Volume 16, Issue 1

In this paper electromagnetic Herglotz dyadics are used to develop a superposition method and apply it for the study of an inversion scheme for electromagnetic scattering in chiral media. The direct scattering problem for the perfect conductor is formulated and Beltrami Herglotz dyadics and dyadic electromagnetic Herglotz pairs are being used. Assuming that the incident electromagnetic field is produced by a superposition of plane incident waves, the scattered field and the corresponding far-field pattern are expressed as the superposition of the scattered fields and the corresponding to them far-field patterns respectively. Far-field operators are defined and studied and an integral equation is posed. The solvability of this equation is related to the solution of the interior perfect conductor boundary value problem. Finally, an inversion scheme is posed and a theorem for it's solvability is proved.

In this paper we prove that the Cauchy data for the Schrödinger equation in the two-dimensional case determines a potential from L p (for p > 2) uniquely. We also obtain a linear inversion formula for smooth potentials.

We consider constrained ill-posed operator equations. The constraints are such that in one case we restrict the domain to functions which are piecewise constant. In another case we allow only functions that attain values in a certain interval. We use Ginzburg–Landau regularization methods for solving these equations. In our numerical examples we consider the inverse conductivity problem which has applications in electrical impedance tomography. We present a numerical implementation along with some results and compare them with modified H 1 -Tikhonov regularization methods.

This paper deals with an inverse problem of identifying a source coefficient in the process of solute transport in 1-D groundwater flow. With help of an adjoint problem, the solute concentration can be made to be monotone on time by some constraints to the known data and the unknown parameters, and an integral identity connecting varies of the known data with corresponding changes of the unknown is constructed. Furthermore, by analyzing the adjoint problem with the variable separation method, a suitable norm for the unknown is well-defined based on the integral identity by which a conditional stability for the inverse problem considered here is obtained.

The shape of a plane acoustical sound-soft obstacle is detected from knowledge of the far field pattern for one time-harmonic incident field. Two methods based on solving a system of integral equations for the incoming wave and the far field pattern are investigated. Properties of the integral operators required in order to apply regularization, i.e. injectivity and denseness of the range, are proved.

This note is concerned with proving convergence rates for nonlinear Tikhonov regularization under mild regularity assumptions on the solution, namely source conditions of logarithmic type. For the choice of the regularization parameter, either a priori or a posteriori parameter choice strategies according to the discrepancy principle can be used. As usual in showing convergence rates for nonlinear ill-posed problems, restrictions on the nonlinearity of the forward operator have to be made unless the initial error is sufficiently smooth.

The methods of constrained approximation in Hilbert spaces of analytic functions are applied to the solution of the inverse problems of detecting cracks or sources in a two-dimensional material by means of boundary measurements. Issues of well-posedness are discussed, and results on continuity and robustness with respect to the given data are established. Constructive and efficient methods for resolution of the above approximation problems are presented. The techniques are illustrated by numerical examples incorporating a further rational approximation step.