Analytic inverse problems with finitely many random measurements
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
@misc{alberti2026analytic,
title = {{Analytic inverse problems with finitely many random measurements}},
author = {Giovanni S. Alberti and Damiano Poletti and Simone Sanna and Matteo Santacesaria},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.14324},
eprint = {2608.14324},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14324},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Analytic inverse problems with finitely many random measurements
AU - Giovanni S. Alberti
AU - Damiano Poletti
AU - Simone Sanna
AU - Matteo Santacesaria
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.14324
UR - https://arxiv.org/abs/2608.14324
AB - 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.
ER -