Analytic inverse problems with finitely many random measurements

Giovanni S. Alberti; Damiano Poletti; Simone Sanna; Matteo Santacesaria

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arXiv · 2608.14324

Analytic inverse problems with finitely many random measurements

Giovanni S. Alberti·Damiano Poletti·Simone Sanna·Matteo Santacesaria

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Abstract

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.

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BibTeX RIS

Giovanni S. Alberti, Damiano Poletti, Simone Sanna, Matteo Santacesaria. 2026-08-14. Analytic inverse problems with finitely many random measurements. https://arxiv.org/abs/2608.14324

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