A number of iterative algorithms to solve integral equations arising in field problems are discussed. We describe the essential features of the Neumann Series, overrelaxation methods, Krylov subspace methods, and the conjugate gradient technique. Proofs of convergence of the conjugate gradient method are directly available when the underlying integral operator is self-adjoint, and in this case the method is equivalent to the Krylov method. However, for non-self-adjoint operators the conjugate gradient method requires an implicit symmetrization which results in poorer convergence than that obtained using the Krylov method. Some convergence results are also available for overrelaxation methods for both self-adjoint and non-self-adjoint operators. Relations between all of the methods will be described and numerical performance will be contrasted using a uniform square error criterion. All the methods are treated in the continuous operator form which is especially useful in using the physical setting to arrive at effective preconditioners.

The problem of two-dimensional scattering of elastic waves by an elastic inclusion can be formulated in terms of a domain integral equation, in which the grad-div operator acts on a vector potential. The vector potential is the spatial convolution of a Green's function with the product of the density and the displacement over the domain of interest. A weak form of the integral equation for the unknown displacement is obtained by testing it with rooftop functions. This method shows excellent numerical performance.

This work is concerned with the inverse problem of determining the density of an elastic inclusion from the knowledge of how the inclusion scatters known incident elastic waves. A modified gradient method which is based on domain–integral representations for the elastic wavefield is used for the solution of the inverse problem. The algorithm employed is an extension of the Kleinman–van den Berg method to elasticity, and involves an iterative determination of both the unknown density and the shape of the inclusion. Some numerical experiments are presented.

This paper is concerned with various variational formulations for the fluid-solid interaction problems. The basic approach here is a coupling of field and boundary integral equation methods. In particular, Garding's inequalities are established in appropriate Sobolev spaces for all the formulations. Existence and uniqueness results of the corresponding weak solutions are given under suitable assumptions.

This paper is concerned with the application of boundary integral equation method to the electromagnetic scattering of a perfect conductor in the three dimensional space. A collocation method is employed for the magnetic field integral equation and error estimates are derived. Far-field patterns and radar cross sections are computed for various wave numbers in the case of sphere. Numerical experiments are compared to those ontained from the Mie series method in order to verify the predicted theoretical results.

A method for reconstructing the complex permittivity of a bounded inhomogeneous object from measured scattered field data is presented. We apply the contrast source inversion (CSI) method to the experiment in which multi-frequency data are available, inspired by the success of the CSI method in case of single-frequency data. The scattering object is considered to be an inhomogeneous lossy dielectric cylinder of arbitrary cross-section embedded in a bounded domain of free space.

We have presented elsewhere the problem of choosing boundary data for the exterior Helmholtz equation in order to optimize a functional of the radiated far field. In this paper, we use asymptotic methods to determine an approximate optimal solution.

This paper presents a simplified method for determining the coefficients in a low frequency expansion of the electric and magnetic fields for the problem of scattering by penetrable homogeneous dielectric (both lossy and lossless) bodies.

The paper considers the application of two different solution methods to the inversion of the Ipswich data available in June, 1996. The first one is derived from diffraction tomography in a cylindrical measurement geometry whereas the second one is based on the modified gradient method specialized to the case of binary targets.

In earlier work, we have given constructive methods to compute the surface currents on a conformal antenna which is required to radiate a maximal amount of energy into a predetermined sector of the far field. More recently, the authors have used asymptotic methods to compute approximate optimal surface currents in the time harmonic two-dimensional electromagnetic case (the Helmholtz equation with impedance boundary condition); for the case of high frequency.In the present work, we extend these asymptotic results to the full three-dimensional time-harmonic electromagnetic case. We obtain a representation of the suboptimal current which explicitly shows the dependence on the total curvature, nx of the radiating structure at each point x.

The cross sectional contour of a sound-soft closed cylindrical obstacle placed in an acoustic planar waveguide modelling a shallow water configuration is retrieved from a limited knowledge of scattered farfield patterns in the water layer at a single frequency. A complete Dirichlet family of fundamental solutions of the corresponding boundary value problem is introduced (Green's functions of the waveguide). Iterative construction of the contour is carried out by minimizing a two-term cost functional. The first term measures how well the data are fitted, the second term how well the boundary condition is satisfied. In practice, a star-shaped contour is sought while the scattered field is taken as a finite weighted sum of Green's functions whose source locations evolve with the retrieved contour. Reconstructions from independently generated synthetic data for both convex and non-convex shapes at a low and a high frequency are shown. Influence of numerical parameters (initial shape, number of Green's functions and sampling nodes of the contour, relative weight of each term in the cost functional) and physical ones (location of sources, location and positioning accuracy of the receivers, measurement noise) is investigated. The good efficiency of this complete family method is confirmed in a demanding situation where, in addition to filtering out of high-spatial-frequency wavefields with range, only finitely many modes are propagated; and where lack of information due to aspect-limited data is not alleviated by frequency diversity.

The second annual AP-S/URSI special session on image reconstruction using real data was held at the 1996 AP-S/URSI International Symposium in Baltimore, Maryland. The participants tested their inversion algorithms on X-band measured data, collected at the USAF Ipswich Measurement Facility. Measurements were made available on a series of complicated objects. Some of the targets were fully known to the participants, while other targets were deliberately presented with a minimum of description: the so-called mystery targets. The targets for 1996 are described with illustrations.

