Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.     

The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions

A general framework is constructed for efficiently and stably evaluating the Hadamard finite-part integrals by composite quadrature rules. Firstly, the integrands are assumed to have the Puiseux expansions at the endpoints with arbitrary algebraic and logarithmic singularities. Secondly, the Euler-Maclaurin expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sums of the Hurwitz zeta function and the generalized Stieltjes constant, which shows that the standard numerical integration formula is not convergent for computing the Hadamard finite-part integrals. Thirdly, the standard quadrature formula is recast in two steps. In step one, the singular part of the integrand is integrated analytically and in step two, the regular integral of the remaining part is evaluated using the standard composite quadrature rule. In this stage, a threshold is introduced such that the function evaluations in the vicinity of the singularity are intentionally excluded, where the threshold is determined by analyzing the roundoff errors caused by the singular nature of the integrand. Fourthly, two practical algorithms are designed for evaluating the Hadamard finite-part integrals by applying the Gauss-Legendre and Gauss-Kronrod rules to the proposed framework. Practical error indicator and implementation involved in the Gauss-Legendre rule are addressed. Finally, some typical examples are provided to show that the algorithms can be used to effectively evaluate the Hadamard finite-part integrals over finite or infinite intervals.     

An asymptotic expansion for product integration applied to Cauchy principal value integrals

Product integration methods for Cauchy principal value integrals based on piecewise Lagrangian interpolation are studied. It is shown that for this class of quadrature methods the truncation error has an asymptotic expansion in integer powers of the step-size, and that a method with an asymptotic expansion in even powers of the step-size does not exist. The relative merits of a quadrature method which employs values of both the integrand and its first derivative and for which the truncation error has an asymptotic expansion in even powers of the step-size are discussed.     

Efficient integration for a class of highly oscillatory integrals

This paper presents some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval. One is a Filon-type method based on the asymptotic expansion. The other is a Clenshaw-Curtis-Filon-type method which is based on a special Hermite interpolation polynomial and can be evaluated efficiently in O(N log N) operations, where N + 1 is the number of Clenshaw-Curtis points in the interval of integration. In addition, we derive the corresponding error bound in inverse powers of the frequency ω for the Clenshaw-Curtis-Filon-type method for the class of highly oscillatory integrals. The efficiency and the validity of these methods are testified by both the numerical experiments and the theoretical results.     

Complex Gaussian quadrature of oscillatory integrals

We construct and analyze Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with non-standard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.     

An Infinite Asymptotic Expansion for the Extreme Zeros of the Pollaczek Polynomials

In this paper, we first establish an integral expression for the Pollaczek polynomials P_n ( x ; a , b ) from a generating function. By applying a canonical transformation to the integral and carrying out a detailed analysis of the integrand, we derive a uniform asymptotic expansion for P_n (cosθ; a , b ) in terms of the Airy function and its derivative, in descending powers of n . The uniformity is in an interval next to the turning point , with M being a constant. The coefficients of the expansion are analytic functions of a parameter that depends only on t where , and not on the large parameter n . From the expansion of the polynomials we obtain an asymptotic expansion in powers of n ^−1/3 for the largest zeros. As a special case, a four-term approximation is provided for comparison and illustration. The method used in this paper seems to be applicable to more general situations.     

Computation of the attractive force of an ellipsoid

A numerical method for computing the attractive force of an ellipsoid is proposed that does not involve separating subdomains with singularities. The sought function is represented as a triple integral such as the inner integral of the kernel can be evaluated analytically with the kernel treated as a weight function. The inner integral is approximated by a quadrature for the product of functions, of which one has an integrable singularity. As a result, the integrand obtained before the second integration has only a weak logarithmic singularity. The subsequent change of variables yields an integrand without singularities. Based on this approach, at each stage of integral evaluation with respect to a single variable, quadrature formulas are derived that do not have singularities at integration nodes and do not take large values at these nodes. For numerical experiments, a rather complicated test function is constructed that is the exact attractive force of an ellipsoid of revolution with an elliptic density distribution.     

Superinterpolation in Highly Oscillatory Quadrature

Asymptotic expansions for oscillatory integrals typically depend on the values and derivatives of the integrand at a small number of critical points. We show that using values of the integrand at certain complex points close to the critical points can actually yield a higher asymptotic order approximation to the integral. This superinterpolation property has interesting ramifications for numerical methods based on exploiting asymptotic behaviour. The asymptotic convergence rates of Filon-type methods can be doubled at no additional cost. Numerical steepest descent methods already exhibit this high asymptotic order, but their analyticity requirements can be significantly relaxed. The method can be applied to general oscillators with stationary points as well, through a simple change of variables.     

Limit Curves for Zeros of Sections of Exponential Integrals

We are interested in studying the asymptotic behavior of the zeros of partial sums of power series for a family of entire functions defined by exponential integrals. The zeros grow on the order of \(O(n)\) , and after rescaling, we explicitly calculate their limit curve. We find that the rate at which the zeros approach the curve depends on the order of the singularities/zeros of the integrand in the exponential integrals. As an application of our findings, we derive results concerning the zeros of partial sums of power series for Bessel functions of the first kind.     

