Introduction

Abstract

It has only been since the mid-1960s that inverse problems has been identified as a proper subfield of mathematics. Prior to this conventional wisdom held it was not an area appropriate for mathematical analysis. This historical prejudice dates back to Hadamard who claimed that the only problems of physical interest were those that had a unique solution depending continuously on the given data. Such problems were well-posed and problems that were not well-posed were labeled ill-posed. In particular, ill-posed problems connected with partial differential equations of mathematical physics were considered to be of purely academic interest and not worthy of serious study. In the meantime, the success of radar and sonar during the Second World War caused scientists to ask the question if more could be determined about a scattering object than simply its location. Such problems are in the category of inverse scattering problems and it was slowly realised that these problems, although of obvious physical interest, were ill-posed mathematically. Similar problems began to present themselves in other areas such as geophysics, medical imaging and non-destructive testing. However, due to the lack of a mathematical theory of inverse problems together with limited computational capabilities, further progress was not possible.