16 - Applications to inverse problems

Summary

The construction of inversion algorithms provide challenging mathematical problems which shape the direction of research in modern science and technology. Medical imaging, non-destructive evaluation and testing, RADAR and SONAR technology, oil exploration, and remote sensing are some areas where mathematical modelling leads to inverse problems of contemporary interest. Keller [199] gives the following definition, “two problems are inverse of one another if the formulation of each involves all or part of the solution of the other,” and continues, “historically, one of the two problems has been studied extensively for some time, while the other is newer and not so well understood. The former is called the direct problem, while the other is called the inverse problem.” Inverse problems are usually not well-posed, most of the time because of a lack of uniqueness and sometimes because of a lack of stability as well. Nevertheless, uniqueness can be secured if a-priori information is available, so that the possible set of solutions is severely restricted. This is the case, for example, when we know that the object we want to reconstruct is an ellipsoid.

In this chapter we will analyze a few inverse problems that are associated with ellipsoidal geometry. In the first three subsections we discuss the inverse problem of identifying an ellipsoid from low-frequency scattering data, from high-frequency time-dependent scattering, and from tomographic data. Following similar approaches, one can identify the thickness of a penetrable ellipsoidal shell surrounding a confocal ellipsoidal core [105].