2024/09/23 by Anton Tiepner, Tiepner, Anton, Lukas Trottner +1
Engineering · Mathematics · #60H15 #62G05 #62H11 #62M05 #Advanced Control Systems Optimization #FOS: Mathematics #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2409.15059
openalex publication_date 2024/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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 Λ+ 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 Λ+ 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 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 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.