Combination Of Orthogonal Polynomials Research Articles

A Christoffel function weighted least squares algorithm for collocation approximations

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial Chaos approximation in uncertainty quantification where a polynomial approximation is formed from a combination of orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density of orthogonality. Our proposed algorithm samples with respect to the equilibrium measure of the parametric domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. 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.

A Computational Spectral Approach to Interest Rate Models

The Polynomial Chaos Expansion (PCE) technique recovers a finite second order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantity $\xi$, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ) method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper we exploit the PCE approach to analyze some equity and interest rate models considering, without loss of generality, the one dimensional case. In particular we will take into account those models which are based on the Geometric Brownian Motion (gBm), e.g. the Vasicek model, the CIR model, etc. We also provide several numerical applications and results which are discussed for a set of volatility values. The latter allows us to test the PCE technique on a quite large set of different scenarios, hence providing a rather complete and detailed investigation on PCE-approximation's features and properties, such as the convergence of statistics, distribution and quantiles. Moreover we give results concerning both an efficiency and an accuracy study of our approach by comparing our outputs with the ones obtained adopting the Monte Carlo approach in its standard form as well as in its enhanced version.

Improved orthogonal polynomial density estimates

Density estimates that are expressible as the product of a base density function and a linear combination of orthogonal polynomials are considered in this paper. More specifically, two criteria are proposed for determining the number of terms to be included in the polynomial adjustment component and guidelines are suggested for the selection of a suitable base density function. A simulation study reveals that these stopping rules produce density estimates that are generally more accurate than kernel density estimates or those resulting from the application of the Kronmal–Tarter criterion. Additionally, it is explained that the same approach can be utilized to obtain multivariate density estimates. The proposed orthogonal polynomial density estimation methodology is applied to several univariate and bivariate data sets, some of which have served as benchmarks in the statistical literature on density estimation.

Application of orthogonal functions to spectrophotometric analysis. The combined polynomial method.

Abstract A general method is outlined for the linear combination of orthogonal polynomials belonging to one group of points to eliminate interferences in spectrophotometric analysis. The method is particularly useful when it is difficult to find a set of wavelengths over which the coefficient of the irrelevant absorption is negligibly small relative to the assay coefficient. The method is illustrated by the determination of salicylamide in the presence of chloroquine.

Spectral Finite-Element Methods for Parametric Constrained Optimization Problems

We present a method to approximate the solution mapping of parametric constrained optimization problems. The approximation, which is of the spectral finite element type, is represented as a linear combination of orthogonal polynomials. Its coefficients are determined by solving an appropriate finite-dimensional constrained optimization problem. We show that, under certain conditions, the latter problem is solvable because it is feasible for a sufficiently large degree of the polynomial approximation and has an objective function with bounded level sets. In addition, the solutions of the finite-dimensional problems converge for an increasing degree of the polynomials considered, provided that the solutions exhibit a sufficiently large and uniform degree of smoothness. Our approach solves, in the case of optimization problems with uncertain parameters, the most computationally intensive part of stochastic finite-element approaches. We demonstrate that our framework is applicable to parametric eigenvalue problems.

Error Estimation for Discontinuous Galerkin Solutions of Two-Dimensional Hyperbolic Problems

We analyze the discretization errors of discontinuous Galerkin solutions of steady two-dimensional hyperbolic conservation laws on unstructured meshes. We show that the leading term of the error on each element is a linear combination of orthogonal polynomials of degrees p and p+1. We further show that there is a strong superconvergence property at the outflow edge(s) of each element where the average discretization error converges as O(h 2p+1) compared to a global rate of O(h p+1). Our analyses apply to both linear and nonlinear conservation laws with smooth solutions. We show how to use our theory to construct efficient and asymptotically exact a posteriori discretization error estimates and we apply these to some examples.

TRANSIENT ACOUSTIC DIFFRACTION AND RADIATION BY AN AXISYMMETRICAL ELASTIC SHELL: A NEW STATEMENT OF THE BASIC EQUATIONS AND A NUMERICAL METHOD BASED ON POLYNOMIAL APPROXIMATIONS

This work deals with an axisymmetrical shell composed of a cylindrical shell closed by two hemispherical shells made of the same material and with the same thickness. The shell is immersed into a homogeneous perfect fluid extending to infinity. The first part is devoted to the establishment of the equations governing the shell vibrations. The method used, which, in the authors' opinion, is not quite new, is based on the expansion of the elasticity equations into a Taylor series of the transverse variable: by using the same degree of expansion, the equations obtained for the cylindrical part and for the spherical parts are consistent (they correspond to the Donnell and Mushtari approximation). The first interest of this analysis is that the continuity conditions along the junction lines between the cylindrical and the spherical parts are immediately obtained. The main problem is to obtain the boundary conditions satisfied by the hemispherical shells displacement at the apexes. Indeed, due to the use of spherical co-ordinates—which is a quite natural choice—the coefficients of the equations become singular at the apexes and boundary conditions are required to express that an apex is a mechanically regular point. The method that is used here enables one to obtain such a result which, to the authors' knowledge, is new. The transient response of the system shell/external fluid is sought as a series of its resonance modes, that is its free oscillations. The main difficulty is to obtain a numerical approximation of the resonance modes: their calculation leads to solving the Fourier transform of the system of homogeneous equations. The numerical method for solving the problem is the following. The acoustic pressure is described by a hybrid layer potential, the density of which is approximated by a linear combination of orthogonal polynomials. Each component of the shell displacement is approximated by a linear combination of polynomial functions: these functions are chosen as linear combinations of orthogonal polynomials which satisfy the same continuity and boundary conditions as the shell displacement components. In the first step, the resonance frequencies are calculated. Then the coefficients of the corresponding resonance mode expansion are deduced. The validity and the efficiency of this approach will be shown in a second article through comparisons between numerical predictions and experimental results.

Methods of constructing generalized polynomial-type moment representations

Algebraic polynomials that occur in the generalized moment representation of a specified number sequence are found in the form of linear combinations of orthogonal polynomials. The coefficients of these linear combinations are computed from a system of algebraic equations and the polynomials that are found satisfy integral and differential equations.

Investigation of an efficient representation of speech spectra for segmentation and classification of speech sounds

A functional representation of speech sounds in orthogonal polynomial space is described and preliminary results are presented. Speech spectra are approximated by a linear combination of orthogonal polynomials which are found to be more efficient than a linear combination of trigonometric functions. The original spectra (100 samples in frequency) and the polynomial approximations are represented by points in their respective Hilbert spaces, the distance between successive points being a measure of the dissimilarity of successive spectra. Segment boundaries are indicated where the distance between successive spectra exceeds a threshold. The effectiveness in segmentation of connected utterances using these spectral forms is compared. Also, representing speech in orthogonal polynomial space appears to be applicable to clustering and separating transformations which yield simple decision boundaries for phoneme classification. Although only one polynomial class is investigated, the procedure is valid for other functional representations of speech data.

Nonnegative expansions of polynomials

Consider expanding a set of orthogonal polynomials { q i ( x ) } \{ {q_i}(x)\} as a combination of orthogonal polynomials { p i ( x ) } \{ {p_i}(x)\} . A sufficient condition to ensure that the expansion coefficients are nonnegative is derived. It is shown by counter-example that it is not a necessary condition.

Nonnegative Expansions of Polynomials

Consider expanding a set of orthogonal polynomials $\{ {q_i}(x)\}$ as a combination of orthogonal polynomials $\{ {p_i}(x)\}$. A sufficient condition to ensure that the expansion coefficients are nonnegative is derived. It is shown by counter-example that it is not a necessary condition.