Global Polynomial Approximation Research Articles

Calculating the minimal/maximal eigenvalue of symmetric parameterized matrices using projection

SummaryIn applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters ω and we are interested in the minimal eigenvalue of a matrix pencil (A,B) with A,B symmetric and B positive definite. If ω can be interpreted as the realization of random variables, one may be interested in statistical moments of the minimal eigenvalue. In order to obtain statistical moments, we need a fast evaluation of the eigenvalue as a function of ω. Because this is costly for large matrices, we are looking for a small parameterized eigenvalue problem whose minimal eigenvalue makes a small error with the minimal eigenvalue of the large eigenvalue problem. The advantage, in comparison with a global polynomial approximation (on which, e.g., the polynomial chaos approximation relies), is that we do not suffer from the possible nonsmoothness of the minimal eigenvalue. The small‐scale eigenvalue problem is obtained by projection of the large‐scale problem. Our main contribution is that, for constructing the subspace, we use multiple eigenvectors and derivatives of eigenvectors. We provide theoretical results and document numerical experiments regarding the beneficial effect of adding multiple eigenvectors and derivatives.

Global optimal polynomial approximation for parametric problems in power systems

The influence of parameters on system states for parametric problems in power systems is to be evaluated. These parameters could be renewable generation outputs, load factor, etc. Polynomial approximation has been applied to express the nonlinear relationship between system states and parameters, governed by the nonlinear and implicit equations. Usually, sampling-based methods are applied, e.g., data fitting methods and sensitivity methods, etc. However, the accuracy and stability of these methods are not guaranteed. This paper proposes an innovative method based on Galerkin method, providing global optimal approximation. Compared to traditional methods, this method enjoys high accuracy and stability. IEEE 9-bus system is used to illustrate its effectiveness, and two additional studies including a 1648-bus system are performed to show its applications to power system analysis.

Numerical Methods for Cauchy Bisingular Integral Equations of the First Kind on the Square

In this paper we investigate the numerical solution of Cauchy bisingular integral equations of the first kind on the square. We propose two different methods based on a global polynomial approximation of the unknown solution. The first one is a discrete collocation method applied to the original equation and then is a “direct” method. The second one is an “indirect” procedure of discrete collocation-type since we act on the so-called regularized Fredholm equation. In both cases, the convergence and the stability of the method is proved in suitable weighted spaces of functions, and the well conditioning of the linear system is showed. In order to illustrate the efficiency of the proposed procedures, some numerical tests are given.

Regularized Fractional Order Integration and Differentiation Via Discrete Orthogonal Polynomials

In this paper, we present a new approach for the regularized fractional order integration and differentiation of signals containing noise. The method combines the highly efficient and accurate piecewise (local) approximation of a signal using discrete orthogonal polynomials, and their fractional order integrals. The method provides a vast improvement in accuracy over the traditional method of using non-regularized formulae, or using global polynomial approximation. The method is demonstrated on a periodic signal over several periods with a high level of synthetic noise.

Global Parametric Polynomial Approximation of Static Voltage Stability Region Boundaries

A novel method for globally approximating the static voltage stability region boundaries (SVSRBs) of power systems is proposed by applying parametric polynomial approximation to the criterion equation which defines the SVSRB. Known as the Galerkin method, the implicit function portraying the SVSRB is described as a linear combination of basis polynomial functions, and the coefficients of the basis functions are obtained by projecting the SVSRB criterion onto each basis function. The approximation guarantees high precision globally in the whole domain of the parameter of interest rather than only in the neighborhood of a point, and the error can be controlled by the degree of polynomial basis functions. In the meantime, analytical expression of the left or right eigenvector of the system's Jacobian matrix corresponding to the zero eigenvalue is obtained in the form of polynomial, which provides valuable information for online voltage stability control or monitoring. Case studies in a 10-bus test system and IEEE 118-bus test system verifies the validity, accuracy, and flexibility of the proposed method.

Sets inℂNwith vanishing global extremal function and polynomial approximation

ou E is est un sous-ensemble de ℂ N tel que la fonction extremale globale V E = 0 sur ℂ N , alors le domaine maximal d'existence G f de f est uniforme, et pour tout compact K C G f . Si, de plus, la suite {d n+1 /d n } est bornee alors G f = ℂ N . Si E est un sous-ensemble ferme de ℂ N alors V E ≡ 0 si et seulement si chaque serie de polynomes homogenes ∑ ∞ j=0 Qj, ayant une sous-suite {s nk } de sommes partielles convergeant ponctuellement sur E, admet des lacunes de type Ostrowski relativement a une sous-suite {n kl } de {n k }. En dimension 1, ces resultats sont dus a J. Muller and A. Yavrian [5].

Preconditioning Stochastic Galerkin Saddle Point Systems

Mixed finite element discretizations of deterministic second-order elliptic PDEs lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations, and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense, and the cost of a matrix vector product is nontrivial. We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed first-order system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of Gaussian random variables. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type for use with the minimal residual method (MINRES). By introducing so-called Kronecker product preconditioners, we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

Algorithm 899

Global polynomial approximation methods applied to piecewise continuous functions exhibit the well-known Gibbs phenomenon. We summarize known methods to remove the Gibbs oscillations and present a collection of Matlab programs that implement the methods. The software features a Graphical User Interface that allows easy access to the postprocessing algorithms for benchmarking and educational purposes.

Edge Detection Free Postprocessing for Pseudospectral Approximations

Pseudospectral Methods based on global polynomial approximation yield exponential accuracy when the underlying function is analytic. The presence of discontinuities destroys the extreme accuracy of the methods and the well-known Gibbs phenomenon appears. Several types of postprocessing methods have been developed to lessen the effects of the Gibbs phenomenon or even to restore spectral accuracy. The most powerful of the methods require that the locations of the discontinuities be precisely known. In this work we discuss postprocessing algorithms that are applicable when it is impractical, or difficult, or undesirable to pinpoint all discontinuity locations.

