Laguerre Polynomial Expansion Research Articles

Linearised Fokker–Planck collision model for gyrokinetic simulations

Abstract We introduce a gyrokinetic, linearised Fokker–Planck collision model that satisfies conservation laws and is accurate at arbitrary collisionalities. The differential test-particle component of the operator is exact; the integral field-particle component is approximated using a spherical harmonic and a modified Laguerre polynomial expansion developed by Hirshman and Sigmar (1976 Phys. Fluids 19 1532). The numerical methods of the implementation in the δf-gyrokinetic code stella (Barnes et al 2019 J. Comput. Phys. 391 365–80) are discussed, and conservation properties of the operator are demonstrated. The collision model is then benchmarked against the collision model of the gyrokinetic solver GS2 in the limiting cases of a reduced test-particle collision operator and energy- and momentum-conserving operator. The accuracy of the full collision model is investigated by solving the parallel Spitzer-Härm problem for the transport coefficients. It is shown that retaining collisional energy flux and higher-order terms in the field-particle operator reduces errors in the transport coefficients from 10%–25% for a simple momentum- and energy-conserving model to under 1%.

A note on the 1-D minimization problem related to solenoidal improvement of the uncertainty principle inequality

This paper gives a second way to solve the one-dimensional minimization problem of the form : minf≢0∫0∞f′′2xμ+1dx∫0∞x2f′2-εf2xμ-1dx∫0∞f′2xμdx2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\min _{f\ ot \\equiv 0}\\frac{\\displaystyle \\int \\limits _0^\\infty \\left( f''\\right) ^2x^{\\mu +1}dx\\int \\limits _0^\\infty \\left( {x}^2\\left( f'\\right) ^2 -\\varepsilon f^2\\right) {{x}}^{\\mu -1}d{x}}{\\displaystyle \\left( \\int \\limits _0^\\infty \\left( f'\\right) ^2 {{x}}^{\\mu }d{x}\\right) ^2} \\end{aligned}$$\\end{document}for scalar-valued functions f on the half line, where f′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f'$$\\end{document} and f′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f''$$\\end{document} are its derivatives and ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} and μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} are positive parameters with ε<μ24.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon < \\frac{\\mu ^2}{4}.$$\\end{document} This problem plays an essential part of the calculation of the best constant in Heisenberg’s uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021).

Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

AbstractIn this paper, we give a criterion to prove boundedness results for several operators from the Hardy‐type space to and also from to the space of functions of bounded mean oscillation , with respect to the probability measure on when is a multi‐index in . We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting.

Variation inequalities for Riesz transforms and Poisson semigroups associated with Laguerre polynomial expansions

In this paper, we establish [Formula: see text]-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.

On optimal convergence rates of Laguerre polynomial expansions for piecewise functions

By applying Hilb type formula and generalized van der Corput type Lemmas, this paper presents optimal decay rates on the expansion coefficients for given functions expanded in the form of Laguerre polynomial series, from which it leads to the convergence rates on the spectral orthogonal projections, and derives new estimates on the errors for Gauss–Laguerre quadrature. Ample numerical experiments are carried out to validate the optimality and accuracy of these theoretical results.

Local gyrokinetic collisional theory of the ion-temperature gradient mode

We present a study of the linear properties of the ion-temperature gradient (ITG) modes with collisions modelled for the first time by the linearized gyrokinetic (GK) Coulomb collision operator (Freiet al.,J. Plasma Phys., vol. 87, issue 5, 2021, 905870501) in the local limit. The study is based on a Hermite–Laguerre polynomial expansion of the perturbed ion distribution function applied to the linearized GK Boltzmann equation, yielding a hierarchy of coupled equations for the expansion coefficients, referred to as gyromoments. We explore the collisionless and high-collisional limits of the gyromoment hierarchy analytically. Parameter scans revealing the dependence of the ITG growth rate on the collisionality modelled using the GK Coulomb operator are reported, showing strong damping at small scales as the collisionality increases and, therefore, the need for a steeper gradient for the ITG onset at high collisionality to overcome the finite Larmor radius (FLR) collisional stabilization. The predictions on the ITG growth rate by the GK Coulomb operator are compared with other collision operator models, such as the Sugama, the Dougherty, as well as the momentum-conserving pitch-angle scattering and the Hirshman–Sigmar–Clarke collision operators derived for the first time in terms of gyromoments. The importance of FLR terms in the collision operators is pointed out by the appearance of a short wavelength ITG branch when collisional FLR terms are neglected, this branch being completely suppressed by FLR collisional effects. Energy diffusion is shown to be important at high collisionality and at small scale lengths. Among the GK collision operators considered in this work, the GK Sugama collision operator yields, in general, the smallest deviation compared with the GK Coulomb collision operator, while the largest deviations are found with the GK Dougherty operator. Convergence studies of the gyromoment method are reported and show that the drifts associated with the gradient and curvature of the magnetic field increase the required number of gyromoments at low collisionality. Nevertheless, the low number of gyromoments necessary for convergence at high collisionality constitutes an attractive numerical and analytical feature of the gyromoment approach to study the plasma dynamics in the boundary of fusion devices.

