Polarization and Moments Tensors with Applications to Inverse Problems and Effective Medium Theory

This book addresses significant recent developments in mathematical analysis and computational methods in impedance imaging and the theory of composite materials. It grows out of a desire to foster communication between these exciting fields. The mathematical problems that appear in these fields are of practical importance and pose high challenges to pure and applied mathematicians. They have attracted a lot of attention of researchers lately. The methods involved come from a wide range of areas of pure and applied mathematics ranging from potential theory to PDEs, to complex analysis, to numerical methods. The unifying thread in this book is the use of generalized polarization and moments tensors that depend only on the geometry and the conductivity or the Lamé parameters of the inclusion. The main approach is based on modern layer potential techniques. Electrical impedance imaging uses measurements of boundary voltage potentials and associated boundary currents to infer information about the internal conductivity profile of an object. Complete information about all voltages and currents is known to uniquely characterize an isotropic conductivity distribution. In its most general form electrical impedance imaging is severely ill-posed and nonlinear. These major and fundamental difficulties can be understood by means of a mean value type theorem in elliptic partial differential equations. The value of the voltage potential at each point inside the region can be expressed as a weighted average of its neighborhood potential where the weight is determined by the conductivity distribution. In this weighted averaging way, the conductivity distribution is conveyed to the boundary potential. Therefore, the boundary data is entangled in the global structure of the conductivity distribution in a highly nonlinear way. This is the main obstacle to finding non-iterative reconstruction algorithms with limited data. If, however, in advance we have additional structural information about the conductivity profile, then we may be able to determine specific features about the conductivity distribution with a satisfactory resolution. One such type of knowledge could be that the body consists of a smooth background containing a number of unknown small inclusions with a significantly different conductivity. The inclusions might in a medical