Effective Conductivity of Nonlinear Two-Phase Media: Homogenization and Two-Point Padé Approximants

The aim of this paper is a study of the quasilinear transport equation, for instance the stationary heat equation. For periodically microheterogeneous media asymptotic homogenization has been performed with the local problem formulated as a minimization problem. The Golden–Papanicolaou integral representation theorem and some bounds developed for the linear equation have been extended. Two-point Padé approximants have been used to calculate bounds. Examples are also provided.     

Anti-Gaussian Padé Approximants

This paper gives a synthesis of Padé approximants and anti-Gaussian quadratures. New rational approximants for Stieltjes series have been constructed. In addition, a three term recurrence relation is given for the numerator and denominator, which is useful when the given functional is not defin ite positive.We give the different algebraic properties of these new polynomials, which are similar to those obtained with the Gaussian quadrature formula. We find an easy definition and several relations with Padé approximants. Finally, some numerical results are given in the last section.     

Formal Orthogonal Polynomials and Newton–Padé Approximants

We proved in [8] that the denominators of Newton–Padé approximants for a formal Newton series are formal orthogonal with respect to linear functionals. The same functional is used along an antidiagonal of the Newton–Padé denominator table. The two linear functionals, corresponding to two adjacent antidiagonals, are linked with a very simple relation. Recurrence relations between denominators are given along an antidiagonal or two adjacent antidiagonals in the normal and non-normal case. The same recurrence relations are also satisfied by the Newton–Padé numerators, which implies another formal orthogonality.     

Two-point Padé approximants for formal Stieltjes series

We prove that certain two-point Padé approximants occupying the diagonal of the Padé table form monotone sequences of lower and upper bounds uniformly converging to a Stieltjes function. The results can be applied to the theory of inhomogeneous media for the calculation of the bounds on the effective transport coefficients of heterogeneous materials.     

Defects and the Convergence of Padé Approximants

A selective survey is given of convergence results for sequences of Padé approximants. Various approaches for dealing with the convergence problems due to `defects" are discussed. Attention is drawn to the close relationship between analyticity properties of a function and the `smoothness" of its Taylor series coefficients. A new theorem on the convergence of horizontal sequences of Padé approximants to functions in the Baker–Gammel–Wills conjecture function class is presented.     

Padé-Type Approximants of Markov and Meromorphic Functions

We study the question of convergence of Padé and Padé-type approximants to functions meromorphic in a domain. As an example we investigate in detail the case of functions of the formwhereμ(z) is a Markov function.     

A direct approach to convergence of multivariate,nonhomogeneous, Padé approximants

We present a direct approach for proving convergence in measure/product capacity of multivariate, nonhomogeneous, Padé approximants. Previous approaches have involved projection onto Padé-type approximation in one variable, and only yielded convergence in (Lebesgue) measure.     

Inequalities for Two-Point Padé Approximants to the Expansions of Stieltjes Functions in a Real Domain

General inequalities satisfied in real domain by two-point Padé approximants (2PPA) to two formal power expansions of Stieltjes functions have been derived. The obtained formulae: (i) generalize the existing inequalities for one-point Padé approximants, (ii) extend the validity of the inequalities reported in J. Comp. Appl. Math. 67 (1996), 59–72, from the particular case 0 k 2 to the general case 0 k , where (u,k) denotes a given number of coefficients of a power expansion of Stieltjes functions at infinity, (iii) clarify the properties of 2PPA estimations of the effective conductivity for regular composites.     

Two-point Padé approximants for the expansions of Stieltjes functions in real domain

The fundamental inequalities for the sequences of subdiagonal and diagonal one-point Padé approximants to Stieltjes function has been extended to the case of certain two-point Padé approximants. The results can be applied to the theory of inhomogeneous media for calculating the bounds for the effective transport coefficients of two-components heterogeneous materials.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

Some Extensions of M-Fractions Related to Strong Stieltjes Distributions

The purpose of this paper is to show the symmetric relations that appear between the coefficients of some even and odd extensions of the M-fractions related to a certain kind of symmetric strong Stieltjes distribution.     

A Comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai Equations

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.     

Bounds and estimates on the effective properties for nonlinear composites

In this paper we derive lower bounds and upper bounds on the effective properties for nonlinear heterogeneous systems. The key result to obtain these bounds is to derive a variational principle, which generalizes the variational principle by P. Ponte Castaneda from 1992. In general, when the Ponte Castaneda variational principle is used one only gets either a lower or an upper bound depending on the growth conditions. In this paper we overcome this problem by using our new variational principle together with the bounds presented by Lukkassen, Persson and Wall in 1995. Moreover, we also present some examples where the bounds are so tight that they may be used as a good estimate of the effective behavior.     