Laguerre-Gramian-based structure-preserving model order reduction for second-order form systems

A new structure-preserving model order reduction technique based on Laguerre-Gramian for second-order form systems is presented in this article. The main task of the proposed approach is to use the Laguerre polynomial expansion of the matrix exponential function to obtain the approximate low-rank decomposition of the Gramians for the equivalent first-order representation of the original second-order form system. The approximate balanced system is generated by a balancing transformation which is directly computed from the expansion coefficients of impulse responses in the space spanned by Laguerre polynomials, without computing the full Gramians for the first-order representation. Then, the reduced second-order model is constructed by truncating the states with small approximate Hankel singular values (HSVs). The above method has a disadvantage that it may unexpectedly result in unstable systems although the original one is stable. Therefore, modified reduction procedure combined with the dominant subspace projection method is presented to alleviate the limitation. Finally, two numerical experiments are provided to demonstrate the effectiveness of the algorithms.

Statistical Feature Extraction and System Identification Algorithms for Partial Discharge Signal Classification Using Laguerre Polynomial Expansion

In this paper, a novel algorithm for partial discharge (PD) pulse waveform analysis and separation based on orthogonal series expansion using Laguerre polynomials is proposed. A system identification technique is also developed to help increase the accuracy rate of the signal classification. Laguerre polynomial series expansions are applied on the PD pulses to effectively extract important information and well-discriminative features. Laguerre expansion coefficients inherently contain information about the amplitude (intensity) and the relaxation time of a dynamic system and this helps direct characterization of PD pulses. The proposed technique is applied to PD pulses acquired from a laboratory set of multiple PD sources. It is shown that this statistical method can classify partial discharge signals with high accuracies, even when the signals are visually indistinguishable. In this research, two methods are developed for system identification. One is based on a deterministic approach in the form of a recursive formula and the other one is a stochastic method based on a group Lasso methodology. The aim of system identification is to improve the classification accuracy by removing the effect of the measurement system from the observed PD waveforms. The accuracy rate of the proposed algorithm is evaluated using the signals that are captured from simultaneous artificial defects in high pressure SF <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</inf> and in low pressure air.

Location recognition of unmanned vehicles based on visual semantic information and geometric distribution

Due to the influence of weather, light, season and angle changes, the appearance of objects changes visually, which makes it difficult for unmanned vehicles that rely on visual positioning to complete their positioning work. This paper proposes a coordinated positioning method that is composed of semantic information and geometric relationships distribution (GRD), which improves the robustness of unmanned vehicle location under the above conditions. First, we improved the FAST-SCNN semantic segmentation network and replaced its fully connected layer with the conv4-3 module to prevent the spatial information of the image from being lost in the fully connected layer. At the same time, the conv4-3 layer contains the richest semantic information, we use image semantic content to create a dense and prominent scene description. These prominent descriptions were learned from a large data set of perceptual changes. The method can accurately segment geometrically stable image regions. We combine the characteristics of these highlighted areas with the existing overall representation to produce a more robust scene descriptor. Second, a method is designed to integrate the matching of semantic labels and geometric distribution relations, which is a new closed loop location recognition label and landmark map. The geometric pair relationship between the ground marks is encoded as a continuous probability density function, the GRD function, which is expressed by a Laguerre polynomial and Fourier series basis expansion. This orthogonal basis representation allows for efficient computation of rotation and translation invariants, which are used to compare signatures and search for potential loop closure candidates. Finally, we evaluate our method with some of the most advanced algorithms, such as OpenSeqSLAM, AlexNet and VSO, to demonstrate its advantages. The experimental results for representative data sets show that the method based on Fast-SCNN is superior to other methods.

