approximate the solution via a satisfactory order of convergence using an operational Tau method. Convergence analysis of this novel method is presented and the expected spectral rate of convergence for the proposed method is established. Numerical results are given which confirm both the theoretical predictions obtained and efficiency of the proposed method.

Discrete galerkin method for fractional integro-differential equations

In this article, we develop a fully Discrete Galerkin(DG) method for solving initial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(GJPs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Modified Jacobi–Bernstein basis transformation and its application to multi-degree reduction of Bézier curves

This paper reports new modified Jacobi polynomials (MJPs). We derive the basis transformation between MJPs and Bernstein polynomials and vice versa. This transformation is merging the perfect Least-square performance of the new polynomials together with the geometrical insight of Bernstein polynomials. The MJPs with indexes corresponding to the number of endpoints constraints are the natural basis functions for Least-square approximation of Bézier curves. Using MJPs leads us to deal with the constrained Jacobi polynomials and the unconstrained Jacobi polynomials as orthogonal polynomials. The MJPs are automatically satisfying the homogeneous boundary conditions. Thereby, the main advantage of using MJPs, in multi-degree reduction of Bézier curves on computer aided geometric design (CAGD), is that the constraints in CAGD are also satisfied and that decreases the steps of multi-degree reduction algorithm. Several numerical results for the multi-degree reduction of Bézier curves on CAGD are given.

Generalized Jacobi functions and their applications to fractional differential equations

In this paper, we consider spectral approximation of fractional differential equations (FDEs). A main ingredient of our approach is to define a new class of generalized Jacobi functions (GJFs), which is intrinsically related to fractional calculus, and can serve as natural basis functions for properly designed spectral methods for FDEs. We establish spectral approximation results for these GJFs in weighted Sobolev spaces involving fractional derivatives. We construct efficient GJF-Petrov-Galerkin methods for a class of prototypical fractional initial value problems (FIVPs) and fractional boundary value problems (FBVPs) of general order, and show that with an appropriate choice of the parameters in GJFs, the resulted linear systems can be sparse and well-conditioned. Moreover, we derive error estimates with convergence rate only depending on the smoothness of data, so truly spectral accuracy can be attained if the data are smooth enough. The idea and results presented in this paper will be useful to deal with more general FDEs associated with Riemann-Liouville or Caputo fractional derivatives.

On Generalized Jacobi–Bernstein Basis Transformation: Application of Multidegree Reduction of Bézier Curves and Surfaces

This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect least-square performance of the GJPs with the geometrical insight of the Bernstein form. Moreover, the GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for least-square approximation of Bézier curves and surfaces. Application to multidegree reduction (MDR) of Bézier curves and surfaces in computer aided geometric design (CAGD) is given.

Comparing low‐frequency and intermittent variability in comprehensive climate models through nonlinear Laplacian spectral analysis

Nonlinear Laplacian spectral analysis (NLSA) is a recently developed technique for spatiotemporal analysis of high‐dimensional data, which represents temporal patterns via natural orthonormal basis functions on the nonlinear data manifold. Through such basis functions, determined efficiently via graph‐theoretic algorithms, NLSA captures intermittency, rare events, and other nonlinear dynamical features which are not accessible through linear approaches (e.g., singular spectrum analysis (SSA)). Here, we apply NLSA to study North Pacific SST monthly data from the CCSM3 and ECHAM5/MPI‐OM models. Without performing spatial coarse graining (i.e., operating in ambient‐space dimensions up to 1.6 × 105after lagged embedding), or seasonal‐cycle subtraction, the method reveals families of periodic, low‐frequency, and intermittent spatiotemporal modes. The intermittent modes, which describe variability in the Western and Eastern boundary currents, as well as variability in the subtropical gyre with year‐to‐year reemergence, are not captured by SSA, yet are likely to have high significance in a predictive context and utility in cross‐model comparisons.

Numerical evolution of general Robinson-Trautman spacetimes: Code tests, wave forms, and the efficiency of the gravitational wave extraction

We present an efficient numerical code based on spectral methods to integrate the field equations of general Robinson-Trautmann spacetimes. The most natural basis functions for the spectral expansion of the metric functions are spherical harmonics. Using the values of appropriate combinations of the metric functions at the collocation points, we have managed to reduce expression swell when the number of spherical harmonics increases. Our numerical code runs with relatively little computational resources and the code tests have shown excellent accuracy and convergence. The code has been applied to situations of physical interest in the context of Robsinson-Trautmann geometries such as: perturbation of the exterior gravitational field of a spheroid of matter; perturbation of an initially boosted black hole; and the nonfrontal collision of two Schwarzschild black holes. In dealing with these processes we have derived analytical lower and upper bounds on the velocity of the resulting black hole and the efficiency of the gravitational wave extraction, respectively. Numerical experiments were performed to determine the forms of the gravitational waves and the efficiency in each situation of interest.

混合カオス信号の独立成分分析とカオス解析

The key ingredient of signal analysis is to find a set of “natural basis functions” which are orthogonal to

Generalized Jacobi polynomials/functions and their applications

We introduce a family of generalized Jacobi polynomials/functions with indexes α , β ∈ R which are mutually orthogonal with respect to the corresponding Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. We establish their basic approximation properties in suitably weighted Sobolev spaces. As an example of their applications, we show that the generalized Jacobi polynomials/functions, with indexes corresponding to the number of homogeneous boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials/functions leads to much simplified analysis, more precise error estimates and well conditioned algorithms.

