An Exactly-Solvable Quasistatic Electricity Inverse Problem: Retrieval of the Complex Permittivity of a Cylinder Taking Account of Nuisance Parameter Uncertainty
This study concerns the 2D inverse problem of the retrieval, using external field data, of either one of the two physical parameters, constituted by the real and imaginary parts of the permittivity, of a z-independent cylindrical dielectric specimen subjected to an external, z-independent, quasistatic electric field. Six other parameters enter into the inverse problem. They are termed nuisance parameters because: 1) they are not retrieved during the inversion and 2) uncertainty as to their actual values can adversely affect the accuracy of the retrieval of the permittivity. This inverse problem is shown to have an exact, mathematically-explicit, solution, both for continuous and discrete input data, whose properties, with respect to the various nuisance parameter uncertainties, are analyzed, first in a mathematical, and subsequently in a numerical manner for noiseless data. It is found that: a) optimal inversion requires data registered at only a small number of sensors, b) the inverse solution, satisfying pre-existing physical constraints, exists and is unique. Moreover, the inverse solution is shown to be unstable with respect to three nuisance parameter uncertainties, the consequence of which is large retrieval inaccuracy for small nuisance parameter uncertainties, acting either individually or in combination.