A numerical method for oscillatory integrals with coalescing saddle points

The value of a highly oscillatory integral is typically determined asymptotically by the behaviour of the integrand near a small number of critical points. These include the endpoints of the integration domain and the so-called stationary points or saddle points -- roots of the derivative of the phase of the integrand -- where the integrand is locally non-oscillatory. Modern methods for highly oscillatory quadrature exhibit numerical issues when two such saddle points coalesce. On the other hand, integrals with coalescing saddle points are a classical topic in asymptotic analysis, where they give rise to uniform asymptotic expansions in terms of the Airy function. In this paper we construct Gaussian quadrature rules that remain uniformly accurate when two saddle points coalesce. These rules are based on orthogonal polynomials in the complex plane. We analyze these polynomials, prove their existence for even degrees, and describe an accurate and efficient numerical scheme for the evaluation of oscillatory integrals with coalescing saddle points

An extended Filon--Clenshaw--Curtis method for high-frequency wave scattering problems in two dimensions

We study the efficient approximation of integrals involving Hankel functions of the first kind which arise in wave scattering problems on straight or convex polygonal boundaries. Filon methods have proved to be an effective way to approximate many types of highly oscillatory integrals, however finding such methods for integrals that involve non-linear oscillators and frequency-dependent singularities is subject to a significant amount of ongoing research. In this work, we demonstrate how Filon methods can be constructed for a class of integrals involving a Hankel function of the first kind. These methods allow the numerical approximation of the integral at uniform cost even when the frequency $\omega$ is large. In constructing these Filon methods we also provide a stable algorithm for computing the Chebyshev moments of the integral based on duality to spectral methods applied to a version of Bessel's equation. Our design for this algorithm has significant potential for further generalisations that would allow Filon methods to be constructed for a wide range of integrals involving special functions. These new extended Filon methods combine many favourable properties, including robustness in regard to the regularity of the integrand and fast approximation for large frequencies. As a consequence, they are of specific relevance to applications in wave scattering, and we show how they may be used in practice to assemble collocation matrices for wavelet-based collocation methods and for hybrid oscillatory approximation spaces in high-frequency wave scattering problems on convex polygonal shapes

Efficient computation of highly oscillatory integrals by using QTT tensor approximation

We propose a new method for the efficient approximation of a class of highly oscillatory weighted integrals where the oscillatory function depends on the frequency parameter $\omega \geq 0$, typically varying in a large interval. Our approach is based, for fixed but arbitrary oscillator, on the pre-computation and low-parametric approximation of certain $\omega$-dependent prototype functions whose evaluation leads in a straightforward way to recover the target integral. The difficulty that arises is that these prototype functions consist of oscillatory integrals and are itself oscillatory which makes them both difficult to evaluate and to approximate. Here we use the quantized-tensor train (QTT) approximation method for functional $m$-vectors of logarithmic complexity in $m$ in combination with a cross-approximation scheme for TT tensors. This allows the accurate approximation and efficient storage of these functions in the wide range of grid and frequency parameters. Numerical examples illustrate the efficiency of the QTT-based numerical integration scheme on various examples in one and several spatial dimensions.Comment: 20 page

On the computation of Gaussian quadrature rules for Chebyshev sets of linearly independent functions

We consider the computation of quadrature rules that are exact for a Chebyshev set of linearly independent functions on an interval $[a,b]$. A general theory of Chebyshev sets guarantees the existence of rules with a Gaussian property, in the sense that $2l$ basis functions can be integrated exactly with just $l$ points and weights. Moreover, all weights are positive and the points lie inside the interval $[a,b]$. However, the points are not the roots of an orthogonal polynomial or any other known special function as in the case of regular Gaussian quadrature. The rules are characterized by a nonlinear system of equations, and earlier numerical methods have mostly focused on finding suitable starting values for a Newton iteration to solve this system. In this paper we describe an alternative scheme that is robust and generally applicable for so-called complete Chebyshev sets. These are ordered Chebyshev sets where the first $k$ elements also form a Chebyshev set for each $k$. The points of the quadrature rule are computed one by one, increasing exactness of the rule in each step. Each step reduces to finding the unique root of a univariate and monotonic function. As such, the scheme of this paper is guaranteed to succeed. The quadrature rules are of interest for integrals with non-smooth integrands that are not well approximated by polynomials

Numerical evaluation of oscillatory integrals via automated steepest descent contour deformation

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder

A new and efficient method for the computation of Legendre coefficients

An efficient procedure for the computation of the coefficients of Legendre expansions is here presented. We prove that the Legendre coefficients associated with a function f(x) can be represented as the Fourier coefficients of an Abel-type transform of f(x). The computation of N Legendre coefficients can then be performed in O(N log N) operations with a single Fast Fourier Transform of the Abel-type transform of f(x).Comment: 5 page