Vandermonde System Research Articles

Performance and accuracy of the basic closure algorithm of quadrature-based moment methods

Quadrature-based moment methods (QBMMs) are numerical methods to close moment equations derived from a population balance equation. There are many variants, most of which are based on the same closure algorithm that can be divided into three subroutines: First, computing the recurrence coefficients of orthogonal polynomials from a moment sequence, second, finding the associated Gaussian quadrature rule by solving an eigenvalue problem, and third, closing the moment equations using that quadrature rule. Since each of these steps may be carried out billions of times in large-scale simulations, knowing the potentials of optimization is the key to making QBMMs as efficient as possible. In this work, we investigate in detail the performance and accuracy of QBMMs with different configurations and over a wide range of input moment sets. Our major findings were the following: Available algorithms to compute the recurrence coefficients are about equally sensitive to the input moment sequence. There are considerable differences in performance, but the contribution to the overall computational costs of the closure algorithm is negligible. The standard method to solve the eigenvalue problem is insensitive to the input moment set in contrast to a less robust but slightly faster alternative that involves solving a Vandermonde system. For relatively simple physical problems, the overall computational costs are mostly determined by the eigenvalue problem, whereas closing the moment equations becomes important in the presence of more complex processes with particle-particle interactions. The complete source code we implemented and used for our numerical investigations is publicly available.

On the numerical computation of bivariate Lagrange polynomials

In this paper we propose a simple procedure for numerically computing the Lagrange interpolation polynomial on a unisolvent set of points in the plane. We suggest the use of the canonical polynomial basis centered at the barycenter of the set of points and the PA=LU decomposition for solving the associated Vandermonde system to compute the coefficients of the Taylor polynomial. We show that the 1-norm condition number of the Vandermonde matrix is an upper bound for the Lebesgue constant of the interpolation node set in the unit disk. Therefore, the analysis of the condition number can be useful to select the unisolvent set of nodes in a set of scattered nodes. Numerical experiments show the efficiency and accuracy of the proposed method.

Generating Nested Quadrature Rules with Positive Weights based on Arbitrary Sample Sets

For the purpose of uncertainty propagation a new quadrature rule technique is proposed that has positive weights, has high degree, and is constructed using only samples that describe the probability distribution of the uncertain parameters. Moreover, nodes can be added to the quadrature rule, resulting in a sequence of nested rules. The rule is constructed by iterating over the samples of the distribution and exploiting the null space of the Vandermonde system that describes the nodes and weights, in order to select which samples will be used as nodes in the quadrature rule. The main novelty of the quadrature rule is that it can be constructed using any number of dimensions, using any basis, in any space, and using any distribution. It is demonstrated both theoretically and numerically that the rule always has positive weights and therefore has high convergence rates for sufficiently smooth functions. The convergence properties are demonstrated by approximating the integral of the Genz test functions. The applicability of the quadrature rule to complex uncertainty propagation cases is demonstrated by determining the statistics of the flow over an airfoil governed by the Euler equations, including the case of dependent uncertain input parameters. The new quadrature rule significantly outperforms classical sparse grid methods.

Mxpfit: A library for finding optimal multi-exponential approximations

mxpfit is a library implemented in C++ to find optimal approximations of functions by multi-exponential sums with complex-valued parameters. The library provides an interface for evaluating exponents and coefficients from sampling data on a uniform grid using the fast Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm originally proposed by Potts and Tasche (2015). The parameters can be estimated efficiently from a sampling data even including noise. A modified balanced truncation algorithm to find the multi-exponential sum with a smaller order is also provided. These features are useful for finding optimal exponential sum approximations of analytic functions or large-scale numerically sampled data set. Program summaryProgram Title: mxpfitProgram Files doi:http://dx.doi.org/10.17632/vjygkwpss4.1Licensing provisions: MITProgramming language: C++Nature of problem: Nonlinear curve fitting by multi-exponential function, frequency estimation of signals, and reduction of multi-exponential sumSolution method: The modified fast Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm originally proposed in [1] is utilized for the parameter estimation of multi-exponential sums from sampled data on a uniform grid. In this method, partial Lanczos bidiagonalization is used to find a low-rank approximation of a Hankel matrix constructed from sampled data. An overdetermined Vandermonde system is solved by the conjugate gradient least square (CGLS) method with preconditioner to obtain the coefficients of the multi-exponential sum. The truncation of the multi-exponential sum is performed by a modified balanced truncation method in [2]. The projection matrix to obtain the truncated system is determined by computing con-eigenvalues of a Gramian matrix, which can be computed efficiently with highly relative accuracy using the algorithm in [3].Additional comments: A C++ compiler supporting the C++11 standard is required to compile programs using the mxpfit library. mxpfit depends on the Eigen (http://eigen.tuxfamily.org/) and FFTW3 (http://www.fftw.org/) libraries. CMake (https://cmake.org/) is used for building example programs.

