具有有限个随机测量的解析逆问题
Analytic inverse problems with finitely many random measurements
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中文总结 AI 辅助
该研究针对严重不适定逆问题，证明随机采样可将所需测量数从随d指数增长降至2d+1，并将结果应用于Calderón问题和逆介质散射问题。
中文摘要 AI 辅助
传统上对无限维逆问题的分析基于连续数据，而实际应用依赖有限个离散测量。近期确定性方法表明，属于d维模型类的未知量可从有限个测量中稳定恢复，但对于严重不适定问题（如Calderón问题和逆散射），已知构造可能需要数量随d指数增长的测量。本文证明，若仅要求精确可识别性，随机采样可大幅减少所需测量数量。通过利用前向映射的解析几何，本文证明：只要无限数据问题在模型类上是单射的，几乎必然地，2d+1个随机标量测量即可唯一确定未知量。本文将结果应用于Calderón问题（含无限维和有限维边界采样），以及基于随机采样远场值的逆介质散射问题。
英文摘要
While infinite-dimensional inverse problems are traditionally analyzed assuming continuous data, practical applications rely on finitely many discrete measurements. Recent deterministic approaches establish that unknowns belonging to a $d$-dimensional model class can be stably recovered from finitely many measurements. However, for severely ill-posed problems, such as the Calderón problem and inverse scattering, the known constructions may require a number of measurements that is exponential in $d$. We show that random sampling reduces this count dramatically if one asks only for exact identifiability. By exploiting the analytic geometry of the forward maps, we prove that, whenever the infinite-data problem is injective on the model class, $2d+1$ random scalar measurements determine the unknown uniquely, almost surely. Applications are given to the Calderón problem, with both infinite- and finite-dimensional boundary sampling, and to inverse medium scattering from randomly sampled far-field values.
发表机构
- University of Genoa（热那亚大学）
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