Harmonic continuation and the magnetostatic inverse problem

The magnetostatic inverse problem is this: given measurements of a static magnetic induction outside some region containing a source, infer the source current density that produced the field. As stated, the magnetostatic inverse problem is ill-posed. That is to say, there is no unique solution. Many different current densities with the source region can produce the same external magnetic field.; There are essentially two ways to "solve" an ill posed problem like the magnetostatic inverse problem. The first approach is probabilistic in nature. In this approach, one assigns probabilities to the various possible solutions and computes an expectation value. The second approach restricts the space of possible solutions by assuming a parametric model for the source. This thesis explores both of these techniques.; The probabilistic approach is found to be successful in problems where the source current density is confined to a given plane. For example, numerical experiments presented in this thesis show that a current flowing around a crack in a thin metal plate can be imaged by inverting measurements of the magnetic induction.; The approach based on parametric models works better for sources in three dimensions. By varying model parameters and iteratively solving a sequence of forward problems, one can often find parameters which minimize the difference between the computed and measured magnetic induction. The usefulness of this approach is limited by the efficiency with which the forward problem can be solved, and hence overly simplified models are often used.; This thesis presents an approach to handling more realistic models. This approach, which relies on the principle of harmonic continuation, is particularly well-suited to the problem of magnetoencephalography in which the magnetic induction outside a patient's head is inverted to yield an image of the biomagnetic source in the brain.