TY - JOUR
T1 - Convex source support in three dimensions
AU - Hanke, Martin
AU - Harhanen, Lauri
AU - Hyvönen, Nuutti
AU - Schweickert, Eva
PY - 2012/3
Y1 - 2012/3
N2 - 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.
AB - 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.
KW - Convex source support
KW - Electrical impedance tomography
KW - Inverse elliptic boundary value problem
KW - Obstacle problem
UR - http://www.scopus.com/inward/record.url?scp=84857685233&partnerID=8YFLogxK
U2 - 10.1007/s10543-011-0338-0
DO - 10.1007/s10543-011-0338-0
M3 - Article
VL - 52
SP - 45
EP - 63
JO - BIT - Numerical Mathematics
JF - BIT - Numerical Mathematics
SN - 0006-3835
IS - 1
ER -