We consider the integration of one-dimensional highly oscillatory functions. Based on analytic continuation, rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand. The accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency, and for the case of an increasing number of points and fixed frequency. These results are then used to obtain quadrature rules for more general oscillatory integrals, i.e., for functions that exhibit some smoothness but that are not analytic. The approach described in this paper is related to the steepest descent method, but it does not employ asymptotic expansions. It can be used for small or moderate frequencies as well as for very high frequencies. The approach is compared with the oscillatory integration techniques recently developed by Iserles and Norsett.

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On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation

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.

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Complex Gaussian quadrature of oscillatory integrals

A new technique to compute the physical optics (PO) integral is presented. The technique consists of a blind code that computes the different contributions (stationary phase points, end points, etc.) numerically. This technique is based on a decomposition of the surface into small triangles and a fast evaluation of each triangle by means of a deformation of the integration path in the complex plane. This algorithm permits a fast and accurate evaluation of the PO integral for smooth large surfaces. The CPU time is almost independent of frequency.

A New Fast Physical Optics for Smooth Surfaces by Means of a Numerical Theory of Diffraction

Saddle point approaches in strong field physics and generation of attosecond pulses

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.

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Superinterpolation in Highly Oscillatory Quadrature

For pt.I see ibid., vol.14, p.973(1988). Products of Green functions for pion-nucleon interactions can most naturally be evaluated by transformation to a momentum-space representation. Such transformations often involve an integral whose integrand is highly oscillatory. By deforming the contour of integration in the complex plane, such an integral can be made to converge rapidly. All oscillatory terms are converted into pure damping terms which decrease exponentially. Examples are given for simple integrals, for the integral of three spherical Bessel functions, and for the Fourier and spherical Bessel function transforms of the Woods-Saxon potential (Fermi-Dirac distribution function). They authors also show that oscillatory principal-value integrals can usually be accurately evaluated using this method. Convergence with respect to the number of Gaussian points is studied. It is shown that the method normally converges much better than ordinary integration along the real axis, especially as the frequency of the oscillations increases relative to the damping. A comparison is made with the method of steepest descent.

http://iopscience.iop.org/article/10.1088/0305-4616/14/7/015/pdf

K T R Davies

Papers

Complex-plane methods for evaluating highly oscillatory integrals in nuclear physics. II