# On statistical Calderón problems

### Kweku Abraham

Université Paris-Saclay, Orsay Cedex, France### Richard Nickl

University of Cambridge, UK

## Abstract

For $D$ a bounded domain in $R_{d},d≥2,$ with smooth boundary $∂D$, the non-linear inverse problem of recovering the unknown conductivity $γ$ determining solutions $u=u_{γ,f}$ of the partial differential equation

from noisy observations $Y$ of the Dirichlet-to-Neumann map $f↦Λ_{γ}(f)=γ∂ν∂u_{γ,f} ∣∣ _{∂D}$, with $∂/∂ν$ denoting the outward normal derivative, is considered. The data $Y$ consists of $Λ_{γ}$ corrupted by additive Gaussian noise at noise level $ε>0$, and a statistical algorithm $γ (Y)$ is constructed which is shown to recover $γ$ in supremum-norm loss at a statistical convergence rate of the order $g(1/ε)_{−δ}$ as $ε→0$. It is further shown that this convergence rate is optimal, up to the precise value of the exponent $δ>0$, in an information theoretic sense. The estimator $γ (Y)$ has a Bayesian interpretation in terms of the posterior mean of a suitable Gaussian process prior and can be computed by MCMC methods.

## Cite this article

Kweku Abraham, Richard Nickl, On statistical Calderón problems. Math. Stat. Learn. 2 (2019), no. 2, pp. 165–216

DOI 10.4171/MSL/14