Generalized trigonometric Fourier-series method with automatic time step control for non-linear point kinetics equations

The trigonometric Fourier-series method (TFS) is generalized to provide approximate solutions for non-linear point kinetics equations with feedback using varying step sizes. This method can provide a very stable solution against the size of the discrete time step allowing much larger step sizes to be used. Systems of the point kinetics equations are solved using Fourier-series expansion over a partition of the total time interval. The approximate solution requires determining the series coefficients over each time step in that partition. These coefficients are determined using the high-order derivatives of the solution vector at the beginning of the time step introducing a system of linear algebraic equations to be solved at each step. This system is similar to the Vandermonde system. Two successive orders of the partial sums are used to estimate the local truncation error. This error and some other constrains are used to produce the largest step size allowable at each step while keeping the error within a specific tolerance. The process of calculating suitable step sizes should be automatic and inexpensive. Convergence and stability of the proposed method are discussed and a new formula is introduced to maintain stability. The proposed method solves the general linear and non-linear kinetics problems. The method has been applied to five different types of reactivities including step/ramp insertions with temperature feedback. The method is seemed to be valid for larger time intervals than those used in the conventional numerical integration, and is thus useful in reducing computing time. Computational results are found to be consistent with the analysis, they demonstrate that the convergence of the iteration scheme can be accelerated and the resulting computing times can be greatly reduced while maintain computational accuracy.

Deterministic sparse FFT for M-sparse vectors

In this paper, we derive a new deterministic sparse inverse fast Fourier transform (FFT) algorithm for the case that the resulting vector is sparse. The sparsity needs not to be known in advance but will be determined during the algorithm. If the vector to be reconstructed is M-sparse, then the complexity of the method is at most $\mathcal {O} (M^{2} \log N)$ if M 2 < N and falls back to the usual $\mathcal {O}(N \log N) $ algorithm for M 2 ≥ N. The method is based on the divide-and-conquer approach and may require the solution of a Vandermonde system of size at most M × M at each iteration step j if M 2 < 2 j . To ensure the stability of the Vandermonde system, we propose to employ a suitably chosen parameter σ that determines the knots of the Vandermonde matrix on the unit circle.

Optimal Caughey series representation of classical damping matrices

A least squares approach to determine the coefficients in a Caughey series representation of a classical damping matrix is presented. Instead of solving an ill-conditioned Vandermonde system of equations involving the modal damping ratios at a set of pivots, the procedure uses a least squares fit between the polynomial representing the damping ratios and the assumed known frequency dependence of the damping ratios. The proposed approach eliminates the large fluctuations of the damping ratios associated with the standard approach. Three alternative procedures are presented: (i) analytical application of the least squares approach for an expansion into power series, (ii) numerical solution of the least squares equations for a dense set of pivots, and (iii) expansion of the damping matrix into series of Legendre polynomials of matrices. In each alternative, the cases of series including or excluding the mass-proportional term are considered separately.

Trigonometric Fourier-series solutions of the point reactor kinetics equations

In this paper, a new method based on the Fourier series is introduced to obtain approximate solutions to the systems of the point kinetics equations. These systems are stiff involving equations with slowly and rapidly varying components. They are solved numerically using Fourier series expansion over a partition of the total time interval. Approximate solution requires determining the series coefficients over each time step in that partition. These coefficients are determined using the high order derivatives of the dependent variables at the beginning of the time step introducing a system of linear algebraic equations to be solved at each step. The obtained algebraic system is similar to the Vandermonde system. Evaluation of the inverse of the Vandermonde matrix is required to determine the coefficients of the Fourier series. Because the obtained Vandermonde matrix has a special structure, due to the properties of the sine and cosine functions, a new formula is introduced to determine its inverse using standard computations. The new method solves the general linear and non-linear kinetics problems with six groups of delayed neutrons. The validity of the algorithm is tested with five different types of reactivities including step reactivity insertion, ramp input, oscillatory reactivity changes, a reactivity as a function of the neutron density and finally temperature feedback reactivity. Comparisons are made with analytical and conventional numerical methods used to solve the point kinetics equations. The results confirm the theoretical analysis and indicate the validity of the method for all applications considered.

