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Uniqueness for tv-regularized inverse problems with source in divergence form

  • L Baratchart
  • D P Hardin
  • C Villalobos-Guillén
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Abstract

Inverse source problems in divergence form consist in finding a vector field with prescribed support S, whose divergence is the Laplacian of some observed potential. In this paper, we assume the unknown vector field is a vector-valued measure, and we study the corresponding least square inversion problems, regularized by penalizing the total variation, without discretizing the criterion nor the unknown. We prove that this problem has a unique minimizer in the case where S is a slender set; i.e., it has zero Lebesgue measure and each connected component of its complement has infinite Lebesgue measure. The proof relies on a refinement of Smirnov's decomposition of divergence-free measures [62] which is of independent interest.

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Cite

BibTeXRIS
BibTeX
@misc{baratchart2026uniqueness,
  title = {{Uniqueness for tv-regularized inverse problems with source in divergence form}},
  author = {L Baratchart and D P Hardin and C Villalobos-Guillén},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11719},
  eprint = {2610.11719},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11719},
}
RIS
TY  - UNPB
TI  - Uniqueness for tv-regularized inverse problems with source in divergence form
AU  - L Baratchart
AU  - D P Hardin
AU  - C Villalobos-Guillén
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.11719
UR  - https://arxiv.org/abs/2610.11719
AB  - Inverse source problems in divergence form consist in finding a vector field with prescribed support S, whose divergence is the Laplacian of some observed potential. In this paper, we assume the unknown vector field is a vector-valued measure, and we study the corresponding least square inversion problems, regularized by penalizing the total variation, without discretizing the criterion nor the unknown. We prove that this problem has a unique minimizer in the case where S is a slender set; i.e., it has zero Lebesgue measure and each connected component of its complement has infinite Lebesgue measure. The proof relies on a refinement of Smirnov's decomposition of divergence-free measures [62] which is of independent interest.
ER  -