BIT Numerical Mathematics

, Volume 52, Issue 1, pp 45–63 | Cite as

Convex source support in three dimensions

  • Martin Hanke
  • Lauri Harhanen
  • Nuutti Hyvönen
  • Eva Schweickert
Article

Abstract

This work extends the algorithm for computing the convex source support in the framework of the Poisson equation to a bounded three-dimensional domain. The convex source support is, in essence, the smallest (nonempty) convex set that supports a source that produces the measured (nontrivial) data on the boundary of the object. In particular, it belongs to the convex hull of the support of any source that is compatible with the measurements. The original algorithm for reconstructing the convex source support is inherently two-dimensional as it utilizes Möbius transformations. However, replacing the Möbius transformations by inversions with respect to suitable spheres and introducing the corresponding Kelvin transforms, the basic ideas of the algorithm carry over to three spatial dimensions. The performance of the resulting numerical algorithm is analyzed both for the inverse source problem and for electrical impedance tomography with a single pair of boundary current and potential as the measurement data.

Keywords

Electrical impedance tomography Convex source support Obstacle problem Inverse elliptic boundary value problem 

Mathematics Subject Classification (2000)

65N21 35R30 

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Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  • Martin Hanke
    • 1
  • Lauri Harhanen
    • 2
  • Nuutti Hyvönen
    • 2
  • Eva Schweickert
    • 1
  1. 1.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany
  2. 2.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland

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