一些高振荡积分、高振荡积分方程的高性能计算

电磁、声波散射问题的研究涉及一类数学物理问题, 此类问题具有深刻的理论价值和重要的应用背景, 亟待解决. 高振荡微分、积分方程是刻画这些问题的重要的数学模型, 其数值计算存在许多挑战性研究课题. 本文从积分方程解法角度出发, 综述了求解这类高振荡问题的一些最新进展, 特别是针对广义Fourier 变换、Bessel 变换的高效算法、高振荡核Volterra 积分方程的数值解法作了详细介绍. 这些数值方法共有特点是振荡频率越高算法精度愈高, 且可望为电磁计算的研究提供一些新的高效算法.     

Numerical approximation of vector-valued highly oscillatory integrals

We present a method for the efficient approximation of integrals with highly oscillatory vector-valued kernels, such as integrals involving Airy functions or Bessel functions. We construct a vector-valued version of the asymptotic expansion, which allows us to determine the asymptotic order of a Levin-type method. Levin-type methods are constructed using collocation, and choosing a basis based on the asymptotic expansion results in an approximation with significantly higher asymptotic order. AMS subject classification (2000)  65D30     

Quadrature methods for multivariate highly oscillatory integrals using derivatives

While there exist effective methods for univariate highly oscillatory quadrature, this is not the case in a multivariate setting. In this paper we embark on a project, extending univariate theory to more variables. Inter alia, we demonstrate that, in the absence of critical points and subject to a nonresonance condition, an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices, a direct counterpart of the univariate case. This provides a convenient avenue towards the generalization of asymptotic and Filon-type methods, as formerly introduced by the authors in a single dimension, to simplices and, more generally, to polytopes. The nonresonance condition is bound to be violated once the boundary of the domain of integration is smooth: in effect, its violation is equivalent to the presence of stationary points in a single dimension. We further explore this issue and propose a technique that often can be used in this situation. Yet, much remains to be done to understand more comprehensively the influence of resonance on the asymptotics of highly oscillatory integrals.

     

Markov cubature rules for polynomial processes

We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using algebraic techniques. Markov cubature rules aid the tractability of path-dependent tasks such as American option pricing in models where the underlying factors are polynomial processes.     

一种非规则积分区域的二重高振荡函数积分方法

在传统L ev in方法与新F ilon型方法的基础上,本文提出了一种求解非规则区域下的二重高振荡函数数值积分方法,通过利用L ev in匹配法将二重积分化为一重积分,并避免了对复杂的m om en ts的求解,能提高计算的效率,且有很高的求积精度.     

Asymptotic expansions for oscillatory integrals using inverse functions

We treat finite oscillatory integrals of the form ∫ _a ^b F(x)e ^ikG(x) dx in which both F and G are real on the real line, are analytic over the open integration interval, and may have algebraic singularities at either or both interval end points. For many of these, we establish asymptotic expansions in inverse powers of k. No appeal to the theories of stationary phase or steepest descent is involved. We simply apply theory involving inverse functions and expansions for a Fourier coefficient ∫ _a ^b φ(t)e ^ikt dt. To this end, we have assembled several results involving inverse functions. Moreover, we have derived a new asymptotic expansion for this integral, valid when , −1<σ ₁<σ ₂<⋅⋅⋅. The authors were supported by the Office of Advanced Scientific Computing Research, Office of Science, US Department of Energy, under Contract DE-AC02-06CH11357.     

On nonoscillating integrals for computing inhomogeneous Airy functions

Integral representations are considered of solutions of the inhomogeneous Airy differential equation . The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain nonoscillating integrals for complex values of . In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.

     

On the calculation of highly oscillatory integrals with an algebraic singularity

In this paper we explore the evaluation of highly oscillatory integrals containing an algebraic singularity based on the change of variable t = x^−r, where r is a positive real number, and the analytical continuation. Robust numerical results demonstrate the accuracy and effectiveness of the proposed approach.     

The construction of numerical integration rules of degree three for product regions

Numerical integration formulas in n-dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n-dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.     

