The dependence of transmissivity estimates on head data is both nonlinear and ill-posed, and so is difficult to analyse and solve. The paper looks at the setting of this problem, in particular in what spaces are the actual data and in what spaces can uniqueness be proven. The paper also looks at the problem of finding a discretization that has at least some of the useful properties found in discretizations of linear ill-posed problems based the singular value decomposition. As definitive answers have not yet been found to the many of the questions raised, the paper is intended to serve more as a platform for further work than as a report on results achieved.

The Lanczos algorithms can be used to find a symmetric tridiagonal matrix from its eigenvalues and the first components of its normalized eigenvectors. The direct application of this method to discretized Strum-Liouville problems is useless since the finite difference eigenvalues behave quite differently asymptotically than the eigenvalues of the continuous Strum-Liouville problem. We suggest a multiplicative asymptotic correction for the discrete equation. The corrected equations can still be solved, at least approximately, by an algorithm similar to the Lanczos algorithm. Numerical experiments show that this approach leads to results of modest accuracy.

This paper is a self-contained account of the application of the Darboux transformation to the one-dimensional SchrOdinger inverse problem, giving results that provide solutions to the Korteweg-de Vries equation. No originality is claimed by the author; the quoted references have been used for the material presented.

The eddy-currents that are induced in a thin conducting sheet by a sinusoidally varying primary magnetic field are investigated in the low frequency limit, when the depth of penetration of the· primary field is much greater than the thickness of the sheet, by setting up a perturbation scheme in terms of the small parameter, $\delta$, which is the ratio of the thickness of the sheet to the length scale of the primary field. The terms of zero order in $\delta$ give the familiar results that were first obtained by Maxwell. The terms of first order in S depend on the tangential as well as the normal component of the primary field and give eddy-current distributions that vary quadratically across the sheet. The boundary conditions that determine the accompanying secondary magnetic field are derived. Detailed results are given when the primary field is that due to a dipole.

The results are relevant to problems in ionospheric physics and geophysical exploration.

This paper illustrates, mainly by example, the links between some existing inverse problem formulations and methodologies, and statistical formulations of the same problems. The examples are from curve fitting, density estimation, image restoration and emission tomography.

For aquifers having a zonation structure with the transmissivity varying smoothly and slowly over each zone, a common approach in determining transmissivity is to presume a known zonation structure and seek a constant approximation to the transmissivity over each zone. However, this procedure is not always acceptable as it may lead to instability in the estimation process.

In this paper, we discuss how one may simultaneously determine the zonation structure and a piecewise constant representation of the transmissivity by adapting the linear functional strategy proposed by Anderssen and Dietrich (1987). The implementation of this idea results in several adaptive and highly parallelizable procedures for the parameter identification problem. Some stability results and a generalization of the method using a Petrov-Galerkin interpretation are also described.