Boundary integral methods in high frequency scattering

In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources

Numerical aspects of special functions

This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions)

Numerical aspects of special functions

This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions)

Stationary Tempering and the Complex Quadrature Problem

In the present paper we describe a stochastic quadrature method that is designed for the evaluation of generalized, complex averages. Motivated by recent advances in sparse sampling techniques, this method is based on a combination of parallel tempering and stationary phase filtering methods. Numerical applications of the resulting ‘‘stationary tempering’’ approach are presented. We also examine inherent structure decomposition from a probabilistic clustering perspective

Correlation Functions for a Chain of Short Range Oscillators

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate $t^{-\frac{1}{3}}$ for position and momentum correlations and as $t^{-\frac{2}{3}}$ for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate $t^{-\frac{1}{4}}$ for position and momentum correlators and with rate $t^{-\frac{1}{2}}$ for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly.Comment: 25 pages, 6 figure

Efficient computation of surface fields excited on a dielectric-coated circular cylinder

Cataloged from PDF version of article.An efficient method to evaluate the surface fields excited on an electrically large dielectric-coated circular cylinder is presented. The efficiency of the method results from the circumferentially propagating representation of the Green’s function as well as its efficient numerical evaluation along a steepest descent path. The circumferentially propagating series representation of the appropriate Green’s function is obtained from its radially propagating counterpart via Watson’s transformation and then the path of integration is deformed to the steepest descent path on which the integrand decays most rapidly. Numerical results are presented that indicate that the representations obtained here are very efficient and valid even for arbitrary small separations of the source and field points. This work is especially useful in the moment-method analysis of conformal microstrip antennas where the mutual coupling effects are important

Correlation Functions for a Chain of Short Range Oscillators

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate t-13 for position and momentum correlations and as t-23 for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate t-14 for position and momentum correlators and with rate t-12 for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly

Airy-function approach to binary black hole merger waveforms: The fold-caustic diffraction model

From numerical simulations of the Einstein equations, and also from gravitational wave observations, the gravitational wave signal from a binary black hole merger is seen to be simple and to possess certain universal features. The simplicity is somewhat surprising given that non-linearities of general relativity are thought to play an important role at the merger. The universal features include an increasing amplitude as we approach the merger, where transition from an oscillatory to a damped regime occurs in a pattern apparently oblivious to the initial conditions. We propose an Airy-function pattern to model the binary black hole (BBH) merger waveform, focusing on accounting for its simplicity and universality. We postulate that the relevant universal features are controlled by a physical mechanism involving: i) a caustic phenomenon in a basic `geometric optics' approximation and, ii) a diffraction over the caustic regularizing its divergence. Universality of caustics and their diffraction patterns account for the observed universal features, as in optical phenomena such as rainbows. This postulate, if true, allows us to borrow mathematical techniques from Singularity (Catastrophe) Theory, in particular Arnol'd-Thom's theorem, and to understand binary mergers in terms of fold caustics. The diffraction pattern corresponding to the fold-caustic is given in terms of the Airy function, which leads (under a `uniform approximation') to the waveform model written in terms of a parameterized Airy function. The post-merger phase does not share the same features of simplicity and universality, and must be added separately. Nevertheless, our proposal allows a smooth matching of the inspiral and post-merger signals by using the known asymptotics of the Airy function