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Results for “Orthogonal polynomials”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

To the Theory of C{sub 0}-Operator Orthogonal Polynomials

Operator orthogonal polynomials are considered whose arguments are generators of strongly continuous semigroups of transformations of class C{sub 0} acting in a Banach space. Earlier such polynomials were considered by the authors in the case of the Chebyshev polynomials of the first and second kind. In this paper, more general classes of operator orthogonal polynomials are considered, which include the Jacobi and Aptekarev polynomials. Integral representations of operator fractional-rational functions and also of Bessel operator-valued functions of an imaginary argument are presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps

Here, we study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Garsia, and Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coefficients of the 3-term recursion relation defining the orthogonal polynomials. We use these graphs to predict whether the singular infinite Bernoulli convolutions belong to the Nevai class. Based on our numerical results, we conjecture that all infinite Bernoulli convolutions with contraction ratios greater than or equal to 1/2 belong to Nevai’s class, regardless of the probability weights assigned to the self-similar measures.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2019

Meshless Local Petrov-Galerkin (MLPG) Method with Orthogonal Polynomials for Euler-Bernoulli Beam Problems

In this paper, the feasibility of orthogonal polynomials in the meshless local Petrov Galerkin method (MLPG) method is studied. The orthogonal polynomials, Chebyshev and Legendre polynomials, are used in this MLPG method as trial functions. The test functions used were power functions with smooth derivatives at their ends. The performance of these methods is studied by applying these methods to Euler-Bernoulli beam problems. The MLPG-Galerkin and Legendre methods passed all the patch tests for simple beam problems. Next the formulations are tested on complex beam problems such as beams with partial loadings and continuous beam problems. Problems with load discontinuities and additional supports require special attention. Near discontinuities, judicious choice of number of nodes and nodal placements are needed to obtain accurate deflections, slopes, moments and shear forces. As polynomial functions are used, the large number of nodes can create a transformation matrix that is ill-conditioned, resulting in problems with the inversion of the matrix. The conditioning worsens as the number of nodes are increased beyond 20. Quadruple precision was needed for models to obtain accurate solutions. Even with quadruple precision the accuracy of the method suffers as the number of nodes is increased beyond 20. This appears to be a drawback of the MLPG-Chebyshev and MLPG-Legendre methods.

Raju, Ivatury S.↗
DOE OSTI2018

Quantitative Boltzmann–Gibbs Principles via Orthogonal Polynomial Duality

We study fluctuation fields of orthogonal polynomials in the context of particle systems with duality. We thereby obtain a systematic orthogonal decomposition of the fluctuation fields of local functions, where the order of every term can be quantified. This implies a quantitative generalization of the Boltzmann–Gibbs principle. In the context of independent random walkers, we complete this program, including also fluctuation fields in non-stationary context (local equilibrium). For other interacting particle systems with duality such as the symmetric exclusion process, similar results can be obtained, under precise conditions on the n particle dynamics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2011

Least-Squares Adaptive Control Using Chebyshev Orthogonal Polynomials

This paper presents a new adaptive control approach using Chebyshev orthogonal polynomials as basis functions in a least-squares functional approximation. The use of orthogonal basis functions improves the function approximation significantly and enables better convergence of parameter estimates. Flight control simulations demonstrate the effectiveness of the proposed adaptive control approach.

Nguyen, Nhan T.↗
DOE OSTI2018

The Basics of Orthogonal Polynomials

To understand the mathematics behind Uncertainty Quantification (UQ), one first needs to understand the basics of orthogonal polynomials, which this report covers.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Floquet operator dynamics and orthogonal polynomials on the unit circle

Operator spreading under stroboscopic time evolution under a unitary is studied. An operator Krylov space is constructed and mapped to orthogonal polynomials on a unit circle (OPUC), as well as to the Krylov space of the edge operator of the Floquet transverse field Ising model with inhomogeneous couplings (ITFIM). The Verblunsky coefficients in the OPUC representation are related to the Krylov angles parametrizing the ITFIM. The relations between the OPUC and spectral functions are summarized and several applications are presented. These include derivation of analytic expressions for the OPUC under persistent m-periodic dynamics, and the numerical construction of the OPUC for autocorrelations of the homogeneous Floquet-Ising model as well as the Z 3 clock model. The numerically obtained Krylov angles of the Z 3 clock model with long-lived period tripled autocorrelations show a spatial periodicity of six, and this observation is used to develop an analytically solvable model for the ITFIM that mimics this behavior.

Floquet systems↗
NASA NTRS1974

Method of Characteristics Calculations and Computer Code for Materials with Arbitrary Equations of State and Using Orthogonal Polynomial Least Square Surface Fits

A numerical scheme using the method of characteristics to calculate the flow properties and pressures behind decaying shock waves for materials under hypervelocity impact is developed. Time-consuming double interpolation subroutines are replaced by a technique based on orthogonal polynomial least square surface fits. Typical calculated results are given and compared with the double interpolation results. The complete computer program is included.

Chang, T. S.↗
DOE OSTI2018

Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

We introduce an extended Kepler–Coulomb quantum model in spherical coordinates. The Schrödinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

The CMV Matrix and the Generalized Lanczos Process

The CMV matrix is the five-diagonal matrix that represents the operator of multiplication by the independent variable in a special basis formed of Laurent polynomials orthogonal on the unit circle C. The article by Cantero, Moral, and Velázquez, published in 2003 and describing this matrix, has attracted much attention because it implies that the conventional orthogonal polynomials on C can be interpreted as the characteristic polynomials of the leading principal submatrices of a certain five-diagonal matrix. The present paper recalls that finite-dimensional sections of the CMV matrix appeared in papers on the unitary eigenvalue problem long before the article by Cantero et al. was published. Moreover, band forms were also found for a number of other situations in the normal eigenvalue problem.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple three-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e. matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

Coutsias, Evangelos A.↗
DOE OSTI2022

Verification of the LOTUS code with C5G7 benchmark

In this study, the verification of the neutron transport code LOTUS through the well-known C5G7 benchmark is performed. LOTUS uses the current coupled collision probability method with the expansion of the flux by orthogonal polynomials (CCCPO) for the solution of the neutron transport equation. The expansion of the flux by orthogonal polynomials allows one to avoid discretization of the calculation regions and significantly decreases the simulation time. The results of the LOTUS calculations are compared with the results of the reference OpenMC Monte Carlo simulations. The results of the comparison demonstrate almost perfect agreement with Monte-Carlo for the second order of the flux expansion. Even though the results obtained in the presented study agree well with the results of reference Monte Carlo calculations, further investigations are necessary for a better understanding of the stability and limitations of the code.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2017

A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditioned $\ell^1$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. We present theoretical analysis to motivate the algorithm and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. In conclusion, numerical examples are also provided to demonstrate that our proposed algorithm leads to comparable or improved accuracy even when compared with Legendre- and Hermite-specific algorithms.

97 MATHEMATICS AND COMPUTING↗