Inverse Boundary Value Problems

In the field of inverse boundary value problems, the connections between research in pure mathematics and practical applications are well highlighted. Usually, the problems which are studied are related to non-invasive testing methods: one wants to determine interior properties of a body by making measurements on the boundary. Typical situations where such problems arise are found in

medical imaging, where one would like to locate possible anomalies within the human body,
geological prospecting, for instance in the search of underground oil fields or mineral deposits, and
nondestructive testing of aircraft parts.

In these situations one uses a probing method whose behaviour is controlled by a partial differential equation, and the interior properties of the medium are determined by the coefficients of the equation. The physical measurements give information on the solutions of the equation with given boundary values, and from this information one should recover the coefficients. A standard example of inverse boundary value problems is the inverse conductivity problem. This is the mathematical basis for Electrical Impedance Tomography (EIT), which is a medical imaging method where one places electrodes around part of the body, applies a voltage to the electrodes, and measures the outcoming current. If $\Omega$ is the part of the body which is imaged and $\sigma$ is the electrical conductivity of the body, then given a voltage

on $\partial \Omega$ the induced potential

inside the body solves the Dirichlet problem

$\displaystyle \nabla \cdot (\sigma \nabla u)$	$\displaystyle = 0$ in $\Omega$ $\displaystyle ,$
$\displaystyle u$	$\displaystyle = f$ on $\partial \Omega$ $\displaystyle .$

The inverse conductivity problem is to determine the coefficient $\sigma$ in $\Omega$ from the boundary measurements, which are given by the voltage to current map $\Lambda_{\sigma}: f \mapsto \sigma \frac{\partial u}{\partial \nu} \big\vert _{\partial \Omega}$ . The RNI Inverse Problems group has worked on the theoretical aspects of several inverse boundary value problems. For the inverse conductivity problem, results have been obtained for nonsmooth conductivities (a case of practical importance) and anisotropic conductivities, where geometry comes into play and the natural setting is a Riemannian manifold. Maxwell's equations have been studied and uniqueness results for related inverse problems have been obtained. Inverse boundary value problems arising in scattering have been an important part of the research. The group has used a blend of methods from partial differential equations, harmonic analysis, complex analysis, and differential geometry. A major result was obtained in 2003 by Kari Astala and Lassi Päivärinta when they solved the original conjencture of Alberto Calderón: for the inverse conductivity problem in two dimensions, any bounded and measurable conductivity is uniquely determined by the boundary measurements. This work introduced quasiconformal techniques in inverse problems.

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