Non-standard commutators for rough oscillatory singular integrals

In this paper, the author studies a class of non-standard commutators with higher order remainders for oscillatory singular integral operators with phases more general than polynomials. For 1 < p < ∞, the L ^p -boundedness of such operators are obtained provided that their kernels belong to the spaces L ^q (S ⁿ⁻¹) for some q > 1.     

On the computation of highly oscillatory multivariate integrals with stationary points

We consider two types of highly oscillatory bivariate integrals with a nondegenerate stationary point. In each case we produce an asymptotic expansion and two kinds of quadrature algorithms: an asymptotic method and a Filon-type method. Our results emphasize the crucial role played by the behaviour at the stationary point and by the geometry of the boundary of the underlying domain. In memory of Germund Dahlquist (1925–2005). AMS subject classification (2000) Primary 65D32     

Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals

We consider the highly oscillatory integral $F(w):={\int }_{-\infty }^{\infty }{e}^{iw(t^{K+2}+e^{i\theta }{t}^p)}g(t)dt$ for large positive values of w, , K and p positive integers with $1\le p\le K$, and $g(t)$ an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when $w\to +\infty$ for general values of K and p in terms of elementary functions, and determine the Stokes lines. For $p\ne 1$, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters K and p; the special case $p=1$ requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals ${\mathrm{\Psi }}_K(x_1,x_2,\text{…},x_K)$ for large values of one of its variables, say $x_p$, and bounded values of the remaining ones. This family of integrals may be written in the form $F(w)$ for appropriate values of the parameters w, θ and the function $g(t)$. Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large $|x_p|$. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.     

Sharp maximal estimates for doubly oscillatory integrals

We study doubly oscillatory integrals

and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on .

     

The estimates of Bessel integrals with oscillatory factors

In this paper, we establish some sharp estimates of Bessel integrals with oscillatory factors . As an application, we obtain the boundedness of the oscillatory singular integral operators with variable kernels.     

A class of oscillatory singular integrals on triebel-lizorkin spaces

The boundedness on Triebel-Lizorkin spaces of oscillatory singular integral operator T in the form e^i｜x｜^aΩ（x）｜x｜^-n is studied,where a∈R,a≠0,1 and Ω∈L^1（S^n-1） is homogeneous of degree zero and satisfies certain cancellation condition. When kernel Ω（x＇ ）∈Llog＋L（S^n-1 ）, the Fp^a,q（R^n） boundedness of the above operator is obtained. Meanwhile ,when Ω（x） satisfies L^1- Dini condition,the above operator T is bounded on F1^0,1 （R^n）.     

Moment-free numerical integration of highly oscillatory functions

^** Email: s.olver{at}damtp.cam.ac.uk The aim of this paper is to derive new methods for numericallyapproximating the integral of a highly oscillatory function.We begin with a review of the asymptotic and Filon-type methodsdeveloped by Iserles and Nørsett. Using a method developedby Levin as a point of departure, we construct a new methodthat utilizes the same information as a Filon-type method, andobtains the same asymptotic order, while not requiring the computationof moments. We also show that a special case of this methodhas the property that the asymptotic order increases with theaddition of sample points within the interval of integration,unlike all the preceding methods whose orders depend only onthe endpoints.     

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.     

Efficient quadrature for highly oscillatory integrals involving critical points

This paper based on the Levin collocation method and Levin-type method together with composite two-point Gauss–Legendre quadrature presents efficient quadrature for integral transformations of highly oscillatory functions with critical points. The effectiveness and accuracy of the quadrature are tested.     

Numerical evaluation of the two-dimensional partition of unity boundary integrals for Helmholtz problems

There has been considerable attention given in recent years to the problem of extending finite and boundary element-based analysis of Helmholtz problems to higher frequencies. One approach is the Partition of Unity Method, which has been applied successfully to boundary integral solutions of Helmholtz problems, providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy. These benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions. In this paper we adapt the numerical steepest descent method to evaluate these integrals for two-dimensional problems. The approach is successful in reducing the computational effort for most integrals encountered. The paper includes some numerical features that are important for successful practical implementation of the algorithm.