Dehomogenization: reconstruction of moments of the spectral measure of the composite

This paper deals with the inverse homogenization or dehomogenization problem of recovering geometric information about the structure of a two-component composite medium from the effective complex permittivity of the composite. The approach is based on the reconstruction of moments of the spectral measure in the Stieltjes analytic representation of the effective property. The moments of the spectral measure are linked to n-point correlation functions of the structure of the composite and thus contain information about the microgeometry. We show that the moments can be uniquely recovered from the measurements of the effective property in a range of frequencies. Two methods of numerical reconstruction of the moments are developed and analyzed. One method, which is referred to as a direct method of moment reconstruction, is based on the solution of the Vandermonde system arising in series expansion of the Stieltjes integral. The second, indirect, method reformulates the problem and reduces it to the problem of reconstruction of the spectral function. This last problem is ill-posed and requires regularization. We show that even though the reconstructed spectral function can be quite sensitive to the choice of the regularization scheme, the moments of the spectral functions can be stably reconstructed. The applicability of these two methods in terms of the choice of data points is also discussed in this paper.

Numerically consistent regularization of force‐based frame elements

AbstractRecent advances in the literature regularize the strain‐softening response of force‐based frame elements by either modifying the constitutive parameters or scaling selected integration weights. Although the former case maintains numerical accuracy for strain‐hardening behavior, the regularization requires a tight coupling of the element constitutive properties and the numerical integration method. In the latter case, objectivity is maintained for strain‐softening problems; however, there is a lack of convergence for strain‐hardening response. To resolve the dichotomy between strain‐hardening and strain‐softening solutions, a numerically consistent regularization technique is developed for force‐based frame elements using interpolatory quadrature with two integration points of prescribed characteristic lengths at the element ends. Owing to manipulation of the integration weights at the element ends, the solution of a Vandermonde system of equations ensures numerical accuracy in the linear‐elastic range of response. Comparison of closed‐form solutions and published experimental results of reinforced concrete columns demonstrates the effect of the regularization approach on simulating the response of structural members. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Statistical study and experimental verification of high-resolution methods in phase-shifting interferometry

The introduction of high-resolution phase-shifting interferometry methods such as annihilation filter, state space, multiple-signal classification, minimum norm, estimation of signal parameter via rotational invariance, and maximum-likelihood estimator have enabled the estimation of phase in an interferogram in the presence of harmonics and noise. These methods are also effective in holographic moiré where incorporating two piezoelectric transducers (PZTs) yields two orthogonal displacement components simultaneously. Typically, when these methods are used, the first step involves estimating the phase steps pixelwise; then the interference phase distribution is computed by designing a Vandermonde system of equations. In this context, we present a statistical study of these methods for the case of single and dual PZTs. The performance of these methods is also compared with other conventional benchmarking algorithms involving the single PZT. The paper also discusses the significant issue of an allowable pair of phase steps in the presence of noise using a robust statistical tool such as the Cramér-Rao bound. Furthermore, experimental validations of these high-resolution methods are presented for the estimation of single phase in holographic interferometry and for the estimation of multiple phases in holographic moiré.

A min-norm approach for estimating phase distribution in an interferogram

Phase distribution in an interferogram is usually computed from several phase shifted intensity images acquired temporally by means of a CCD camera. These phase shifts are commonly accomplished by translation of a mirror with a piezoelectric transducer (PZT). In reality the PZT motion is not exact. Thus the aim of this paper is to present a novel approach for estimating the phase steps that are imparted to the PZT in the presence of nonsinusoidal waveforms and random noise. The method functions by designing an autocovariance matrix from the intensity registered on the CCD for N data frames. The eigendecomposition of an autocovariance matrix yields the signal and noise subspaces. The phase step values are estimated pixelwise from the noise subspace. This approach provides the flexibility of using arbitrary phase steps, a feature most commonly attributed to generalized phase shifting algorithms. Once the phase steps are estimated the Vandermonde system of equations is applied to estimate the phase distribution.

From an approximate to an exact absolute polynomial factorization

We propose an algorithm for computing an exact absolute factorization of a bivariate polynomial from an approximate one. This algorithm is based on some properties of the algebraic integers over Z and is certified. It relies on a study of the perturbations in a Vandermonde system. We provide a sufficient condition on the precision of the approximate factors, depending only on the height and the degree of the polynomial.

