Joe Coyle

Recent Developments in Inverse Acoustic Scattering Theory

We survey some of the highlights of inverse scattering theory as it has developed over the last 15 years, with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves. Included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects.

Hierarchic hp-edge element families for Maxwell's equations on hybrid quadrilateral/triangular meshes

We construct and study a set of hierarchic basis functions for the Galerkin discretisation of the space H(curl;Φ#169;) suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non-uniform order polynomial approximation. We investigate the conditioning and dispersive behaviour of the elements. In addition, numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time-harmonic Maxwells equations that have singularities.

Numerical validation of the linear sampling method

The problem of visualizing scattering objects from far-field data can be addressed by a simple method, named linear sampling method (LSM), which requires the solution of ill-conditioned linear systems. In the present paper we perform a computational and experimental validation of the method, which is implemented by means of four different regularization algorithms. The effectiveness of the LSM when coupled with these algorithms is tested in the case of both simulated and real data. Furthermore a criterion for the choice of a level curve optimally approximating the profile of the scatterers is provided.

Locating the support of objects contained in a two-layered background medium in two dimensions

In this paper the inverse problem of determining the boundaries of bounded objects buried in a two-layered medium using the regularized sampling method is considered. The objects are taken to be either impenetrable obstacles or penetrable anisotropic scatterers and contained in the absorbing layer of a two-layered medium. Measurements of the scattered field are restricted to the layer not containing the scatterer.

Conditioning of Hierarchic p-Version Nédélec Elements on Meshes of Curvilinear Quadrilaterals and Hexahedra

The conditioning of a set of hierarchic basis functions for p-version edge element approximation of the space H(curl) is studied. Theoretical bounds are obtained on the location of the eigenvalues and on the growth of the condition numbers for the mass, curl-curl, and stiffness matrices that naturally arise from Galerkin approximation of Maxwells equations. The theory is applicable to meshes of curvilinear quadrilaterals or hexahedra in two and three dimensions, respectively, including the case in which the local order of approximation is nonuniform. Throughout, the theory is illustrated with numerical examples that show that the theoretical asymptotic bounds are sharp and are attained within the range of practical computation.

Scattering of Time-Harmonic Electromagnetic Waves by Anisotropic Inhomogeneous Scatterers or Impenetrable Obstacles

We investigate an overlapping solution technique to compute the scattering of time-harmonic electromagnetic waves in two dimensions. The technique can be used to compute waves scattered by penetrable anisotropic inhomogeneous scatterers or impenetrable obstacles. The major focus is on implementing the method using finite elements. We prove existence of a unique solution to the discretized problem and derive an optimal convergence rate for the scheme, which is verified by numerical examples.

Computing Maxwell eigenvalues by using higher order edge elements in three dimensions

We use recently proposed hierarchic basis functions and a tetrahedral partitioning to compute Maxwell eigenvalues on a bounded polygonal domain in /spl Ropf//sup 3/, using a p-version finite-element procedure based on edge elements. The problem formulation requires a set of basis functions that are H(curl)-conforming and another compatible set that is H/sup 1/-conforming. In this preliminary study, we employ a uniform order of approximation throughout the domain.

Postprocessing of the Linear Sampling Method by Means of Deformable Models

The linear sampling method is a qualitative procedure for the visualization of both impenetrable and inhomogeneous scatterers, which requires the regularized solution of a linear ill-posed integral equation of the first kind. An open issue in this technique is the one of determining the optimal scatterer profile from the visualization maps in an automatic manner. In the present paper this problem is addressed in two steps. First, linear sampling is optimized by using a new regularization algorithm for the solution of the integral equation, which provides more accurate maps for different levels of the noise affecting the data. Then an edge detection technique based on active contours is applied to the optimized maps. Our computation exploits a recently introduced implementation of the linear sampling method, which enhances both the accuracy and the numerical effectiveness of the approach.

Computation of Maxwell eigenvalues on curvilinear domains using hp -version Nédélec elements

In this paper we present and numerically verify theoretical bounds on the growth of the conditioning number for an H(curl)-conforming basis suitable for variable order approximation on curvilinear quadrilateral or hexahedral meshes. These bounds are given explicitly in terms of the maximum polynomial degree of approximation employed throughout the mesh. Additionally, numerical examples demonstrating the use of the basis in the context of electromagnetic eigenvalue problems on curved domains with reentrant comers are given. These examples also serve as a preliminary investigation of hp-refinement in computing eigenvalues corresponding to both singular and non-singular eigenfunctions on curvilinear domains.