The Construction of cubature rules for multivariate highly oscillatory integrals

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule.

The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.

     

Asymptotics of orthogonal polynomials with asymptotic Freud-like weights

We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.     

Bounds on the error of fejer and clenshaw-curtis type quadrature for analytic functions

We consider the problem of integrating a function f : [-1,1] → R which has an analytic extension to an open disk D^r of radius r and center the origin, such that for any . The goal of this paper is to study the minimal error among all algorithms which evaluate the integrand at the zeros of the n-degree Chebyshev polynomials of first or second kind (Fejer type quadrature formulas) or at the zeros of (n-2)-degree Chebyshev polynomials jointed with the endpoints -1,1 (Clenshaw-Curtis type quadrature formulas), and to compare this error to the minimal error among all algorithms which evaluate the integrands at n points. In the case r > 1, it is easy to prove that Fejer and Clenshaw-Curtis type quadrature are almost optimal. In the case r = 1, we show that Fejer type formulas are not optimal since the error of any algorithm of this type is at least about n^-2. These results hold for both the worst-case and the asymptotic settings.     

Nyström discretization of parabolic boundary integral equations

A Nyström method for the discretization of thermal layer potentials is proposed and analyzed. The method is based on considering the potentials as generalized Abel integral operators in time, where the kernel is a time dependent surface integral operator. The time discretization is the trapezoidal rule with a corrected weight at the endpoint to compensate for singularities of the integrand. The spatial discretization is a standard quadrature rule for surface integrals of smooth functions. We will discuss stability and convergence results of this discretization scheme for second-kind boundary integral equations of the heat equation. The method is explicit, does not require the computation of influence coefficients, and can be combined easily with recently developed fast heat solvers.     

非线性积分方程迭代配置法的渐近展开及其外推

非线性积分方程迭代配置法的渐近展开及其外推韩国强（华南理工大学计算机工程与科学系）ＡＳＹＭＰＴＯＴＩＣＥＲＲＯＲＥＸＭＮＳＩＯＮＳＡＮＤＥＸＴＲＡＰＯＬＡＴＩＯＮＦＯＲＴＨＥＩＴＥＲＡＴＥＤＣＯＬＬＯＣＡＴＩＯＮＭＥＴＨＯＤＳＯＦＮＯＮＬＩＮＥＡＲＩ...     

On expansions in orthogonal polynomials

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; finally, the integrand is transformed to accelerate the convergence of the Taylor expansion of the kernel, allowing for rapid computation using Fast Fourier Transform. In the current paper we demonstrate that the first two steps remain valid in the general setting of orthogonal polynomials on the real line with finite support, orthogonal polynomials on the unit circle and Laurent orthogonal polynomials on the unit circle.     

On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation

The main theme of this paper is the construction of efficient, reliable and affordable error bounds for two families of quadrature methods for highly oscillatory integrals. We demonstrate, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations. Moreover, in place of derivatives it is possible to use finite difference approximations, with spacing inversely proportional to frequency. This renders the computation of error bounds even cheaper and, more importantly, leads to a new family of quadrature methods for highly oscillatory integrals that can attain arbitrarily high asymptotic order without computation of derivatives. AMS subject classification (2000) Primary 65D30, secondary 34E05.Received June 2004. Accepted October 2004. Communicated by Lothar Reichel.     

Asymptotic Approximations to the Nodes and Weights of Gauss–Hermite and Gauss–Laguerre Quadratures

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a stand‐alone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than 100, the asymptotic methods are enough for a double precision accuracy computation (15–16 digits) of the nodes and weights of the Gauss–Hermite and Gauss–Laguerre quadratures.     

HIGH ORDER NUMERICAL METHODS TO TWO DIMENSIONAL HEAVISIDE FUNCTION INTEGRALS

In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimensional Heaviside function integral can be rewritten as a one dimensional ordinary integral with the integrand being a one dimensional Heaviside function integral which is smooth on several subsets of the integral interval. Thus the two dimensional Heaviside function integral is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional Heaviside function integrals. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms. Numerical examples are presented showing that the second- to fourth-order methods implemented in this paper achieve or exceed the expected accuracy.     

A combined Filon/asymptotic quadrature method for highly oscillatory problems

A crossing between the asymptotic expansion of an oscillatory integral and Filon-type methods is obtained by applying a Filon-type method to the error term in the asymptotic expansion, which is in itself an oscillatory integral. The efficiency of the approach is investigated through analysis and numerical experiments, revealing a method which in many cases performs better than the Filon-type method. It is shown that considerable savings in terms of the required number of potentially expensive moments can be expected. The case of multivariate oscillatory integrals is discussed briefly. AMS subject classification (2000)  65D30     

Interpolatory product quadratures for Cauchy principal value integrals with Freud weights

Summary. We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions. Received July 15, 1997 / Revised version received August 25, 1998