Orthogonal Polynomial Basis Research Articles

Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion

We discuss the arbitrary polynomial chaos (aPC), which has been subject of research in a few recent theoretical papers. Like all polynomial chaos expansion techniques, aPC approximates the dependence of simulation model output on model parameters by expansion in an orthogonal polynomial basis. The aPC generalizes chaos expansion techniques towards arbitrary distributions with arbitrary probability measures, which can be either discrete, continuous, or discretized continuous and can be specified either analytically (as probability density/cumulative distribution functions), numerically as histogram or as raw data sets. We show that the aPC at finite expansion order only demands the existence of a finite number of moments and does not require the complete knowledge or even existence of a probability density function. This avoids the necessity to assign parametric probability distributions that are not sufficiently supported by limited available data. Alternatively, it allows modellers to choose freely of technical constraints the shapes of their statistical assumptions. Our key idea is to align the complexity level and order of analysis with the reliability and detail level of statistical information on the input parameters. We provide conditions for existence and clarify the relation of the aPC to statistical moments of model parameters. We test the performance of the aPC with diverse statistical distributions and with raw data. In these exemplary test cases, we illustrate the convergence with increasing expansion order and, for the first time, with increasing reliability level of statistical input information. Our results indicate that the aPC shows an exponential convergence rate and converges faster than classical polynomial chaos expansion techniques.

Computation of the canonical polynomials and applications to some optimal control problems

The Tau method is a numerical technique that consists in constructing polynomial approximate solutions for ordinary differential equations. This method has two approaches: operational and recursive. The former converts the differential problem to a matrix problem and produces approximations in terms of a prescribed orthogonal polynomials basis. In the recursive approach, we construct approximate solutions in terms of a special set of polynomials {Qk(t); k = 0, 1, 2...} called canonical polynomials basis. In some cases, the Qk’s can be obtained explicitly through a recursive formula. But no analogous formulae are reported in the literature for the general cases. In this paper, utilizing the operational Tau method, we develop an algorithm that allows to generate those canonical polynomials iteratively and explicitly. In addition, we demonstrate the capability of the operational Tau method in treating quadratic optimal control problems governed by ordinary differential equations.

Spectral Analysis Methods for the Robust Measurement of the Flexural Rigidity of Biopolymers

The mechanical properties of biopolymers can be determined from a statistical analysis of the ensemble of shapes they exhibit when subjected to thermal forces. In practice, extracting information from fluorescence microscopy images can be challenging due to low signal/noise ratios and other artifacts. To address these issues, we develop a suite of tools for image processing and spectral data analysis that is based on a biopolymer contour representation expressed in a spectral basis of orthogonal polynomials. We determine biopolymer shape and stiffness using global fitting routines that optimize a utility function measuring the amount of fluorescence intensity overlapped by such contours. This approach allows for filtering of high-frequency noise and interpolation over sporadic gaps in fluorescence. We use benchmarking to demonstrate the validity of our methods, by analyzing an ensemble of simulated images generated using a simulated biopolymer with known stiffness and subjected to various types of image noise. We then use these methods to determine the persistence lengths of taxol-stabilized microtubules. We find that single microtubules are well described by the wormlike chain polymer model, and that ensembles of chemically identical microtubules show significant heterogeneity in bending stiffness, which cannot be attributed to sampling or fitting errors. We expect these approaches to be useful in the study of biopolymer mechanics and the effects of associated regulatory molecules.

Orthogonal Polynomials and Expansions for a Family of Weight Functions in Two Variables

Orthogonal polynomials for a family of weight functions on [−1,1]2, $$\mathcal{W}_{\alpha,\beta,\gamma}(x,y) = |x+y|^{2\alpha+1}|x-y|^{2\beta+1} \bigl(1-x^2\bigr)^{\gamma}\bigl(1-y^2\bigr)^{\gamma},$$ are studied and shown to be related to the Koornwinder polynomials defined on the region bounded by two lines and a parabola. In the case of γ=±1/2, an explicit basis of orthogonal polynomials is given in terms of Jacobi polynomials, and a closed formula for the reproducing kernel is obtained. The latter is used to study the convergence of orthogonal expansions for these weight functions.

