Convex source support in three dimensions

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 Communicated by Lothar Reichel.

Optimal Depth-Dependent Distinguishability Bounds for Electrical Impedance Tomography in Arbitrary Dimension

“…It is worth noting that the formulas (2.13) and (2.14) below generalize the two-dimensional result in [18, Proposition 2.1]; see Appendix A. Moreover, an equivalent characterization for three spatial dimensions can be found in [20].…”

“…for any u ∈ D (Ω), all φ ∈ C ∞ c (Ω * ), and with ·, · denoting the dual pairing of (D , C ∞ c ). The definition (2.8) coincides with that of the distributional Kelvin transformation in [20], and it can be motivated as follows: for any u ∈ L 1 loc (Ω) and φ ∈ C ∞ c (Ω * ), we have (Ω). The same conclusion also applies to (2.9).…”

“…However, we need to consider inversions with respect to arbitrary spheres in our analysis, leading to the employment of translated and dilated versions of the classic Kelvin transformation (cf. [8,20]). A central property of all Kelvin transformations, relating them to EIT and also explaining their use in potential theory, is that a function is harmonic if and only if its Kelvin transformation is harmonic.…”

Sweep data of electrical impedance tomography

“…Because the reconstruction method is essentially the same as the one presented in [12] for standard EIT data and subsequently in [14] for the backscatter data of EIT, we will skip many of the details and only outline the main ideas. Note also that the algorithm itself generalizes to a three-dimensional setting [9]; the restriction on the dimension of D is due to the (complex) analysis of sections 3 and 4.…”

Box constraints and weighted sparsity regularization for identifying sources in elliptic PDEs

“…In [18] a more general case is considered for the Helmholtz equation, and the author of that paper shows that it is possible to recover the support function of a polygonal shaped support, see also [19]. Furthermore, during the last three decades very advanced investigations have been presented to further clarify the possibilities for computing the support of sources (or scattering objects) from boundary (or far field) data, see, e.g., [11,12,13,14,15,16,21,23,24,26]. Several of these papers address the important task of computing the so-called convex source (or scattering) support.…”