Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters

Abstract

Abstract. We consider electromagnetic interrogation problems for complex materials involving distributions of polarization mechanisms and also distributions for the parameters in these mechanisms. A theoretical and computational framework for such problems is given. Computational results for specific problems with multiple Debye mechanisms are given in the case of discrete, uniform, log-normal, and log-bi-Gaussian distributions. Introduction.For at least the past century [52,53], scientific investigators have sought to understand what happens to electromagnetic fields (and how to mathematically model the associated phenomena) when they are introduced into complex materials such as biotissue and more general dielectrics, conductors and magnetics. More specifically, a fundamental question is how to model dispersion and dissipation of the fields in these complex materials. This has most often led to the use of Maxwell's equations in a nonvacuum environment which entails constitutive relationships for polarization (in dielectrics), magnetization (in magnetic materials) and conductivity. We focus here on modeling polarization in dielectric materials for which we develop a new modeling framework. Even though we treat only polarization as our dispersive mechanism in our formulation (adopting Ohm's law for conductivity and considering nonmagnetic materials), the approach is sufficiently general so as to be readily extended to treat magnetization and conductivity in materials (each in some type of convolution representation involving susceptibility kernels, e.g., see [2,3]). We develop a framework that allows not only uncertainty (through

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