A numerical study of the probe method

The goal of this work is the numerical realization of the probe method suggested by Ikehata for the detection of an obstacle D in inverse scattering. The main idea of the method is to use probes in the form of point source (., z) with source point z to define an indicator function (I) over cap (z) which can be reconstructed from Cauchy data or far. eld data. The indicator function boolean AND (I) over cap (z) can be shown to blow off when the source point z tends to the boundary aD, and this behavior can be used to find D. To study the feasibility of the probe method we will use two equivalent formulations of the indicator function. We will carry out the numerical realization of the functional and show reconstructions of a sound-soft obstacle.

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.

The sediment trapping efficiency of the macro-tidal Daly Estuary, tropical Australia

Field studies were carried out on the water and sediment dynamics in the tropical, macro-tidal, Daly Estuary. The estuary is shallow, very-turbid, about 100 km long, and the entrance is funnel-shape. In the wet, high flow season, normal tidal ranges can be suppressed in the estuary, depending on inflow rates, and freshwater becomes dominant up to the mouth. At that time a fraction of the fine sediment load is exported offshore as a bottom-tagging nepheloid layer after the sediment falls out of suspension of the thin, near-surface, river plume. The remaining fraction and the riverine coarse sediment form a large sediment bar 10 km long, up to 6 m in height and extending across the whole width of the channel near the mouth. This bar, as well as shoals in the estuary, partially pond the mid- to upper-estuary. This bar builds up from the deposition of riverine sediment during a wet season with high runoff and can raise mean water level by up to 2 m in the upper estuary in the low flow season. This ponding effect takes about three successive dry years to disappear by the sediment forming the bar being redistributed all over the estuary by tidal pumping of fine and coarse sediment in the dry season, which is the low flow season. The swift reversal of the tidal currents from ebb to flood results in macro-turbulence that lasts about 20 min. Bed load transport is preferentially landward and occurs only for water currents greater than 0.6 m s(-1). This high value of the threshold velocity suggests that the sand may be cemented by the mud. The Daly Estuary thus is a leaky sediment trap with an efficiency varying both seasonally and inter-annually. (c) 2006 Elsevier Ltd. All rights reserved.

High-resolution, unstructured meshes for hydrodynamic models of the Great Barrier Reef, Australia

Accuracy and mesh generation are key issues for the high-resolution hydrodynamic modelling of the whole Great Barrier Reef. Our objective is to generate suitable unstructured grids that can resolve topological and dynamical features like tidal jets and recirculation eddies in the wake of islands. A new strategy is suggested to refine the mesh in areas of interest taking into account the bathymetric field and an approximated distance to islands and reefs. Such a distance is obtained by solving an elliptic differential operator, with specific boundary conditions. Meshes produced illustrate both the validity and the efficiency of the adaptive strategy. Selection of refinement and geometrical parameters is discussed. (c) 2006 Elsevier Ltd. All rights reserved.

Abstracts of communications: Proceedings of the thirty-nineth meeting of the Agricultural Research Modellers' Group

This group, which is concerned with the applications of mathematics to agricultural science, was formed in 1970 and has since met at approximately yearly intervals in London for one-day meetings. The thirty-ninth meeting of the group, chaired by Professor N. Crout of the University of Nottingham, was held in the Kohn Centre at the Royal Society, 6 Carlton House Terrace, London on Friday, 30 March 2007 when the following papers were read.

A new solution of the rendering equation with stratified Monte Carlo approach

This paper is turned to the advanced Monte Carlo methods for realistic image creation. It offers a new stratified approach for solving the rendering equation. We consider the numerical solution of the rendering equation by separation of integration domain. The hemispherical integration domain is symmetrically separated into 16 parts. First 9 sub-domains are equal size of orthogonal spherical triangles. They are symmetric each to other and grouped with a common vertex around the normal vector to the surface. The hemispherical integration domain is completed with more 8 sub-domains of equal size spherical quadrangles, also symmetric each to other. All sub-domains have fixed vertices and computable parameters. The bijections of unit square into an orthogonal spherical triangle and into a spherical quadrangle are derived and used to generate sampling points. Then, the symmetric sampling scheme is applied to generate the sampling points distributed over the hemispherical integration domain. The necessary transformations are made and the stratified Monte Carlo estimator is presented. The rate of convergence is obtained and one can see that the algorithm is of super-convergent type.

Approximate Riemann solutions of the two-dimensional shallow-water equations

A finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. An extension to the two-dimensional equations with source terms, is included. The scheme is applied to a dam-break problem with cylindrical symmetry.