Adaptive finite element methods for the solution of inverse problems in optical tomography

  • Bangerth W
  • Joshi A
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Optical tomography attempts to determine a spatially variable coefficient
in the interior of a body from measurements of light fluxes at the
boundary. Like in many other applications in biomedical imaging,
computing solutions in optical tomography is complicated by the fact
that one wants to identify an unknown number of relatively small
irregularities in this coefficient at unknown locations, for example
corresponding to the presence of tumors. To recover them at the resolution
needed in clinical practice, one has to use meshes that, if uniformly
fine, would lead to intractably large problems with hundreds of millions
of unknowns. Adaptive meshes are therefore an indispensable tool.
In this paper, we will describe a framework for the adaptive finite
element solution of optical tomography problems. It takes into account
all steps starting from the formulation of the problem including
constraints on the coefficient, outer Newton-type nonlinear and inner
linear iterations, regularization, and in particular the interplay
of these algorithms with discretizing the problem on a sequence of
adaptively refined meshes. We will demonstrate the efficiency and
accuracy of these algorithms on a set of numerical examples of clinical
relevance related to locating lymph nodes in tumor diagnosis.

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  • Wolfgang Bangerth

  • Amit Joshi

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