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At least 109 records · Page 6

All sources NASA NTRS DOE OSTI

DOE OSTI2019

A GENERAL FRAMEWORK FOR ENHANCING SPARSITY OF GENERALIZED POLYNOMIAL CHAOS EXPANSIONS

Compressive sensing has become a powerful addition to uncertainty quantification when only limited data are available. In this paper, we provide a general framework to enhance the sparsity of the representation of uncertainty in the form of generalized polynomial chaos expansion. We use an alternating direction method to identify new sets of random variables through iterative rotations so the new representation of the uncertainty is sparser. Consequently, we increase both the efficiency and accuracy of the compressive-sensing-based uncertainty quantification method. We demonstrate that the previously developed iterative method to enhance the sparsity of Hermite polynomial expansion is a special case of this general framework. Moreover, we use Legendre and Chebyshev polynomial expansions to demonstrate the effectiveness of this method with applications in solving stochastic partial differential equations and high-dimensional (O(100)) problems.

Yang, Xiu↗

DOE OSTI2017

Generation and application of multivariate polynomial quadrature rules

The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second main contribution comes in the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1972

Polynomial approximations of thermodynamic properties of arbitrary gas mixtures over wide pressure and density ranges

Computer programs for flow fields around planetary entry vehicles require real-gas equilibrium thermodynamic properties in a simple form which can be evaluated quickly. To fill this need, polynomial approximations were found for thermodynamic properties of air and model planetary atmospheres. A coefficient-averaging technique was used for curve fitting in lieu of the usual least-squares method. The polynomials consist of terms up to the ninth degree in each of two variables (essentially pressure and density) including all cross terms. Four of these polynomials can be joined to cover, for example, a range of about 1000 to 11000 K and 0.00001 to 1 atmosphere (1 atm = 1.0133 x 100,000 N/m sq) for a given thermodynamic property. Relative errors of less than 1 percent are found over most of the applicable range.

Allison, D. O.↗

NASA NTRS1973

Polynomial weights and code constructions.

Study of certain polynomials with the 'weight-retaining' property that any linear combination of these polynomials with coefficients in a general finite field has Hamming weight at least as great as that of the minimum-degree polynomial included. This fundamental property is used in applications to Reed-Muller codes, a new class of 'repeated-root' binary cyclic codes, two new classes of binary convolutional codes derived from binary cyclic codes, and two new classes of binary convolutional codes derived from Reed-Solomon codes.

Massey, J. L.↗

NASA NTRS1975

An algorithm and computer program to locate real zeros of real polynomials

A method for reliably extracting real zeros of real polynomials using an expanded two-point secant and bisection method is formed into an algorithm for a digital computer, and a computer program based on this algorithm is presented. The results obtained with the program show that the proposed method compares favorably with the Laguerre, Newton-Raphson, and Jenkins-Traub methods when the polynomial has all real zeros, and is more efficient when the polynomial has complex zeros.

Hedgley, D. R., Jr.↗

NASA NTRS1977

Modular polynomial arithmetic in partial fraction decomposition

Algorithms for general partial fraction decomposition are obtained by using modular polynomial arithmetic. An algorithm is presented to compute inverses modulo a power of a polynomial in terms of inverses modulo that polynomial. This algorithm is used to make an improvement in the Kung-Tong partial fraction decomposition algorithm.

Abdali, S. K.↗

NASA NTRS1977

Evaluation of the Tensor Polynomial and Hoffman strength theories for composite materials

The Hoffman theory and the Tensor Polynomial (Tsai-Wu) theory with the stress interaction term set equal to zero have been found to be preferred alternatives to the general Tensor Polynomial theory for predicting strength of filamentary composite laminae. These theories were used to predict failure of off-axis boron/epoxy and E-glass/epoxy test specimens and gave excellent agreement with available experimental results. A numerical experiment was also performed to estimate the errors for ten different composite systems under six different loadings. The maximum error in predicted failure loads among all cases was below 10 percent. These results suggest that the Hoffman failure theory and the Tensor Polynomial theory with the stress interaction term equal to zero can predict failure of practical filamentary composite materials under general biaxial loading with sufficient accuracy for engineering applications.

Narayanaswami, R.↗

NASA NTRS1980

Discrete Tchebycheff orthonormal polynomials and applications

Discrete Tchebycheff orthonormal polynomials offer a convenient way to make least squares polynomial fits of uniformly spaced discrete data. Computer programs to do so are simple and fast, and appear to be less affected by computer roundoff error, for the higher order fits, than conventional least squares programs. They are useful for any application of polynomial least squares fits: approximation of mathematical functions, noise analysis of radar data, and real time smoothing of noisy data, to name a few.