Dynamics of quantum coherence in a spin-star system: Bipartite initial state and coherence distribution

We investigate the transient dynamics of quantum coherence for a system of two central spins in a spin-star environment by employing a numerical procedure based on a Laguerre polynomial expansion scheme. The dynamics of the total, local, and global coherence are calculated for different values of the anisotropy parameter, the system-bath interaction strengths, and temperature for different initial bipartite states. Significant dynamical features of quantum coherence are found as follows: (i) an $X$ state can only have global coherence; (ii) a state with only initial local coherence gains global coherence during the course of evolution by the induced correlations between the two-qubit system and the common bath; (iii) an incoherent state gains coherence by interacting with an external bath. We find there are two primary ways to gain coherence for an incoherent state: one is by interacting with the external quantum bath and the other is through interconversion of other quantum properties such as purity into coherence. Finally, we demonstrate that our results for the system in an infinite bath also hold qualitatively when the system is in contact with a finite bath.

Quasi-Optimal Rank-Structured Approximation to Multidimensional Parabolic Problems by Cayley Transform and Chebyshev Interpolation

Abstract In the present paper we propose and analyze a class of tensor approaches for the efficient numerical solution of a first order differential equation ψ ′ ⁢ ( t ) + A ⁢ ψ = f ⁢ ( t ) {\psi^{\prime}(t)+A\psi=f(t)} with an unbounded operator coefficient A. These techniques are based on a Laguerre polynomial expansions with coefficients which are powers of the Cayley transform of the operator A. The Cayley transform under consideration is a useful tool to arrive at the following aims: (1) to separate time and spatial variables, (2) to switch from the continuous “time variable” to “the discrete time variable” and from the study of functions of an unbounded operator to the ones of a bounded operator, (3) to obtain exponentially accurate approximations. In the earlier papers of the authors some approximations on the basis of the Cayley transform and the N-term Laguerre expansions of the accuracy order 𝒪 ⁢ ( e - N ) {\mathcal{O}(e^{-N})} were proposed and justified provided that the initial value is analytical for A. In the present paper we combine the Cayley transform and the Chebyshev–Gauss–Lobatto interpolation and arrive at an approximation of the accuracy order 𝒪 ⁢ ( e - N ) {\mathcal{O}(e^{-N})} without restrictions on the input data. The use of the Laguerre expansion or the Chebyshev–Gauss–Lobatto interpolation allows to separate the time and space variables. The separation of the multidimensional spatial variable can be achieved by the use of low-rank approximation to the Cayley transform of the Laplace-like operator that is spectrally close to A. As a result a quasi-optimal numerical algorithm can be designed.

Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation

In this article, we study sharp estimates for the Schrödinger operator via the framework of orthogonal polynomials. We use Hermite and Laguerre polynomial expansions to produce sharp Strichartz estimates for even exponents. In particular, for radial initial data in dimension |$2$|⁠, we establish an interesting connection of the Strichartz norm with a combinatorial problem about words with four letters. We use spherical harmonics and Gegenbauer polynomials to prove a sharpened weighted inequality for the Schrödinger equation that is maximized by radial functions.

Impact of the neoclassical distribution function on turbulent impurity and momentum fluxes: fluid model and gyrokinetic simulations

The impact of the neoclassical background on turbulent impurity transport is investigated by means of gyrokinetic simulations supported by fluid equations. The latter are derived, using a Laguerre polynomials expansion of the first order neoclassical distribution function, and analytical expressions of the turbulent momentum flux and impurity transport coefficients are assessed. Comparisons of gyrokinetic simulations including this neoclassical background (coupling between the codes GKW and NEO) and the fluid model are used to identify the main mechanisms behind the modification of the turbulent transport channels and benchmark the numerical implementation. These mechanisms include a modification of the parallel dynamics of the main ions and direct contributions stemming from the asymmetry in the parallel velocity space of the neoclassical distribution function. The latter which is found dominant for turbulent impurity transport, increases with increasing collisionality, , , impurity mass, safety factor and aspect ratio. These contributions to momentum and impurity fluxes are also found to depend on the directions of the toroidal magnetic field and plasma current.

Numerical solution of DGLAP equations using Laguerre polynomials expansion and Monte Carlo method.

