Division of Applied Mathematics
Department of Mathematics and Computer Science
The University of Akron
L. Borcea
Rice - Texas
Network approximations for low frequency electromagnetic transport in high contrast conductive and dielectric media.
by Liliana Borcea
Collaborators: James Berryman and George C. Papanicolaou
We consider inverse problems of imaging the electrical conductivity and permittivity of high contrast, isotropic, two-dimensional media. We start with the static (zero frequency) problem and show that the Neumann to Dirichlet map associated with imaging a large class of high contrast media is asymptotically equivalent to the Neumann to Dirichlet map of a resistor network. We present a constructive proof which shows how to define the asymptotic network and its Neumann to Dirichlet map, given arbitrary current excitations at the boundary of the high contrast continuum. Next, we consider low frequencies and we distinguish two types of problems: The ``complex conductivity'' problem that is relevant for dielectric media and the ``inductive problem'' that arises in good conductors. The complex conductivity problem is described by a system of elliptic equations, with high contrast coefficients, that is obtained from Maxwell's equations by neglecting the magnetic term
, where is the frequency, is the magnetic permeability and is the magnetic field. The inductive problem, on the other hand, neglects the displacement current in Maxwell's equations. For both problems, we show that, in the asymptotic limit of infinitely high contrast, electromagnetic transport can be accurately described in terms of currents through resistor-capacitor and resistor-capacitor-inductor networks. We show how to construct these networks for a large class of high contrast conductivities and permittivities. Furthermore, we show that in high contrast inversion, where the boundary data is always contaminated by noise, the imaging problem is basically equivalent to the determination of the asymptotic networks. Some discussion will be given about imaging static, resistor networks.
, where 
is the frequency, 
is the magnetic permeability and 
is the magnetic field. The inductive problem, on the other hand, neglects the displacement current in Maxwell's equations. For both problems, we show that, in the asymptotic limit of infinitely high contrast, electromagnetic transport can be accurately described in terms of currents through resistor-capacitor and resistor-capacitor-inductor networks. We show how to construct these networks for a large class of high contrast conductivities and permittivities. Furthermore, we show that in high contrast inversion, where the boundary data is always contaminated by noise, the imaging problem is basically equivalent to the determination of the asymptotic networks. Some discussion will be given about imaging static, resistor networks.