On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

It is premature to include non-CO2 effects of aviation in emission trading schemes

The recent G8 Gleneagles climate statement signed on 8 July 2005 specifically mentions a determination to lessen the impact of aviation on climate [Gleneagles, 2005. The Gleneagles communique: climate change, energy and sustainable development. http://www.fco.gov.uk/Files/kfile/PostG8_Gleneagles_Communique.pdf]. In January 2005 the European Union Emission Trading Scheme (ETS) commenced operation as the largest multi-country, multi-sector ETS in the world, albeit currently limited only to CO2 emissions. At present the scheme makes no provision for aircraft emissions. However, the UK Government would like to see aircraft included in the ETS and plans to use its Presidencies of both the EU and G8 in 2005 to implement these schemes within the EU and perhaps internationally. Non-CO2 effects have been included in some policy-orientated studies of the impact of aviation but we argue that the inclusion of such effects in any such ETS scheme is premature; we specifically argue that use of the Radiative Forcing Index for comparing emissions from different sources is inappropriate and that there is currently no metric for such a purpose that is likely to enable their inclusion in the near future. (c) 2005 Elsevier Ltd. All rights reserved.

A practical approach to goal modelling for time-constrained projects

Goal modelling is a well known rigorous method for analysing problem rationale and developing requirements. Under the pressures typical of time-constrained projects its benefits are not accessible. This is because of the effort and time needed to create the graph and because reading the results can be difficult owing to the effects of crosscutting concerns. Here we introduce an adaptation of KAOS to meet the needs of rapid turn around and clarity. The main aim is to help the stakeholders gain an insight into the larger issues that might be overlooked if they make a premature start into implementation. The method emphasises the use of obstacles, accepts under-refined goals and has new methods for managing crosscutting concerns and strategic decision making. It is expected to be of value to agile as well as traditional processes.