Geometric methods for anisotopic inverse boundary value problems.

Lionheart, William R.B. (2004) Geometric methods for anisotopic inverse boundary value problems. In: New Analytic and Geometric Methods in Inverse Problems Lectures given at the EMS Summer School and Conference held in Edinburgh, Scotland 2000. Springer, Berlin, pp. 337-351. ISBN 3540406824

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Abstract

Electromagnetic fields have a natural representation as differential forms. Typically the measurement of a field involves an integral over a submanifold of the domain. Differential forms arise as the natural objects to integrate over submanifolds of each dimension. We will see that the (possibly anisotropic) material response to a field can be naturally associated with a Hodge star operator. This geometric point of view is now well established in computational electromagnetism, particularly by Kotiuga, and by Bossavit and and others. The essential point is that Maxwell’s equations can be formulated in a context independent of the ambient Euclidean metric. This approach has theoretical elegance and leads to simplicity of computation. In this paper we will review the geometric formulation of the (scalar) anisotropic inverse conductivity problem, amplifying some of the geometric points made in Uhlmann’s paper in this volume. We will go on to consider generalizations of this anisotropic inverse boundary value problem to systems of Partial Differential Equation, including the result of Joshi and the author on the inverse boundary value problem for harmonic k-forms.

Item Type:	Book Section
Uncontrolled Keywords:	Anisotropy, Maxwell's equations, electromagnetics, inverse problem
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations MSC 2010, the AMS's Mathematics Subject Classification > 53 Differential geometry MSC 2010, the AMS's Mathematics Subject Classification > 78 Optics, electromagnetic theory
Depositing User:	Prof WRB Lionheart
Date Deposited:	28 Nov 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/659

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