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

Analysis and Applications of Orthogonal Polynomials with Zeros in the Complex Plane

Oscillatory integrals such as Fourier transforms arise in many fields of sciences and engineering. Examples are acoustics, electromagnetism, mechanics and seismology. Numerical techniques to approximate integrals are well known. Classical quadrature rules are accurate when the integrand is smooth and slowly varying. However those methods fail for highly oscillatory cases. Cauchy's Theorem allows us to deform the path of integration into the complex plane. The problem is then solved by deforming the path so that the integrand is not oscillatory anymore. This is done by the method of steepest descent. One efficient technique to approximate non oscillatory integrals along the real line is Gaussian quadrature. This approach crucially relies on orthogonal polynomials. That method is effectively modified for some type of oscillatory integrals within the \textit{``numerical method of steepest descent''.} However when different paths of steepest descent come too close to each other, this generates singularities and thus the failure of the method. In this thesis we develop a numerical method which extends the existing numerical method of steepest descent. The proposed method also uses orthogonal polynomials. Experiments illustrate the failure of the new method for some choices of parameter values, which is caused by the non existence of the orthogonal polynomial under the particular conditions. This immediately shows the importance of the second focus of the thesis: a mathematically rigorous analysis of orthogonal polynomials. Orthogonal polynomials and their limiting zero distribution in particular are a research topic in analysis. We use techniques from potential theory to prove when the zeros of the polynomials accumulate on one analytic arc, and when they split into two separate groups. Since the numerical method preferentially uses low degree polynomials, these polynomials are also analyzed carefully. This allows us to derive conditions for the existence of the orthogonal polynomials as well as connections to classical families of orthogonal polynomials.nrpages: 158status: publishe

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. Unfortunately, uniform asymptotic expansions are difficult to compute numerically, they do not have controllable error and in this case they are not numerically stable. 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.status: publishe