Simon N. Chandler-Wilde

Timo Betcke, Simon N. Chandler-Wilde, Ivan G. Graham et al.

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(γk)$, for some $γ>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

David P. Hewett, Stephen Langdon, Simon N. Chandler-Wilde

We propose and analyse a hybrid numerical-asymptotic $hp$ boundary element method for time-harmonic scattering of an incident plane wave by an arbitrary collinear array of sound-soft two-dimensional screens. Our method uses an approximation space enriched with oscillatory basis functions, chosen to capture the high frequency asymptotics of the solution. We provide a rigorous frequency-explicit error analysis which proves that the method converges exponentially as the number of degrees of freedom $N$ increases, and that to achieve any desired accuracy it is sufficient to increase $N$ in proportion to the square of the logarithm of the frequency as the frequency increases (standard boundary element methods require $N$ to increase at least linearly with frequency to retain accuracy). Our numerical results suggest that fixed accuracy can in fact be achieved at arbitrarily high frequencies with a frequency-independent computational cost, when the oscillatory integrals required for implementation are computed using Filon quadrature. We also show how our method can be applied to the complementary "breakwater" problem of propagation through an aperture in an infinite sound-hard screen.

Simon N. Chandler-Wilde, Ratchanikorn Chonchaiya, Marko Lindner

In this paper we develop and apply methods for the spectral analysis of non-self-adjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). As a major application to illustrate our methods we focus on the "hopping sign model" introduced by J.Feinberg and A.Zee (Phys. Rev. E 59 (1999), 6433-6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random $\pm 1$'s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and $p$-norm $\eps$-pseudospectra ($\eps>0$, $p\in [1,\infty]$) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum $Σ$. We also propose a sequence of inclusion sets for $Σ$ which we show is convergent to $Σ$, with the $n$th element of the sequence computable by calculating smallest singular values of (large numbers of) $n\times n$ matrices. We propose similar convergent approximations for the 2-norm $\eps$-pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.

Mohammad Alazah, Simon N. Chandler-Wilde, Scott La Porte

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel ({\em J. Math. Phys.} {\bf 34} (1956), 298--307) and Hunter and Regan ({\em Math. Comp.} {\bf 26} (1972), 539--541). We construct approximations which we prove are exponentially convergent as a function of $N$, the number of quadrature points, obtaining explicit error bounds which show that accuracies of $10^{-15}$ uniformly on the real line are achieved with N=12, this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement.

Simon N. Chandler-Wilde, Marko Lindner

We study the spectra and pseudospectra of finite and infinite tridiagonal random matrices, in the case where each of the diagonals varies over a separate compact set, say $U,V,W\subset\mathbb{C}$. Such matrices are sometimes termed stochastic Toeplitz matrices $A_+$ in the semi-infinite case and stochastic Laurent matrices $A$ in the bi-infinite case. Their spectra, $Σ=$ spec $A$ and $Σ_+=$ spec $A_+$, are independent of $A$ and $A_+$ as long as $A$ and $A_+$ are pseudoergodic (in the sense of E.B. Davies, Commun. Math. Phys., 2001), which holds almost surely in the random case. This was shown in Davies (2001) for $A$; that the same holds for $A_+$ is one main result of this paper. We give upper and lower bounds on $Σ$ and $Σ_+$, and we explicitly compute a set $G$ that fills the gap between the two in the sense that $Σ\cup G=Σ_+$. We show that invertibility of one operator $A_+$ implies invertibility - and uniform boundedness of the inverses - of all finite square matrices with three diagonals in $U, V$ and $W$. This implies that the so-called finite section method for the approximate solution of a system $A_+x=b$ is applicable as soon as $A_+$ is invertible, and that the same method for estimating the spectrum of $A_+$ does not suffer from spectral pollution. Both results illustrate that tridiagonal stochastic Toeplitz operators share important properties of (classical) Toeplitz operators. One of our main tools is a new version of the Coburn lemma for classical Toeplitz operators, saying that a stochastic tridiagonal Toeplitz operator, if Fredholm, is always injective or surjective. In the final part we bound and compare the norms, and the norms of inverses, of bi-infinite, semi-infinite and finite tridiagonal matrices over $U$, $V$ and $W$. This allows the study of the resolvent norms, and hence the pseudospectra, of these operators and matrices.

Simon N. Chandler-Wilde, David P. Hewett

We consider time-harmonic acoustic scattering by planar sound-soft (Dirichlet) and sound-hard (Neumann) screens. In contrast to previous studies, in which the domain occupied by the screen is assumed to be Lipschitz or smoother, we consider screens occupying an arbitrary bounded open set in the plane. Thus our study includes cases where the closure of the domain occupied by the screen has larger planar Lebesgue measure than the screen, as can happen, for example, when the screen has a fractal boundary. We show how to formulate well-posed boundary value problems for such scattering problems, our arguments depending on results on the coercivity of the acoustic single-layer and hypersingular boundary integral operators, and on properties of Sobolev spaces on general open sets which appear to be new. Our analysis teases out the explicit wavenumber dependence of the continuity and coercivity constants of the boundary integral operators, viewed as mappings between fractional Sobolev spaces, this in part extending previous results of Ha-Duong. We also consider the complementary problem of propagation through a bounded aperture in an infinite planar screen.

Simon N. Chandler-Wilde, Stephen Langdon

We describe some recent advances in the numerical solution of acoustic scattering problems. A major focus of the paper is the efficient solution of high frequency scattering problems via hybrid numerical-asymptotic boundary element methods. We also make connections to the unified transform method due to A.S. Fokas and co-authors, analysing particular instances of this method, proposed by J.A. DeSanto and co-authors, for problems of acoustic scattering by diffraction gratings.