Multivariate change estimation for a stochastic heat equation from local measurements

We study a stochastic heat equation with piecewise constant diffusivity ϑ having a jump at a hypersurface Γ that splits the underlying space [0,1]d, d≥2, into two disjoint sets Λ−∪Λ+. Based on multiple spatially localized measurement observations on a regular δ-grid of [0,1]d, we propose a joint M-estimator for the diffusivity values and the set Λ+ that is inspired by statistical image reconstruction methods. We study convergence of the domain estimator Λ̂+ in the vanishing resolution level regime δ→0 and with respect to the expected symmetric difference pseudometric. As a first main finding we give a characterization of the convergence rate for Λ̂+ in terms of the complexity of Γ measured by the number of intersecting hypercubes from the regular δ-grid. Furthermore, for the special case of domains Λ+ that are built from hypercubes from the δ-grid, we demonstrate that perfect identification with probability tending to one is possible with a slight modification of the estimation approach. Implications of our general results are discussed under two specific structural assumptions on Λ+. For a β-Hölder smooth boundary fragment Γ, the set Λ+ is estimated with rate δβ. If we assume Λ+ to be convex, we obtain a δ-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 Λ+ anchored on the δ-grid.