On the Uniqueness of a Nonlinear Discrete Calderón Problem

Abstract

The discrete Calderón problem aims at recovering the conductivity on the edges of a graph from boundary measurements, encoded by the discrete Dirichlet-to-Neumann (DtN) map. The problem has been intensively studied in the linear case on square lattices since the seminal works of Curtis and Morrow. In this work, we investigate a nonlinear analogue of the discrete Calderón problem for a semilinear second-order elliptic equation on square lattices. We establish three uniqueness results for the conductivity recovery. First, we show that conductivity-dependent corner excitations allow a layer-by-layer reconstruction of the conductivity. Second, we study the linearization of the nonlinear DtN map at an arbitrary background boundary datum and prove that the conductivity and the background potential are uniquely determined by one pair of nonlinear Cauchy data and the linearized DtN map. Third, we show that the linearized data can be replaced by finitely many nonlinear measurements, which uniquely determine the conductivity.

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BibTeX RIS

BibTeX

@misc{beretta2026uniqueness,
  title = {{On the Uniqueness of a Nonlinear Discrete Calderón Problem}},
  author = {Elena Beretta and Maolin Deng and Alberto Gandolfi and Bangti Jin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15175},
  eprint = {2609.15175},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15175},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - On the Uniqueness of a Nonlinear Discrete Calderón Problem
AU  - Elena Beretta
AU  - Maolin Deng
AU  - Alberto Gandolfi
AU  - Bangti Jin
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15175
UR  - https://arxiv.org/abs/2609.15175
AB  - The discrete Calderón problem aims at recovering the conductivity on the edges of a graph from boundary measurements, encoded by the discrete Dirichlet-to-Neumann (DtN) map. The problem has been intensively studied in the linear case on square lattices since the seminal works of Curtis and Morrow. In this work, we investigate a nonlinear analogue of the discrete Calderón problem for a semilinear second-order elliptic equation on square lattices. We establish three uniqueness results for the conductivity recovery. First, we show that conductivity-dependent corner excitations allow a layer-by-layer reconstruction of the conductivity. Second, we study the linearization of the nonlinear DtN map at an arbitrary background boundary datum and prove that the conductivity and the background potential are uniquely determined by one pair of nonlinear Cauchy data and the linearized DtN map. Third, we show that the linearized data can be replaced by finitely many nonlinear measurements, which uniquely determine the conductivity.
ER  -

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