The modified gradient algorithm has been shown to provide a stable method for reconstructing complex refractive indices (acoustic and electromagnetic) of bounded isotropic inhomogeneities in a variety of 2D problems where the size of the inhomogeneity is of the order of one to three wavelengths. The essential features of the method will be summarized including the use of regularization techniques for resolving discontinuities in the refractive index. The method involves the iterative construction of a global optimizer of a functional consisting of the error in satisfying an integral form of the field equation (the Lippmann-Schwinger equation) and the discrepancy between measured and predicted data. The optimizer is a function pair consisting of the refractive index and the field within the inhomogeneity. The extension of this method to 3D problems and multifrequency data is described. While the extension to three dimensions presents no basic theoretical difficulties, the computational problem is much more complicated. A way to ameliorate this complication by adjusting the integral form of the field equation is described. When data is available at more than one frequency the algorithm must be further modified and these modifications are given. In this case it is necessary to have a priori information on the dispersion relation in the inhomogeneity, for example, in electromagnetics an assumption that the medium is Maxwellian. Results of numerical experiments will be presented to illustrate both the strengths and weaknesses of the method.

This paper describes a simple algorithm for reconstructing the complex index of refraction of a bounded object immersed in a known background from a knowledge of how the object scatters known incident radiation. The method described here is versatile accommodating both spatially and frequency varying incident fields and allowing a priori information about the scatterer to be introduced in a simple fashion. Numerical results show that this new algorithm outperforms the modified gradient approach which until now has been one of the most effective reconstruction algorithms available.

We develop a new algorithm for solving inverse acoustic scattering problems. In particular we show that this algorithm can reproduce scattering shapes efficiently, using synthetic data, from only one incident wave in the acoustically hard case and using at most two incident waves for the acoustically soft problem. In order to test the inversion algorithm we generate synthetic data using a technique which combines the distributed source method and the fundamental solution of the Helmholtz equation in order to calculate the scattered field for each of these problems. Numerical results for three-dimensional axially symmetric shapes are compared with those obtained previously by other authors.

For pt.II see ibid., vol.39, p.29-32 (1997). We describe results obtained in image reconstruction using the Ipswich data sets IPS009-IPS012. In van den Berg et al. (1995, 1997) and van den Berg and Kleinman (1996), we employed versions of the modified gradient method to reconstruct the shape, location, and index of refraction of known and unknown scatterers, both dielectric and perfectly conducting, from the measured scattered field data contained in IPS001-IPS008. In the present paper, we employ a new inversion method, the contrast source inversion (CSI) method, for the reconstructions. We include here a brief description of the method, given in greater detail in van den Berg and Kleinman. In the case of the new Ipswich experiments, we have 36 angles of incidence, equidistantly distributed around the object. The unknown scatterer is assumed to be located somewhere in a known, bounded, test domain D (taken to be square), and the scattered field is measured on a domain S (taken to be a circle) containing the test domain D in its interior. In the case of the Ipswich experiments, S was taken to be in the far zone of the scattered field, and measurements were made at 18 angles of observation, equidistantly distributed over a semicircle. For each experiment, this semicircle started with the forward-scattering angle.

A method for reconstructing the complex permittivity of a bounded inhomogeneous object from measured scattered field data is presented. Consider the scattering object to be an inhomogeneous lossy dielectric cylinder of arbitrary cross-section embedded in a bounded domain D of free space. The incident excitation consists of electromagnetic waves with the electric field vector polarized along the cylinder axis. To reconstruct the permittivity and conductivity of an unknown scatterer from a knowledge of the scattered field we assume that the object is illuminated successively by a number of line sources at different locations and different frequencies. For each excitation we then have a scalar problem. We assume that, for the description of the material dispersion, the Maxwell model holds. We then introduce the frequency dependent contrast.

We are concerned herein with inverse scattering problems in stratified media and aspect-limited data configurations. In such configurations, the sources and receivers of the probing waves are located in a medium different from the one which contains the object under test. This results in a lack of information which enhances the inherent ill-posedness of the inverse problem. To make the problem more tractable, we assume that the test object is homogeneous with known constitutive parameters so that the inverse problem consists of reconstructing its shape and location. This non-linear inverse problem is solved using the modified gradient method in which the a priori information is introduced as a binary constraint. A cooling parameter is introduced at the same time, which allows us to control the evolution of the iterative process. The effectiveness of this algorithm is studied for three different physical applications.

This paper is devoted to the analysis of the numerical solution of the exterior Neumann problem for the Helmholtz equation formulated as a hypersingular integral equation. Three boundary element Galerkin methods for the solution of the screen problem are investigated when the boundary is an open surface (screen): standard h-version, augmented h-version, and h-p version. Their convergence is proven and a detailed discussion of a posteriors error estimates based on the residual error method is presented. An adaptive boundary element algorithm based on the estimates is also presented.

The total variation minimization method for deblurring noisy data is shown to be effective in dramatically increasing the resolution in a modified gradient approach to index of refraction reconstruction from measured scattered field data. Numerical evidence is presented which shows that by including the total variation in the functional to be minimized the reconstructions of piecewise constant profiles are considerably sharpened. The stability of the modified gradient method with respect to noise is apparently also enhanced. Furthermore, the presence of the total variation does not appear to adversely effect the established effectiveness of the modified gradient method in reconstructing smooth profiles.