散度形式源的TV正则化逆问题的唯一性
Uniqueness for TV-regularized inverse problems with source in divergence form
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中文总结 AI 辅助
本文针对散度形式的逆源问题，研究以总变差正则化的最小二乘逆问题，证明当支撑集S为细长集时该问题存在唯一极小元，其证明基于Smirnov散度自由测度分解的改进。
中文摘要 AI 辅助
散度形式的逆源问题旨在寻找具有指定支撑集S的向量场，其散度为某个观测势的拉普拉斯算子。本文假设未知向量场为向量值测度，研究对应的最小二乘逆问题，通过惩罚总变差进行正则化，既不离散化准则也不离散化未知量。我们证明当S为细长集（即其勒贝格测度为零，且其补集的每个连通分支均具有无限勒贝格测度）时，该问题存在唯一极小元。证明依赖于Smirnov散度自由测度分解的改进，该改进具有独立研究价值。
英文摘要
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.
发表机构
- INRIA（法国国家信息与自动化研究所）
- Vanderbilt University（范德堡大学）
- University of Vienna（维也纳大学）
机构由 AI 辅助整理，请以论文原文为准。
