On the convergence of the no response test

The no response test is a new scheme in inverse problems for partial differential equations which was recently proposed in [D. R. Luke and R. Potthast, SIAM J. Appl. Math., 63 (2003), pp. 1292–1312] in the framework of inverse acoustic scattering problems. The main idea of the scheme is to construct special probing waves which are small on some test domain. Then the response for these waves is constructed. If the response is small, the unknown object is assumed to be a subset of the test domain. The response is constructed from one, several, or many particular solutions of the problem under consideration. In this paper, we investigate the convergence of the no response test for the reconstruction information about inclusions D from the Cauchy values of solutions to the Helmholtz equation on an outer surface $\partial\Omega$ with $\overline{D} \subset \Omega$. We show that the one‐wave no response test provides a criterion to test the analytic extensibility of a field. In particular, we investigate the construction of approximations for the set of singular points $N(u)$ of the total fields u from one given pair of Cauchy data. Thus, the no response test solves a particular version of the classical Cauchy problem. Also, if an infinite number of fields is given, we prove that a multifield version of the no response test reconstructs the unknown inclusion D. This is the first convergence analysis which could be achieved for the no response test.

Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

An integral equation method for a boundary value problem arising in unsteady water wave problems

In this paper we consider the 2D Dirichlet boundary value problem for Laplace’s equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al [J. Int. Equ. Appl. 15 (2003) pp. 1-35]. This then leads to an existence proof for the boundary value problem. Keywords. Boundary integral equation method, Water waves, Laplace’s

Problem structuring methods: theorizing the benefits of deconstructing sustainable development projects

Problem structuring methods or PSMs are widely applied across a range of variable but generally small-scale organizational contexts. However, it has been argued that they are seen and experienced less often in areas of wide ranging and highly complex human activity-specifically those relating to sustainability, environment, democracy and conflict (or SEDC). In an attempt to plan, track and influence human activity in SEDC contexts, the authors in this paper make the theoretical case for a PSM, derived from various existing approaches. They show how it could make a contribution in a specific practical context-within sustainable coastal development projects around the Mediterranean which have utilized systemic and prospective sustainability analysis or, as it is now known, Imagine. The latter is itself a PSM but one which is 'bounded' within the limits of the project to help deliver the required 'deliverables' set out in the project blueprint. The authors argue that sustainable development projects would benefit from a deconstruction of process by those engaged in the project and suggest one approach that could be taken-a breakout from a project-bounded PSM to an analysis that embraces the project itself. The paper begins with an introduction to the sustainable development context and literature and then goes on to illustrate the issues by grounding the debate within a set of projects facilitated by Blue Plan for Mediterranean coastal zones. The paper goes on to show how the analytical framework could be applied and what insights might be generated.

Putting the pieces back together again: an illustration of the problem of interpreting development indicators using an African case study

Indicators are commonly recommended as tools for assessing the attainment of development, and the current vogue is for aggregating a number of indicators together into a single index. It is claimed that such indices of development help facilitate maximum impact in policy terms by appealing to those who may not necessarily have technical expertise in data collection, analysis and interpretation. In order to help counter criticisms of over-simplification, those advocating such indices also suggest that the raw data be provided so as to allow disaggregation into component parts and hence facilitate a more subtle interpretation if a reader so desires. This paper examines the problems involved with interpreting indices of development by focusing on the United Nations Development Programmes (UNDP) Human Development Index (HDI) published each year in the Human Development Reports (HDRs). The HDI was intended to provide an alternative to the more economic based indices, such as GDP, commonly used within neo-liberal development agendas. The paper explores the use of the HDI as a gauge of human development by making comparisons between two major political and economic communities in Africa (ECOWAS and SADC). While the HDI did help highlight important changes in human development as expressed by the HDI over 10 years, it is concluded that the HDI and its components are difficult to interpret as methodologies have changed significantly and the 'averaging' nature of the HDI could hide information unless care is taken. The paper discusses the applicability of alternative models to the HDI such as the more neo-populist centred methods commonly advocated for indicators of sustainable development. (C) 2003 Elsevier Ltd. All rights reserved.

Making 'dirty' nations look clean? The nation state and the problem of selecting and weighting indices as tools for measuring progress towards sustainability

Pressing global environmental problems highlight the need to develop tools to measure progress towards "sustainability." However, some argue that any such attempt inevitably reflects the views of those creating such tools and only produce highly contested notions of "reality." To explore this tension, we critically assesses the Environmental Sustainability Index (ESI), a well-publicized product of the World Economic Forum that is designed to measure 'sustainability' by ranking nations on league tables based on extensive databases of environmental indicators. By recreating this index, and then using statistical tools (principal components analysis) to test relations between various components of the index, we challenge ways in which countries are ranked in the ESI. Based on this analysis, we suggest (1) that the approach taken to aggregate, interpret and present the ESI creates a misleading impression that Western countries are more sustainable than the developing world; (2) that unaccounted methodological biases allowed the authors of the ESI to over-generalize the relative 'sustainability' of different countries; and, (3) that this has resulted in simplistic conclusions on the relation between economic growth and environmental sustainability. This criticism should not be interpreted as a call for the abandonment of efforts to create standardized comparable data. Instead, this paper proposes that indicator selection and data collection should draw on a range of voices, including local stakeholders as well as international experts. We also propose that aggregating data into final league ranking tables is too prone to error and creates the illusion of absolute and categorical interpretations. (c) 2004 Elsevier Ltd. All rights reserved.

Ensemble-based data assimilation and the localisation problem

The “butterfly effect” is a popularly known paradigm; commonly it is said that when a butterfly flaps its wings in Brazil, it may cause a tornado in Texas. This essentially describes how weather forecasts can be extremely senstive to small changes in the given atmospheric data, or initial conditions, used in computer model simulations. In 1961 Edward Lorenz found, when running a weather model, that small changes in the initial conditions given to the model can, over time, lead to entriely different forecasts (Lorenz, 1963). This discovery highlights one of the major challenges in modern weather forecasting; that is to provide the computer model with the most accurately specified initial conditions possible. A process known as data assimilation seeks to minimize the errors in the given initial conditions and was, in 1911, described by Bjerkness as “the ultimate problem in meteorology” (Bjerkness, 1911).

A wavenumber independent boundary element method for an acoustic scattering problem

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous. at terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials ( of degree.) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O( N-(v+1) log(1/2) N), where the number of degrees of freedom is proportional to N logN. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.