Multivariate change estimation for a stochastic heat equation from local measurements

Abstract: We study a stochastic heat equation with piecewise constant diffusivity $\theta$ having a jump at a hypersurface $\Gamma$ that splits the underlying space $[0,1]^d$, $d\geq2,$ into two disjoint sets $\Lambda_-\cup\Lambda_+.$ Based on multiple spatially localized measurement observations on a regular $\delta$-grid of $[0,1]^d$, we propose a joint M-estimator for the diffusivity values and the set $\Lambda_+$ that is inspired by statistical image reconstruction methods. We study convergence of the domain estimator $\hat{\Lambda}+$ in the vanishing resolution level regime $\delta \to 0$ and with respect to the expected symmetric difference pseudometric. As a first main finding we give a characterization of the convergence rate for $\hat{\Lambda}+$ in terms of the complexity of $\Gamma$ measured by the number of intersecting hypercubes from the regular $\delta$-grid. Furthermore, for the special case of domains $\Lambda_+$ that are built from hypercubes from the $\delta$-grid, we demonstrate that perfect identification with overwhelming probability is possible with a slight modification of the estimation approach. Implications of our general results are discussed under two specific structural assumptions on $\Lambda_+$. For a $\beta$-H\"older smooth boundary fragment $\Gamma$, the set $\Lambda_+$ is estimated with rate $\delta^\beta$. If we assume $\Lambda_+$ to be convex, we obtain a $\delta$-rate. While our approach only aims at optimal domain estimation rates, we also demonstrate consistency of our diffusivity estimators, which is strengthened to a CLT at minimax optimal rate for sets $\Lambda_+$ anchored on the $\delta$-grid.