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Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method
We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form where is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations
in the space of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator under consideration, with an emphasis on the function space setting . Firstly, under which conditions is a Fredholm operator, and, secondly, when is the finite section method applicable to
Positive-energy D-bar method for acoustic tomography: a computational study
A new computational method for reconstructing a potential from the
Dirichlet-to-Neumann map at positive energy is developed. The method is based
on D-bar techniques and it works in absence of exceptional points -- in
particular, if the potential is small enough compared to the energy. Numerical
tests reveal exceptional points for perturbed, radial potentials.
Reconstructions for several potentials are computed using simulated
Dirichlet-to-Neumann maps with and without added noise. The new reconstruction
method is shown to work well for energy values between and ,
smaller values giving better results
Refraction-corrected ray-based inversion for three-dimensional ultrasound tomography of the breast
Ultrasound Tomography has seen a revival of interest in the past decade,
especially for breast imaging, due to improvements in both ultrasound and
computing hardware. In particular, three-dimensional ultrasound tomography, a
fully tomographic method in which the medium to be imaged is surrounded by
ultrasound transducers, has become feasible. In this paper, a comprehensive
derivation and study of a robust framework for large-scale bent-ray ultrasound
tomography in 3D for a hemispherical detector array is presented. Two
ray-tracing approaches are derived and compared. More significantly, the
problem of linking the rays between emitters and receivers, which is
challenging in 3D due to the high number of degrees of freedom for the
trajectory of rays, is analysed both as a minimisation and as a root-finding
problem. The ray-linking problem is parameterised for a convex detection
surface and three robust, accurate, and efficient ray-linking algorithms are
formulated and demonstrated. To stabilise these methods, novel
adaptive-smoothing approaches are proposed that control the conditioning of the
update matrices to ensure accurate linking. The nonlinear UST problem of
estimating the sound speed was recast as a series of linearised subproblems,
each solved using the above algorithms and within a steepest descent scheme.
The whole imaging algorithm was demonstrated to be robust and accurate on
realistic data simulated using a full-wave acoustic model and an anatomical
breast phantom, and incorporating the errors due to time-of-flight picking that
would be present with measured data. This method can used to provide a
low-artefact, quantitatively accurate, 3D sound speed maps. In addition to
being useful in their own right, such 3D sound speed maps can be used to
initialise full-wave inversion methods, or as an input to photoacoustic
tomography reconstructions
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The PML for rough surface scattering
In this paper we investigate the use of the perfectly matched layer (PML) to truncate a time harmonic rough
surface scattering problem in the direction away from the scatterer. We prove existence and uniqueness of the solution of the truncated problem as well as an error estimate depending on the thickness and composition of the layer. This global error estimate predicts a linear rate of convergence (under some conditions on the relative size of the real and imaginary parts of the PML function) rather than the usual exponential rate. We then consider scattering by a half-space and show that the solution
of the PML truncated problem converges globally at most quadratically (up to logarithmic factors), providing support for our general theory. However we also prove exponential convergence on compact subsets. We continue by proposing an iterative correction method for the PML truncated problem and, using our estimate for the PML approximation, prove convergence of this method. Finally we provide some numerical results in 2D.(C) 2008 IMACS. Published by Elsevier B.V. All rights reserved
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Boundary integral methods in high frequency scattering
In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources
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