Rotational invariance approach for the evaluation of multiple phases in interferometry in the presence of nonsinusoidal waveforms and noise

Incorporation of two phase-shifting devices in a holographic moiré configuration not only renders the interferometer compatible with automated measurements but also allows for simultaneous measurement of multiple phase information in the interferometer. However, simultaneous handling of multiple phase steps and subsequent simultaneous determination of multiple phase distributions requires the introduction of novel tools in phase-shifting interferometry. In this context, the aim of this paper is to propose a subspace invariance approach to address these issues. This approach takes advantage of the rotational invariance of signal subspaces spanned by two temporally displaced data sets formed from the intensity fringes recorded temporally on pixels of the CCD camera. The method first identifies the arbitrary phase steps imparted to the piezoactuator devices. The estimated phase steps are subsequently applied in the linear Vandermonde system of equations to determine the phase distributions. The method also allows for handling nonsinusoidal wavefronts. Since the phase steps are extracted at every point on the interferogram, the method is applicable to configurations that use spherical beams. The robustness of the method is investigated by adding white Gaussian noise during the simulations.

High-resolution frequency estimation technique for recovering phase distribution in interferometers

An integral approach to phase measurement is presented. First, the use of a high-resolution technique for the pixelwise detection of phase steps is proposed. Next, the robustness of the algorithm that is developed is improved by incorporation of a denoising procedure during spectral estimation. The pixelwise knowledge of phase steps is then applied to the Vandermonde system of equations for retrieval of phase values at each pixel point. Conceptually, our proposal involves the design of an annihilating filter that has zeros at the frequencies associated with the polynomial that describes the fringe intensity. The parametric estimation of this annihilating filter yields the desired spectral information embedded in the signal, which in our case represents the phase steps. The proposed method offers the advantage of extracting the interference phase of nonsinusoidal waveforms in the presence of miscalibration error of the piezoelectric transducer. In addition, in contrast to previously reported methods, this method does not require the application of selective phase steps between data frames for nonsinusoidal waveforms.

Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm

The paper proposes a novel approach for estimating multiple phases in holographic moiré. The need to design such an algorithm is necessitated by the development of optical configurations containing two phase stepping devices, e.g. PZTs, with a view to measure simultaneously two phase distributions. The approach consists of first applying minimum-norm algorithm to extract phase steps imparted to the PZTs. Salient feature of the algorithm lies in its ability to handle nonsinusoidal waveforms and noise. This approach also provides the flexibility of using arbitrary phase steps, a feature most commonly attributed to generalized phase shifting interferometry. Once the phase steps are estimated for each PZT, the Vandermonde system of equations is designed to estimate the phase distributions.

An iterative error-free algorithm to solve Vandermonde systems

The paper describes an algorithm to solve Vandermonde system of linear equations. The algorithm is error-free, i.e., it does not introduce either rounding-off errors or errors caused by finite length of computer word into the solution. Cost analysis and its comparison to alternative error-free method are included.

Minimum variance linear estimation of amplitudes for exponential signal models

This article presents the minimum variance consistent linear estimator for amplitude parameters in exponential signal models. A simple heuristic algorithm is presented to compute the weighting matrix that minimizes error variance; the resulting weighted least squares estimator accounts for the statistics of pole estimation errors. Additionally, analysis of binary diagonal weighting matrices demonstrates that for unweighted least-squares, the amplitude variance is reduced by truncating the Vandermonde system of equations.

Bounds for the Structured Backward Errors of Vandermonde Systems

Consider the primal Vandermonde system V(a)x=b and the dual Vandermonde system V(a)Tx=b, whereV(a) is the Vandermonde matrix defined in terms of the vector a =(\alpha_1, \ldots, \alpha_n)^T$ with distinct scalars $\alpha_1,\ldots, \alpha_n \in {\cal C}$. In view of the special structure of the matrix V(a), we define structured backward errors (SBEs) of the Vandermonde systems and describe a technique for obtaining upper and lower bounds for the SBEs. The results are illustrated by numerical examples.

Recursive solution of Löwner-Vandermonde systems of equations. I

An O( n 2) algorithm for the solution of a linear system the n × n coefficient matrix of which consists of a Vandermonde and a Löwner part is presented. In the first part of the algorithm the corresponding Vandermonde system is solved. In the second part the last rows of the Vandermonde matrix are successively replaced by rows of the Löwner structure. The algorithm is convenient if the Vandermonde part is dominant. The main tool is the connection of such systems with rational interpolation problems.