Minimax estimation of linear functionals in the Darcy inverse problem

@misc{liu2026minimax,
  title = {{Minimax estimation of linear functionals in the Darcy inverse problem}},
  author = {Yi Liu and Xiaotian Chang and Qirui Hu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01290},
  eprint = {2611.01290},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01290},
}

TY  - UNPB
TI  - Minimax estimation of linear functionals in the Darcy inverse problem
AU  - Yi Liu
AU  - Xiaotian Chang
AU  - Qirui Hu
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01290
UR  - https://arxiv.org/abs/2611.01290
AB  - We study estimation of linear functionals ∫ fψ of the conductivity f in the Darcy problem ∇·(f∇ u)=g from observations of u in Gaussian white noise of level N^−1/2. When the potential has a single nondegenerate minimum, we determine the minimax rates over Sobolev balls in the linearised experiment. Unless a compatibility condition along the gradient flow holds, the rate is governed by an effective dimension T of the Hessian at the minimum, not by the ambient dimension: it is N^−(2s+T)/(4s+8) for T<4, has an elbow at T=4 with an extra logarithmic factor, and is parametric for T>4. In the nonlinear experiment we show that these lower bounds persist when compatibility fails at the reference and T≤4, and we prove the minimax rate on classes with isotropic critical Hessians, and the parametric rate near references with T>4. In dimension at most four, no nonzero linear functional is uniformly √N-estimable on a Sobolev neighbourhood of the constant conductivity, even when it is regular there.
ER  -