Lear, W. M.↗

NASA NTRS1983

A parallel-pipeline architecture of the fast polynomial transform for computing a two-dimensional cyclic convolution

It is pointed out that the two-dimensional cyclic convolution is a useful tool for many two-dimensional digital signal processing applications. Two important applications are related to spaceborne high-resolution synthetic aperture radar (SAR) processing and image processing. Nussbaumer and Quandalle (1978) showed that a radix-2 polynomial transform analogous to the conventional radix-2 FFT algorithm can be used to compute a two-dimensional cyclic convolution. On the basis of results reported by Arambepola and Rayner (1979), a radix-2 polynomial transform can be defined to compute a multidimensional cyclic convolution. Truong et al. (1981) used the considered ideas together with the Chinese Theorem to further reduce the complexity of the radix-2 fast polynomial transform (FPT). Reed et al. (1981) demonstrated that such a new FPT algorithm is significantly faster than the FFT algorithm for computing a two-dimensional convolution. In the present investigation, a parallel-pipeline architecture is considered for implementing the FPT developed by Truong et al.

Truong, T. K.↗

NASA NTRS1985

FIT: Computer Program that Interactively Determines Polynomial Equations for Data which are a Function of Two Independent Variables

A computer program for interactively developing least-squares polynomial equations to fit user-supplied data is described. The program is characterized by the ability to compute the polynomial equations of a surface fit through data that are a function of two independent variables. The program utilizes the Langley Research Center graphics packages to display polynomial equation curves and data points, facilitating a qualitative evaluation of the effectiveness of the fit. An explanation of the fundamental principles and features of the program, as well as sample input and corresponding output, are included.

Arbuckle, P. D.↗

NASA NTRS1987

Polynomial approximation of functions of matrices and its application to the solution of a general system of linear equations

During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described.

Tal-Ezer, Hillel↗

NASA NTRS1987

Inverse of polynomial matrices in the irreducible form

An algorithm is developed for finding the inverse of polynomial matrices in the irreducible form. The computational method involves the use of the left (right) matrix division method and the determination of linearly dependent vectors of the remainders. The obtained transfer function matrix has no nontrivial common factor between the elements of the numerator polynomial matrix and the denominator polynomial.

Chang, Fan R.↗

NASA NTRS1987

Polynomial compensation, inversion, and approximation of discrete time linear systems

The least-squares transformation of a discrete-time multivariable linear system into a desired one by convolving the first with a polynomial system yields optimal polynomial solutions to the problems of system compensation, inversion, and approximation. The polynomial coefficients are obtained from the solution to a so-called normal linear matrix equation, whose coefficients are shown to be the weighting patterns of certain linear systems. These, in turn, can be used in the recursive solution of the normal equation.

Baram, Yoram↗

NASA NTRS1988

Stability margins for Hurwitz polynomials

The authors treat the robust stability issue using the characteristic polynomial, for two different cases: first in coefficient space with respect to perturbations in the coefficient of the characteristic polynomial; and then for a control system containing perturbed parameters in the transfer function description of the plant. In coefficient space, a simple expression is first given for the l-(squared) stability margin for both the monic and nonmonic cases. Following this, a method is given to find the l(infinity) margin, and the method is extended to reveal much larger stability regions. In parameter space the authors consider all single-input (multi-output) or single-output (multi-input) systems with a fixed controller and a plant described by a set of transfer functions which are ratios of polynomials with variable coefficients. A procedure is presented to calculate the radius of the largest stability ball in the space of these variable parameters. The calculation serves as a stability margin for the control system. The formulas that result are quasi-closed-form expressions for the stability margin and are computationally efficient.

Chapellat, Herve↗

NASA NTRS1989

Chebyshev polynomials are not always optimal

The problem is that of finding among all polynomials of degree at most n and normalized to be 1 at c the one with minimal uniform norm on Epsilon. Here, Epsilon is a given ellipse with both foci on the real axis and c is a given real point not contained in Epsilon. Problems of this type arise in certain iterative matrix computations and, in this context, it is generally believed and widely referenced that suitably normalized Chebyshev polynomials are optimal for such constrained approximation problems. It is shown that this is not true in general. Moreover, sufficient conditions are derived which guarantee that Chebyshev polynomials are optimal. Some numerical examples are also presented.

Fischer, Bernd↗

NASA NTRS1989

Tests for Hurwitz and Schur properties of convex combination of complex polynomials

Given that two polynomials having complex coefficients are Hurwitz (Schur) a test based on the application of the resultant theory is advanced to determine whether the set formed from the convex combination of the specified polynomials is also Hurwitz (Schur). The procedure, on the one hand, provides a unified approach to the relevant testing procedure for both continuous- and discrete-time systems and, on the other hand, generalizes earlier results restricted to the case of polynomials with real coefficients.

Bose, N. K.↗

NASA NTRS1989

Optimal Chebyshev polynomials on ellipses in the complex plane

The design of iterative schemes for sparse matrix computations often leads to constrained polynomial approximation problems on sets in the complex plane. For the case of ellipses, we introduce a new class of complex polynomials which are in general very good approximations to the best polynomials and even optimal in most cases.

Fischer, Bernd↗

NASA NTRS1993

Conventional modeling of the multilayer perceptron using polynomial basis functions

A technique for modeling the multilayer perceptron (MLP) neural network, in which input and hidden units are represented by polynomial basis functions (PBFs), is presented. The MLP output is expressed as a linear combination of the PBFs and can therefore be expressed as a polynomial function of its inputs. Thus, the MLP is isomorphic to conventional polynomial discriminant classifiers or Volterra filters. The modeling technique was successfully applied to several trained MLP networks.

Chen, Mu-Song↗