Better Global Polynomial Approximation for Image Rectification

When using images to locate objects, there is the problem of correcting for distortion and misalignment in the images. An elegant way of solving this problem is to generate an error correcting function that maps points in an image to their corrected locations. We generate such a function by fitting a polynomial to a set of sample points. The objective is to identify a polynomial that passes “sufficiently close” to these points with “good” approximation of intermediate points. In the past, it has been difficult to achieve good global polynomial approximation using only sample points. We report on the development of a global polynomial approximation algorithm for solving this problem.

BETTER GLOBAL POLYNOMIAL APPROXIMATION FOR IMAGE RECTIFICATION

When using images to locate objects, there is the problem of correcting for distortion and misalignment in the images. An elegant way of solving this problem is to generate an error correcting function that maps points in an image to their corrected locations. We generate such a function by fitting a polynomial to a set of sample points. The objective is to identify a polynomial that passes sufficiently close to these points with approximation of intermediate points. In the past, it has been difficult to achieve good global polynomial approximation using only sample points. We report on the development of a global polynomial approximation algorithm for solving this problem. Key Words: Polynomial approximation, interpolation, image rectification.

On the formulation of linear driving force approximations for adsorption and desorption of multicomponent gaseous mixtures in sorbent particles

A generalized linear driving force (LDF) approximation for multicomponent adsorption-based separations is formulated in this study. It is based on the three-parameter dusty-gas model and accounts for bulk diffusion, Knudsen diffusion, Knudsen flow and viscous flow in the sorbent particle. No restrictions are imposed on the type of adsorption isotherm. The generalized LDF approximation equations are derived using global or piecewise continuous polynomial approximations for the intraparticle partial pressure profiles and applying the subdomain method of weighted residuals to the mass balance equations. Similar expressions are derived using the Fickian diffusion model and the Fickian diffusion/convection model. The validity of the various approximations is examined by comparing their predictions to the exact solution of the corresponding intraparticle models. Since the generalized LDF approximation permits the inclusion of convective coupling among the various species, it can satisfactorily predict the overshoot in the uptake of the less adsorbable/faster moving species in a multicomponent mixture. This phenomenon cannot be described by existing LDF approximations, since their derivation is based on transport and adsorption models that do not account for the coupling of the fluxes of the components of the multicomponent mixture. For short cycle times, the accuracy of the approximation increases with the degree of the polynomial. However, a generalized LDF approach, based on a fourth degree global polynomial approximation, may suffice in most applications.

Global polynomial approximation for Symm's equation on polygons

Summary. To solve 1D linear integral equations on bounded intervals with nonsmooth input functions and solutions, we have recently proposed a quite general procedure, that is essentially based on the introduction of a nonlinear smoothing change of variable into the integral equation and on the approximation of the transformed solution by global algebraic polynomials. In particular, the new procedure has been applied to weakly singular equations of the second kind and to solve the generalized air foil equation for an airfoil with a flap. In these cases we have obtained arbitrarily high orders of convergence through the solution of very-well conditioned linear systems. In this paper, to enlarge the domain of applicability of our technique, we show how the above procedure can be successfully used also to solve the classical Symm's equation on a piecewise smooth curve. The collocation method we propose, applied to the transformed equation and based on Chebyshev polynomials of the first kind, has shown to be stable and convergent. A comparison with some recent numerical methods using splines or trigonometric polynomials shows that our method is highly competitive.

Calculation of the aberration spot: Improved numerical algorithm

In this paper a numerical algorithm for calculation of the point spread function from the complex light amplitude known on the output pupil of the imaging system is described. The first step of this algorithm includes a global polynomial approximation of the input data given in a number of points distributed arbitrarily over the entire output pupil. In the next step the numerical calculation of the diffraction integral is performed. After transition to polar coordinates and dividing the region of integration into a number of infinitesimal subregions of square shape the exponent in the integrand is approximated locally by a polynomial and then the whole integrand is approximated by a polynomial multiplied by the exponential function with linear exponent. Finally the respective integration is performed analytically. The analysis of the accuracy obtained as well as the results of exemplary calculations prove that the described algorithm is very useful in the analysis of image quality of such imaging elements as hybrid (diffractive–refractive) lenses.

Global Symplectic Polynomial Approximation of Area-Preserving Maps

Different methods are proposed and tested for transforming a nonlinear differential system, and more particularly a Hamiltonian one, into a map without having to integrate the whole orbit as in the well known Poincaré map technique . We construct a synthetic map, based on a global fitting, which satisfies the symplectic condition. Taking the Standard Map as a model problem we point our attention on methods suitable for comparing the model map and its synthetic counterpart. We test the agreement of the fitting on finer scales through the visual representation, the computation of the rotation number and the measure of the local distribution of the Lyapunov characteristic exponents. Comparing these results with those obtained by Froeschlé and Petit using a method based on Taylor interpolation, we show that the symplectic character is a crucial condition for the recovering of the finest details of a dynamical system. On the other hand the global character of our method makes the generalization to any system of differential equations difficult.

Asymptotic Properties of Collocation Matrix Norms 1: Global Polynomial Approximation

On montre qu'une matrice reliee a la solution collocation en n points d'equations differentielles d'ordre m en utilisant des polynomes globaux a une norme maximum qui tend, quand n→∞, vers la norme maximum d'un operateur relie a l'equation differentielle si les points collocation sont choisis aux zeros de certains polynomes orthogonaux