A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory

We investigate Fréchet differentiability of the scattered field with respect to variation in the boundary in the case of time–harmonic acoustic scattering by an unbounded, sound–soft, one–dimensional rough surface. We rigorously prove the differentiability of the scattered field and derive a characterization of the Fréchet derivative as the solution to a Dirichlet boundary value problem. As an application of these results we give rigorous error estimates for first–order perturbation theory, justifying small perturbation methods that have a long history in the engineering literature. As an application of our rigorous estimates we show that a plane acoustic wave incident on a sound–soft rough surface can produce an unbounded scattered field.

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.

A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces

We propose a Nystr¨om/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces. Stability and convergence of the method is established with convergence rates dependent on the smoothness of components of the kernel. The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f, and superalgebraic convergence is established in the case when f is infinitely smooth. Numerical results are presented illustrating this behavior for the case when f is periodic (the diffraction grating case). The Nystr¨om method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.

A uniqueness result for scattering by infinite rough surfaces

Consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane with an unbounded, piecewise Lyapunov boundary. This problem models time-harmonic electromagnetic scattering in transverse magnetic polarization by one-dimensional rough, perfectly conducting surfaces. A radiation condition is introduced for the problem, which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi-periodic, and allows a variety of incident fields including an incident plane wave to be included in the results obtained. We show in this paper that the boundary value problem for the scattered field has at most one solution. For the case when the whole boundary is Lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle.

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

Uniqueness results for direct and inverse scattering by infinite surfaces in a lossy medium

We consider the Dirichlet boundary-value problem for the Helmholtz equation, Au + x2u = 0, with Imx > 0. in an hrbitrary bounded or unbounded open set C c W. Assuming continuity of the solution up to the boundary and a bound on growth a infinity, that lu(x)l < Cexp (Slxl), for some C > 0 and S~< Imx, we prove that the homogeneous problem has only the trivial salution. With this resnlt we prove uniqueness results for direct and inverse problems of scattering by a bounded or infinite obstacle.

A high frequency $hp$ boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

Mobile-to-mobile MIMO transmit-receive diversity systems: analysis and performance in three-dimensional double-correlated channels

Mobile-to-mobile (M-to-M) communications are expected to play a crucial role in future wireless systems and networks. In this paper, we consider M-to-M multiple-input multiple-output (MIMO) maximal ratio combining system and assess its performance in spatially correlated channels. The analysis assumes double-correlated Rayleigh-and-Lognormal fading channels and is performed in terms of average symbol error probability, outage probability, and ergodic capacity. To obtain the receive and transmit spatial correlation functions needed for the performance analysis, we used a three-dimensional (3D) M-to-M MIMO channel model, which takes into account the effects of fast fading and shadowing. The expressions for the considered metrics are derived as a function of the average signal-to-noise ratio per receive antenna in closed-form and are further approximated using the recursive adaptive Simpson quadrature method. Numerical results are provided to show the effects of system parameters, such as distance between antenna elements, maximum elevation angle of scatterers, orientation angle of antenna array in the x–y plane, angle between the x–y plane and the antenna array orientation, and degree of scattering in the x–y plane, on the system performance. Copyright © 2011 John Wiley & Sons, Ltd.

Performance analysis of MIMO MRC in 3D mobile-to-mobile double-correlated channels

In this paper, we consider multiple-input multiple- output (MIMO) maximal ratio combining (MRC) systems and assess the system performance in terms of average symbol error probability (SEP), outage probability and ergodic capacity in double-correlated Rayleigh-and-Lognormal fading channels. In order to derive the receive and transmit correlation functions needed for the performance analysis, a three-dimensional (3D) MIMO mobile-to-mobile (M-to-M) channel model, which takes into account the effects of fast fading and shadowing is used. Numerical results are provided to show the effects of system parameters, such as maximum elevation angle of scatterers, orientation angle of antenna array in the x-y plane, angle between x-y plane and the antenna array orientation, and degree of scattering in the x-y plane, on the system performance.

Detection of flooded urban areas in high resolution Synthetic Aperture Radar images using double scattering

Flooding is a particular hazard in urban areas worldwide due to the increased risks to life and property in these regions. Synthetic Aperture Radar (SAR) sensors are often used to image flooding because of their all-weather day-night capability, and now possess sufficient resolution to image urban flooding. The flood extents extracted from the images may be used for flood relief management and improved urban flood inundation modelling. A difficulty with using SAR for urban flood detection is that, due to its side-looking nature, substantial areas of urban ground surface may not be visible to the SAR due to radar layover and shadow caused by buildings and taller vegetation. This paper investigates whether urban flooding can be detected in layover regions (where flooding may not normally be apparent) using double scattering between the (possibly flooded) ground surface and the walls of adjacent buildings. The method estimates double scattering strengths using a SAR image in conjunction with a high resolution LiDAR (Light Detection and Ranging) height map of the urban area. A SAR simulator is applied to the LiDAR data to generate maps of layover and shadow, and estimate the positions of double scattering curves in the SAR image. Observations of double scattering strengths were compared to the predictions from an electromagnetic scattering model, for both the case of a single image containing flooding, and a change detection case in which the flooded image was compared to an un-flooded image of the same area acquired with the same radar parameters. The method proved successful in detecting double scattering due to flooding in the single-image case, for which flooded double scattering curves were detected with 100% classification accuracy (albeit using a small sample set) and un-flooded curves with 91% classification accuracy. The same measures of success were achieved using change detection between flooded and un-flooded images. Depending on the particular flooding situation, the method could lead to improved detection of flooding in urban areas.

Control and analysis of oriented thin films of lipid inverse bicontinuous cubic phases using grazing incidence small-angle X‑ray scattering

Lipid cubic phases are complex nanostructures that form naturally in a variety of biological systems, with applications including drug delivery and nanotemplating. Most X-ray scattering studies on lipid cubic phases have used unoriented polydomain samples as either bulk gels or suspensions of micrometer-sized cubosomes. We present a method of investigating cubic phases in a new form, as supported thin films that can be analyzed using grazing incidence small-angle X-ray scattering (GISAXS). We present GISAXS data on three lipid systems: phytantriol and two grades of monoolein (research and industrial). The use of thin films brings a number of advantages. First, the samples exhibit a high degree of uniaxial orientation about the substrate normal. Second, the new morphology allows precise control of the substrate mesophase geometry and lattice parameter using a controlled temperature and humidity environment, and we demonstrate the controllable formation of oriented diamond and gyroid inverse bicontinuous cubic along with lamellar phases. Finally, the thin film morphology allows the induction of reversible phase transitions between these mesophase structures by changes in humidity on subminute time scales, and we present timeresolved GISAXS data monitoring these transformations.

Electron doping and phonon scattering in Ti1+xS2 thermoelectric compounds

Bulk polycrystalline samples in the series Ti1+xS2 (x = 0 to 0.05) were prepared using high temperature synthesis from the elements and spark plasma sintering. X-ray structure analysis shows that the lattice constant c expands as titanium intercalates between TiS2 slabs. For x=0, a Seebeck coefficient close to -300 μV/K is observed for the first time in TiS2 compounds. The decrease in electrical resistivity and Seebeck coefficient that occurs upon Ti intercalation (Ti off stoichiometry) supports the view that charge carrier transfer to the Ti 3d band takes place and the carrier concentration increases. At the same time, the thermal conductivity is reduced by phonon scattering due to structural disorder induced by Ti intercalation. Optimum ZT values of 0.14 and 0.48 at 300K and 700K, respectively, are obtained for x=0.025.

A high frequency boundary element method for scattering by a class of nonconvex obstacles

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.