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Uniqueness for tv-regularized inverse problems with source in divergence form
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- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2610.11719
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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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