Qualitative Approaches in Inverse Scattering Theory

Inverse scattering theory has been a particularly active area in applied
mathematics for the past twenty five years. The aim of research in this field
has been to not only detect but also to identify unknown objects through
the use of acoustic, electromagnetic or elastic waves. Mathematically, such
problems lead to nonlinear and severely ill-posed equations. Until a few
years ago, essentially all existing algorithms for target identification were
based on either a weak scattering approximation or on the use of nonlinear
optimization techniques. In recent years alternative methods for imaging
have been developed which avoid incorrect model assumptions inherent in
weak scattering approximations and, as oppose to nonlinear optimization
techniques, do not require a priori information. Such methods come under
the general title of qualitative methods in inverse scattering theory.
This lecture will provide a basic introduction of qualitative methods together
with numerical examples showing the practicality of these approaches
to the problem of target identification and finally discuss some related open
problems and opportunities.