On nonstandard Padé approximants suitable for effective properties of two-phase composite materials

This article investigates the existence of the nonstandard Padé approximants introduced by Cherkaev and Zhang [D.-L. Zhang and E. Cherkaev, Reconstruction of spectral function from effective permittivity of a composite material using rational function approximations, J. Comput. Phys. 228 (2009), pp. 5390–5409] for approximating the spectral function of composites from effective properties at different frequencies. The spectral functions contain information on microstructure of composites. Since this reconstruction problem is ill-posed Cherkaev [Inverse homogenization for evaluation of effective properties of a mixture, Inverse Probl. 17 (2001), pp. 1203–1218], the well-performed Padé approach is noteworthy and deserves further investigations. In this article, we validate the assumption that the effective dielectric component of interest of all two-phase composites can be approximated by Padé approximants whose denominator has nonzero power one term. We refer to this as the nonstandard Padé approximant, in contrast to the standard approximants whose denominators have nonzero constant terms. For composites whose spectral function assumes infinitely many different values such as the checkerboard microstructure, the proof is carried by using classical results for Markov–Stieltjes functions (also referred to as Stieltjes functions) Golden and Papanicolaou [Bounds on effective parameters of heterogeneous media by analytic continuation, Commun. Math. Phys. 90 (1983), pp. 473–491] and Cherkaev and Ou [De-homogenization: Reconstruction of moments of the spectral measure of the composite, Inverse Probl. 24 (2008), p. 065008]. However, it is well-known that spectral functions for microstructure such as rank-n laminates assume only finitely many different values, i.e. the measure in the Markov–Stieltjes function is supported at only finitely many points. For this case, we cannot find any existence results for nonstandard Padé approximants in the literature. The proof for this case is the focus of this article. It is done by utilizing a special product decomposition of the coefficient matrix of the Padé system. The results in this article can be considered as an extension of the Padé theory for Markov–Stieltjes functions whose spectral function take infinitely many different values to those taking only finitely many values. In the literature, the latter is usually excluded from the definition of Markov–Stieltjes functions because they correspond to rational functions, hence convergence of their Padé approximants is trivial. However, from an inverse problem point of view, we need to assure both the existence and convergence of the nonstandard Padé approximants, for all microstructures. The results in this article provide a mathematical foundation for applying the Padé approach for reconstructing the spectral functions of composites whose microstructure is not a priori known.     

Computation of special functions by Padé approximants with orthogonal polynomial denominators

This paper deals with the computation of special functions of mathematics and physics in the complex domain using continued fraction (one-point or two-point Padé) approximants. We consider three families of continued fractions (Stieltjes fractions, real J-fractions and non-negative T-fractions) whose denominators are orthogonal polynomials or Laurent polynomials. Orthogonality of these denominators plays an important role in the analysis of errors due to numerical roundoff and truncation of infinite sequences of approximants. From the rigorous error bounds described one can determine the exact number of significant decimal digits contained in the approximation of a given function value. Results from computational experiments are given to illustrate the methods.Research supported in part by the National Science Foudation under Grant No. DMS-9302584.     

On the convergence of certain sequences of rational approximants to meromorphic functions in several variables

In a previous paper, the author introduced a new class of multivariate rational interpolants, which are called Optimal Padé-type Approximants (OPTA). There, for this class of rational interpolants, which extends classical univariate Padé Approximants, a direct extension of the “de Montessus de Ballore's Theorem” for meromorphic functions in several variables is established. In the univariate case, this theorem ensures uniform convergence of a row of Pade Approximants when the denominator degree equals the number of poles (counting multiplicities) in a certain disc. When one overshoots the number of poles when fixing the denominator degree, convergence in measure or capacity has been proved and, under certain additional restrictions, the uniform convergence of a subsequence of the row. The author tackles the latter case and studies its generalization to functions in several variables by using OPTA.     

Homogenization of the Maxwell Equations: Case I. Linear Theory

The Maxwell equations in a heterogeneous medium are studied. Nguetseng's method of two-scale convergence is applied to homogenize and prove corrector results for the Maxwell equations with inhomogeneous initial conditions. Compactness results, of two-scale type, needed for the homogenization of the Maxwell equations are proved.     

The maximal accuracy of stable difference schemes for the wave equation

We consider three time-level difference schemes, symmetric in time and space, for the solution of the wave equation,u _tt =c ² u _xx , given by

Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann–Hilbert problem, are presented. The first method uses a scalar Riemann–Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Riemann–Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.