Abstract
We consider the problem of inversion of elastic light scattering (ELS) measurements from polymeric emulsions, to obtain its particle size distribution (PSD) and its refractive index. The mathematical formulation results in a nonlinear inverse problem. A Fredholm integral equation of the first kind appears with an unknown parameter in its kernel. We discuss the existence, uniqueness, and stability of the generalized solutions of the problem when it is stated as a minimization problem with a least square functional. First, we assume that the PSD is known, and for this case we prove that the solution exists and is unique as long as the relation between the measurements and the parameter is by an injective function. Then, we use this result to state sufficient conditions for the complete problem. The analysis of existence and uniqueness of the solution for the problem in hand is supported by numerical simulation. The Phillips–Tikhonov regularization method is proposed to stabilize the problem when noisy-data is available.
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