Differential qd Algorithm with Shifts for Rank-Structured Matrices

Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable, in particular, in the symmetric tridiagonal case where the differential qd algorithm with shifts (dqds) proposed by Fernando and Parlett enjoys often faster convergence while preserving high relative accuracy. In eigenvalue computations for rank-structured matrices, the QR algorithm is also a popular choice since, in the symmetric case, the rank structure is preserved. In the unsymmetric case, however, the QR algorithm destroys the rank structure and, hence, LR-type algorithms come in to play once again. In the current paper we adapt several variants of qd algorithms to quasi-separable matrices. Remarkably, one of them, when applied to Hessenberg matrices, becomes a direct generalization of the dqds algorithm for tridiagonal matrices. Therefore, it can be applied to such important matrices as companion and confederate and provides an alternative algorithm for finding roots of a polynomial represented in a basis of orthogonal polynomials. Results of preliminary numerical experiments are presented.

Estimation of Approximating Rate for Neural Network inLwp Spaces

A class of Soblove type multivariate function is approximated by feedforward network with one hidden layer of sigmoidal units and a linear output. By adopting a set of orthogonal polynomial basis and under certain assumptions for the governing activation functions of the neural network, the upper bound on the degree of approximation can be obtained for the class of Soblove functions. The results obtained are helpful in understanding the approximation capability and topology construction of the sigmoidal neural networks.

A New Spectral Element Method for Pricing European Options Under the Black–Scholes and Merton Jump Diffusion Models

We present a new spectral element method for solving partial integro-differential equations for pricing European options under the Black–Scholes and Merton jump diffusion models. Our main contributions are: (i) using an optimal set of orthogonal polynomial bases to yield banded linear systems and to achieve spectral accuracy; (ii) using Laguerre functions for the approximations on the semi-infinite domain, to avoid the domain truncation; and (iii) deriving a rigorous proof of stability for the time discretizations of European put options under both the Black–Scholes model and the Merton jump diffusion model. The new method is flexible for handling different boundary conditions and non-smooth initial conditions for various contingent claims. Numerical examples are presented to demonstrate the efficiency and accuracy of the new method.

Spatially explicit estimation of fish length histograms, with application to anchovy habitats in the Bay of Biscay

AbstractPetitgas, P., Doray, M., Massé, J., and Grellier, P. 2011. Spatially explicit estimation of fish length histograms, with application to anchovy habitats in the Bay of Biscay. – ICES Journal of Marine Science, 68: 2086–2095. Fish length frequency histograms from research surveys are of prime importance for identifying habitats of different life stages, as well as for stock assessment. However, no method has thus far been available for mapping these histograms as spatially varying curves. Here, a procedure is applied to map spatially connected curves, and detail is given on how it can be applied to map the length frequency histograms. At each sample location, a fish length frequency histogram is given as a vector of non-independent values. The histogram is first modelled as a polynomial expansion on the basis of orthogonal polynomials. Then, the polynomial coefficients are mapped by co-kriging, after fitting a model of co-regionalization. The length frequency map is finally derived by linearly combining maps of polynomial coefficients. An estimation variance associated with the map is also derived. Maps of anchovy length distributions are produced by applying the method to midwater trawl length data from the PELGAS acoustic surveys in the Bay of Biscay. This novel approach extends the application of kriging techniques to curves or functions, opening new perspectives for mapping more complex information than just the values of fish density.

Orthogonal polynomial representation of imaginary-time Green’s functions

We study the expansion of single-particle and two-particle imaginary-time Matsubara Green's functions of quantum impurity models in the basis of Legendre orthogonal polynomials. We discuss various applications within the dynamical mean-field theory (DMFT) framework. The method provides a more compact representation of the Green's functions than standard Matsubara frequencies and therefore significantly reduces the memory-storage size of these quantities. Moreover, it can be used as an efficient noise filter for various physical quantities within the continuous-time quantum Monte Carlo impurity solvers recently developed for DMFT and its extensions. In particular, we show how to use it for the computation of energies in the context of realistic DMFT calculations in combination with the local density approximation to the density functional theory (LDA+DMFT) and for the calculation of lattice susceptibilities from the local irreducible vertex function.

Direct calculation of modal parameters from matrix orthogonal polynomials

The object of this paper is to introduce a new technique to derive the global modal parameter (i.e. system poles) directly from estimated matrix orthogonal polynomials. This contribution generalized the results given in Rolain et al. (1994) [5] and Rolain et al. (1995) [6] for scalar orthogonal polynomials to multivariable (matrix) orthogonal polynomials for multiple input multiple output (MIMO) system. Using orthogonal polynomials improves the numerical properties of the estimation process. However, the derivation of the modal parameters from the orthogonal polynomials is in general ill-conditioned if not handled properly. The transformation of the coefficients from orthogonal polynomials basis to power polynomials basis is known to be an ill-conditioned transformation. In this paper a new approach is proposed to compute the system poles directly from the multivariable orthogonal polynomials. High order models can be used without any numerical problems. The proposed method will be compared with existing methods (Van Der Auweraer and Leuridan (1987) [4] Chen and Xu (2003) [7]). For this comparative study, simulated as well as experimental data will be used.

