Sharp minimax rates for a Darcy conductivity
Let $D$ be the open unit ball of $\mathbb{R}^d$, $d \ge 2$, fix an integer $\alpha > \max\{d/2 + 2, 2d - 1\}$, and fix $0 < r < 1$ and $R > \|1\|_{H^\alpha(D)}$. Let $$\mathcal{F}_{\alpha,r}(R) = \left\{f \in H^\alpha(D) : \inf_D f > r,\ \|f\|_{H^\alpha} \le R,\ f|_{\partial D} = 1,\ \partial_n^j f|_{\partial D} = 0 \ (1 \le j \le \alpha - 1)\right\},$$ with boundary derivatives understood as Sobolev traces. For $f$ in this class let $u_f$ solve $\nabla \cdot (f \nabla u_f) = 1$ in $D$ with $u_f = 0$ on $\partial D$, and observe the Gaussian white-noise experiment $$Y_\varepsilon(h) = \langle u_f, h \rangle_{L^2(D)} + \varepsilon W(h), \qquad h \in L^2(D),$$ where $W$ is isonormal Gaussian white noise and $\varepsilon \downarrow 0$. Put $$\mathcal{R}_\varepsilon = \inf_{\widehat f} \sup_{f \in \mathcal{F}_{\alpha,r}(R)} \mathbb{E}_f \|\widehat f(Y_\varepsilon) - f\|_{L^2(D)},$$ the infimum ranging over measurable $L^2(D)$-valued estimators. Nickl, van de Geer and Wang proved $\mathcal{R}_\varepsilon \lesssim \varepsilon^{2(\alpha - 1)/(2\alpha + 2 + d)}$ with penalised least squares estimators, and that the same power of $\varepsilon$ is a minimax lower bound for estimating $u_f$ in $H^2$ loss.
Problem. Determine the sharp asymptotic order of $\mathcal{R}_\varepsilon$ as $\varepsilon \downarrow 0$, including any logarithmic factors, with matching upper and lower bounds up to constants depending on the fixed class parameters.
Source: Stated by AIM — Open Applied Problems, after Nickl, van de Geer and Wang (2020), §3.2, who leave the optimal rates for estimating the conductivity to later work; see also Giordano and Nickl (2020), §2.3.2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- ST — Statistics Theory
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, minimax rates, statistical inverse problems
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{nickl2020,
author = {Nickl, Richard and van de Geer, Sara and Wang, Sven},
title = {Convergence rates for penalized least squares estimators in {PDE} constrained regression problems},
journal = {SIAM/ASA J. Uncertain. Quantif.},
volume = {8},
number = {1},
year = {2020},
pages = {374--413},
doi = {10.1137/18M1236137},
eprint = {1809.08818},
archivePrefix = {arXiv},
}
@article{giordano2020,
author = {Giordano, Matteo and Nickl, Richard},
title = {Consistency of {B}ayesian inference with {G}aussian process priors in an elliptic inverse problem},
journal = {Inverse Problems},
volume = {36},
number = {8},
year = {2020},
pages = {085001},
doi = {10.1088/1361-6420/ab7d2a},
eprint = {1910.07343},
archivePrefix = {arXiv},
}
@misc{aim-sharp-minimax-rates-for-a-darcy-conductivity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp minimax rates for a Darcy conductivity}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/666-sharp-minimax-rates-for-a-darcy-conductivity.md},
note = {Entry 666 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01125, "Sharp minimax rates for a Darcy conductivity", clopen.solutions.