Solving the Inverse Scattering Problem: The Linear Sampling Method

Abstract

Scattering occurs when a wave is forced to deviate from its original trajectory by a material object in its path; reconstructing this object, or scatterer, from measurements of the outgoing scattered wave, is referred to as the inverse scattering problem. Mathematically, this problem is modelled by the Helmholtz equation, the time-independent component of the scalar wave function, and, for a penetrable scatterer, classical methods such as the Born approximation and constrained optimization techniques are discussed in this paper, to motivate the requirement of a new method. Although the classical methods are highly efficient, they require a significant amount of a priori data, which is not always available; the Linear Sampling Method (LSM), however, was introduced by Andreas Kirsch and David Colton as a rapid method that requires only a limited number of assumptions. The LSM is discussed in detail and even applied to existing data via a python programme. To conclude, the classical Helmholtz equation is compared to the quantum physical Schrödinger equation; although these equations seem to be rather similar, no analogy between the two could be drawn to apply the LSM to the latter. Regardless, the author notes that the similarities should not be ignored and suggests that it may be possible to modify the LSM to apply to the Schrödinger equation.