Uncertainty analysis of vibrational frequencies of an incompressible liquid in a rectangular tank with and without a baffle using polynomial chaos expansion

Polynomial chaos expansion (PCE) with Latin hypercube sampling (LHS) is employed for calculating the vibrational frequencies of an inviscid incompressible fluid partially filled in a rectangular tank with and without a baffle. Vibration frequencies of the coupled system are described through their projections on the PCE which uses orthogonal basis functions. PCE coefficients are evaluated using LHS. Convergence on the coefficient of variation is used to find the orthogonal polynomial basis function order which is employed in PCE. It is observed that the dispersion in the eigenvalues is more in the case of a rectangular tank with a baffle. The accuracy of the PCE method is verified with standard MCS results and is found to be more efficient.

ORTHOGONAL BASES FOR POLYNOMIAL REGRESSION WITH DERIVATIVE INFORMATION IN UNCERTAINTY QUANTIFICATION

We discuss the choice of polynomial basis for approximation of uncertainty propagation through complex simulation models with capabilityto outputderivative information.Ourwork ispart of a larger research effortinuncertaintyquantification using sampling methods augmented with derivative information. The approach has new challenges compared with standard polynomial regression. In particular, we show that a tensor product multivariate orthogonal polynomial basis of an arbitrary degree may no longer be constructed. We provide sufficient conditions for an orthonormal set of this type to exist, a basis for the space it spans. We demonstrate the benefits of the basis in the propagation of material uncertainties through a simplified model of heat transport in a nuclear reactor core. Compared with the tensor product Hermite polynomial basis, the orthogonal basis results in a better numerical conditioning of the regression procedure, a modest improvement in approximation error when basis polynomials are chosen a priori, and a significant improvement when basis polynomials are chosen adaptively, using a stepwise fitting procedure.

DESIGN UNDER UNCERTAINTY EMPLOYING STOCHASTIC EXPANSION METHODS

Non-intrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) methods are attractive techniques for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability. PCE estimates coefficients for known orthogonal polynomial basis functions based on a set of response function evaluations, using sampling, linear regression, tensor-product quadrature, or Smolyak sparse grid approaches. SC, on the other hand, forms interpolation functions for known coefficients, and requires the use of structured collocation point sets derived from tensor product or sparse grids. When tailoring the basis functions or interpolation grids to match the forms of the input uncertainties, exponential convergence rates can be achieved with both techniques for general probabilistic analysis problems. Once PCE or SC representations have been obtained for a response metric of interest, analytic expressions can be derived for the moments of the expansion and for the design derivatives of these moments, allowing for efficient design under uncertainty formulations involving moment control (e.g., robust design). This paper presents two approaches for moment design sensitivities, one involving response function expansions over both design and uncertain variables and one involving response derivative expansions over only the uncertain variables. These approaches present a trade-off between increased dimensionality in the expansions (and therefore increased simulation runs required to construct them) with global expansion validity versus increased data requirements per simulation with local expansion validity. Given this capability for analytic moments and their sensitivities, we explore bilevel, sequential, and multifidelity formulations for OUU. Initial experiences with these approaches is presented for a number of benchmark test problems.

Linear finite elements with orthogonal discontinuous basis functions for explicit earthquake ground motion modeling

The dynamic explicit finite element method is commonly used in earthquake ground motion modeling. In this method, the element mass matrix is approximately lumped, which may lead to numerical dispersion. On the other hand, the orthogonal finite element method, based on orthogonal polynomial basis functions, naturally derives a lumped diagonal mass matrix and can be applied to dynamic explicit finite element analysis. In this paper, we propose finite elements based on orthogonal discontinuous basis functions, the element mass matrices of which are lumped without approximation. Orthogonal discontinuous basis functions are used to improve the accuracy and reduce the numerical dispersion in earthquake ground motion modeling. We present a detailed formulation of the 4-node tetrahedral and 8-node hexahedral elements. The relationship between the proposed finite elements and conventional finite elements is investigated, and the solutions obtained from the conventional explicit finite element method are compared with analytical solutions to verify the numerical dispersion caused by the lumping approximation. Comparison of solutions obtained with the proposed finite elements to analytical solutions demonstrates the usefulness of the technique. Examples are also presented to illustrate the effectiveness of the proposed method in earthquake ground motion modeling in the actual three-dimensional crust structure. Copyright © 2010 John Wiley & Sons, Ltd.