We investigate the numerical solutions of the DGLAP evolution equations at the LO and NLO approximations, using the Laguerre polynomials expansion. The theoretical framework is based on Furmanski et al.’s articles. What makes the content of this paper different from other works, is that all calculations in the whole stages to extract the evolved parton distributions, are done numerically. The employed techniques to do the numerical solutions, based on Monte Carlo method, has this feature that all the results are obtained in a proper wall clock time by computer. The algorithms are implemented in FORTRAN and the employed coding ideas can be used in other numerical computations as well. Our results for the evolved parton densities are in good agreement with some phenomenological models. They also indicate better behavior with respect to the results of similar numerical calculations.

A Laguerre expansion method for the field particle portion in the linearized Coulomb collision operator

The spherical coordinates expressions of the Rosenbluth potentials are applied to the field particle portion in the linearized Coulomb collision operator. The Sonine (generalized Laguerre) polynomial expansion formulas for this operator allowing general field particles' velocity distributions are derived. An important application area of these formulas is the study of flows of thermalized particles in NBI-heated or burning plasmas since the energy space structure of the fast ions' slowing down velocity distribution cannot be expressed by usual orthogonal polynomial expansions, and since the Galilean invariant property and the momentum conservation of the collision must be distinguished there.

Analyzing the parton densities and constructing the xF3 structure function, using the Laguerre polynomials expansion and Monte Carlo calculations

We consider the neutrino-nucleon scattering to analyse the xF3 structure function. We solve first numerically the DGLAP evolution equation, using the Laguerre polynomials expansion and Monte Carlo calculations. What we get for the evolved parton densities is in good agreement with the fitting GRSV and CETQ parameterizations. We then construct the xF3 structure function in the Bjorken x-space using the Laguerre polynomials expansion while we achieve this structure function initially in the Mellin moment space. The relations which convert the structure function from the Mellin moment space to the Bjorken x-space, using the Laguerre polynomials, are introduced for the first time in this article. These relations enable us to do the fitting and extract the QCD cut-off parameter which is in good agreement with what is expected. To confirm the calculation we reconstruct the xF3 structure function directly in Bjorken x-space using the evolved parton densities which are obtained, themselves, based on the Laguerre expansion. At this stage, to get the evolved parton densities, we employ the QCD cut-off parameter which we get from the fitting. The results for the xF3 structure function, using the two different described methods, are in good agreement with each other and also with the available experimental data which confirms the validity of our calculations.

A numerical method to extract [formula omitted] using hypergeometrical functions

The solutions for DGLAP evolution equations are considered numerically based on Laguerre polynomials expansion. We prepare some FORTRAN codes and subroutines based on Monte Carlo method which yields us reasonable results for parton densities at high energy scales. The techniques to write the codes can be used for other numerical computations. Our results for parton densities are in good agreement with GRSV and CTEQ phenomenological groups. They are also better with respect to what have been represented by other people, especially for sea and gluon densities. All the stages of computations have been done completely in a numerical way. Our results have sufficient stability after doing few iterations. This indicates that Laguerre polynomials contain reliable algebraic strucute to use them for the numerical solutions. In continuation by comparison the evolved nucleon structure, based on Jaccobi polynomials expansion with the similar function which are arising from the evolved parton densities using the Laguerre polynomials expansion, we are able to extract the numerical value of QCD cut off parameter.

Computing derivative-based global sensitivity measures using polynomial chaos expansions

In the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol׳ indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol׳ indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The derivative-based global sensitivity measures (DGSMs) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. Closed-form expressions for Hermite, Legendre and Laguerre polynomial expansions are given. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis.

Laguerre and Disk Polynomial Expansions with Nonnegative Coefficients

We establish Wiener type theorems and Paley type theorems for Laguerre polynomial expansions and disk polynomial expansions with nonnegative coefficients.

Asymptotics on Laguerre or Hermite polynomial expansions and their applications in Gauss quadrature

In this paper, we present asymptotic analysis on the coefficients of functions expanded in forms of Laguerre or Hermite polynomial series, which shows the decay of the coefficients and derives new error bounds on the truncated series. Moreover, by applying the asymptotics, new estimates on the errors for Gauss–Laguerre, Radau–Laguerre and Gauss–Hermite quadrature are deduced. These results show that Gauss–Laguerre-type and Gauss-Hermite-type quadratures are nearly of same convergence rates.