| Abstract: |
| The Calder\`on problem consists in recovering an unknown parameter of a partial differential equation from boundary measurements of its solution. While global uniqueness is well-established for the full Dirichlet-to-Neumann operator, corresponding to infinitely many boundary measurements, practical applications, such as EIT, operate with finite and discrete data.
In this realistic setting, the unknown parameter is assumed to lie in a finite-dimensional space, and the boundary data are restricted to a finite family of Dirichlet-Neumann pairs. This reduction leads to a fundamental question regarding sample complexity: what is the minimal number of measurements M required to reconstruct a d-dimensional unknown? Due to the non-linearity and severe ill-posedness of the problem, existing deterministic results often yield sample complexity estimates that are exponential in d.
In this work, we address this question using randomized boundary data. Instead of deterministically choosing a finite number of boundary data from a fixed orthonormal basis on the boundary of the domain, we take random linear combinations of such basis elements. We prove that this approach ensures almost sure uniqueness with a number of measurements M that is merely proportional to the dimension of the parameter space d, significantly improving upon current deterministic bounds. |
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