Online Segmentation of Time Series Based on Polynomial Least-Squares Approximations

The paper presents SwiftSeg, a novel technique for online time series segmentation and piecewise polynomial representation. The segmentation approach is based on a least-squares approximation of time series in sliding and/or growing time windows utilizing a basis of orthogonal polynomials. This allows the definition of fast update steps for the approximating polynomial, where the computational effort depends only on the degree of the approximating polynomial and not on the length of the time window. The coefficients of the orthogonal expansion of the approximating polynomial-obtained by means of the update steps-can be interpreted as optimal (in the least-squares sense) estimators for average, slope, curvature, change of curvature, etc., of the signal in the time window considered. These coefficients, as well as the approximation error, may be used in a very intuitive way to define segmentation criteria. The properties of SwiftSeg are evaluated by means of some artificial and real benchmark time series. It is compared to three different offline and online techniques to assess its accuracy and runtime. It is shown that SwiftSeg-which is suitable for many data streaming applications-offers high accuracy at very low computational costs.

An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis

The main purpose of this paper is to provide an efficient numerical approach for the fractional differential equations (FDEs) based on a spectral Tau method. An extension of the operational approach of the Tau method with the orthogonal polynomial bases is proposed to convert FDEs to its matrix–vector multiplication representation. The fractional derivatives are described in the Caputo sense. The spectral rate of convergence for the proposed method is established in the L 2 norm. We tested our procedure on several examples and observed that the obtained numerical results confirm the theoretical prediction of the exponential rate of convergence.

A Prediction Method for Nonlinear Correlative Time Series

A nonlinear correlative time series prediction method is presented in this paper.It is based on the mutual information of time series and orthogonal polynomial basis neural network. Inputs of network are selected by mutual information, and orthogonal polynomial basis is used as active function.The network is trained by an error iterative learning algorithm.This proposed method for nonlinear time series is tested using two well known time series prediction problems:Gas furnace data time series and Mackey-Glass time series.

Robust, efficient computational methods for axially symmetric optical aspheres

Whether in design or the various stages of fabrication and testing, an effective representation of an asphere's shape is critical. Some algorithms are given for implementing tailored polynomials that are ideally suited to these needs. With minimal coding, these results allow a recently introduced orthogonal polynomial basis to be employed to arbitrary orders. Interestingly, these robust and efficient methods are enabled by the introduction of an auxiliary polynomial basis.

Cross-section parameterization of the pebble bed modular reactor using the dimension-wise expansion model

This paper discusses the use of the dimension-wise expansion model for cross-section parameterization. The components of the model were approximated with tensor products of orthogonal polynomials. As we demonstrate, the model for a specific cross-section can be built in a systematic way directly from data without any a priori knowledge of its structure. The methodology is able to construct a finite basis of orthogonal polynomials that is required to approximate a cross-section with pre-specified accuracy. The methodology includes a global sensitivity analysis that indicates irrelevant state parameters which can be excluded from the model without compromising the accuracy of the approximation and without repetition of the fitting process. To fit the dimension-wise expansion model, Randomised Quasi-Monte-Carlo Integration and Sparse Grid Integration methods were used. To test the parameterization methods with different integrations embedded we have used the OECD PBMR 400 MW benchmark problem. It has been shown in this paper that the Sparse Grid Integration achieves pre-specified accuracy with a significantly (up to 1–2 orders of magnitude) smaller number of samples compared to Randomised Quasi-Monte-Carlo Integration.

A new second-order accurate time discretization method for the nonlinear cubic Schrödinger equation

A new second-order accurate semi-analytical time discretization method is introduced for the numerical solution of the one-dimensional nonlinear cubic Schrödinger equation. This method is based on the combination of the method of lines, Crank–Nicolson method, Newton method and Lanczos’ Tau method. It is a self-starting averaged two-time-level scheme that has proved to be stable, accurate and energy conservative for long time integration periods. At each time level, approximate solutions are sought on a segmented spatial interval as finite expansions in terms of a given orthogonal polynomial basis mapped appropriately onto each spatial subsegment. We have carried out numerical simulation concerning several cases for the propagation, collision and the bound states of solitons. Accurate results have been obtained using Chebyshev and Legendre polynomials. These results are well comparable with other published results obtained by the use of various standard numerical methods.