An S-Plus function to calculate relative risks and adjusted means for regression models using natural splines

We provide for generalized linear regression models that use natural cubic splines to model predictors an S-Plus function to calculate relative risks (RR), log relative risk (log RR), mean percent change (MPC) for continuous covariates modeled using a logarithmic link as well as adjusted means differences (MD) for the identity link. The function makes explicit use of the natural spline basis functions, the estimated coefficients for each natural spline basis function, and the fitted correlation matrix for the estimated coefficients and can thus accommodate any number of degrees of freedom. The main function produces a publication–quality graph of all of these quantities as compared to a user-specified reference value as well as the associated confidence limits. In another function, specific values of these statistics comparing a vector of values of the independent variable to the reference value can be calculated rather than plotted.

Reconstruction of 2D PET data with Monte Carlo generated system matrix for generalized natural pixels

In discrete detector PET, natural pixels are image basis functions calculated from the response of detector pairs. By using reconstruction with natural pixel basis functions, the discretization of the object into a predefined grid can be avoided. Here, we propose to use generalized natural pixel reconstruction. Using this approach, the basis functions are not the detector sensitivity functions as in the natural pixel case but uniform parallel strips. The backprojection of the strip coefficients results in the reconstructed image. This paper proposes an easy and efficient way to generate the matrix M directly by Monte Carlo simulation. Elements of the generalized natural pixel system matrix are formed by calculating the intersection of a parallel strip with the detector sensitivity function. These generalized natural pixels are easier to use than conventional natural pixels because the final step from solution to a square pixel representation is done by simple backprojection. Due to rotational symmetry in the PET scanner, the matrix M is block circulant and only the first blockrow needs to be stored. Data were generated using a fast Monte Carlo simulator using ray tracing. The proposed method was compared to a listmode MLEM algorithm, which used ray tracing for doing forward and backprojection. Comparison of the algorithms with different phantoms showed that an improved resolution can be obtained using generalized natural pixel reconstruction with accurate system modelling. In addition, it was noted that for the same resolution a lower noise level is present in this reconstruction. A numerical observer study showed the proposed method exhibited increased performance as compared to a standard listmode EM algorithm. In another study, more realistic data were generated using the GATE Monte Carlo simulator. For these data, a more uniform contrast recovery and a better contrast-to-noise performance were observed. It was observed that major improvements in contrast recovery were obtained with MLEM when the correct system matrix was used instead of simple ray tracing. The correct modelling was the major cause of improved contrast for the same background noise. Less important factors were the choice of the algorithm (MLEM performed better than ART) and the basis functions (generalized natural pixels gave better results than pixels).

Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials

We extend the definition of the classical Jacobi polynomials withindexes ?, β>?1 to allow ? and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials leads to much simplified analysis, more precise error estimates and well conditioned algorithms.

Smoothing spline ANOVA models for large data sets with Bernoulli observations and the randomized GACV

We propose the randomized Generalized Approximate Cross Validation (ranGACV) method for choosing multiple smoothing parameters in penalized likelihood estimates for Bernoulli data. The method is intended for application with penalized likelihood smoothing spline ANOVA models. In addition we propose a class of approximate numerical methods for solving the penalized likelihood variational problem which, in conjunction with the ranGACV method allows the application of smoothing spline ANOVA models with Bernoulli data to much larger data sets than previously possible. These methods are based on choosing an approximating subset of the natural (representer) basis functions for the variational problem. Simulation studies with synthetic data, including synthetic data mimicking demographic risk factor data sets is used to examine the properties of the method and to compare the approach with the GRKPACK code of Wang (1997c). Bayesian “confidence intervals” are obtained for the fits and are shown in the simulation studies to have the “across the function” property usually claimed for these confidence intervals. Finally the method is applied to an observational data set from the Beaver Dam Eye study, with scientifically interesting results.

Comparison of basis functions in soft x-ray tomography and observation of poloidal asymmetries in impurity density

Natural basis functions (NBF), also known as natural pixels in the literature, have been applied in tomographic reconstructions of soft x-ray (SXR) measurements in the JET (Joint European Torus) tokamak. The results are compared with those obtained with local basis functions (LBF), and those obtained with a conventional constrained-optimization tomography method. Truncated singular value decomposition is used as the inversion method. Reconstructions without a priori information, such as the NBF reconstructions, are, as can be expected, less good than reconstructions in which a priori information is used, as in the conventional method, but the reconstructions are shown to be reliable by means of phantom simulations. Reconstructions with the same number of LBFs as NBFs without a priori information are comparable to NBF reconstructions, although the latter seem to be somewhat better. No significant changes in results, apart from smaller reconstruction errors, are obtained if the measuring system has more regular or complete coverage than the JET SXR system. The various tomography methods are used to assess whether a newly observed in-out asymmetry in the SXR emission during the injection of nickel into an RF-heated plasma, with the peak on the inboard side, is real. A possible explanation for the asymmetry in emissivity is an increased nickel density on the inboard side as a result of an RF-induced increase of the hydrogen-minority density on the outboard side.