Nonparametric Diffusivity Estimation for the Stochastic Heat Equation from Noisy Observations

TL;DR

This work addresses nonparametric estimation of a spatially varying diffusivity $\vartheta$ in the stochastic heat equation driven by space-time white noise, using noisy observations. It introduces a two-step localization strategy that first estimates the latent state $X$ via local averaging and then performs a locally linear regression to recover $\vartheta(x_0)$, accounting for static measurement noise $\varepsilon$. A central contribution is the combination of a regression-based estimator with an instrumental-variable construction, supported by a rigorous uniform Trotter--Kato semigroup approximation to control the impact of localized domain inflation on the heat semigroup; this yields dimension-dependent rates: a parametric rate $O_p(\varepsilon^{1/2+d/4})$ in high dimensions and Lipschitz-type rates in lower dimensions, with a general nonparametric rate $O_p(\varepsilon^{(2+d)\gamma/(4\gamma+2d)})$ for Hölder regularity $\gamma$. The theory is complemented by numerical experiments illustrating the estimator’s accuracy and the nonstandard scaling induced by static noise, highlighting practical viability for diffusivity imaging in SPDEs.

Abstract

We estimate nonparametrically the spatially varying diffusivity of a stochastic heat equation from observations perturbed by additional noise. To that end, we employ a two-step localization procedure, more precisely, we combine local state estimates into a locally linear regression approach. Our analysis relies on quantitative Trotter--Kato type approximation results for the heat semigroup that are of independent interest. The presence of observational noise leads to non-standard scaling behaviour of the model. Numerical simulations illustrate the results.

Figures (2)

• Figure 1: Left: Realization of a stochastic heat equation with spatially heterogeneous diffusivity. Center: Same trajectory, but with additional static noise ($\varepsilon=0.16$) that is locally averaged for better visibility. Right: Smoothed trajectory with static noise, obtained by testing with $(K_{k, x})_{\varepsilon, x_0}$ for $x_0=0.5$ and $\varepsilon=0.0016$, i.e. $\delta=0.04$.
• Figure 2: Left: Reconstruction of a spatially varying diffusivity at different points in space. Right: RMSE vs. $\varepsilon$, formal comparison of the convergence rate of the Lipschitz (${\gamma}=1$) and the parametric (${\gamma}=\infty$) estimation problem.

Theorems & Definitions (90)

• Lemma 2.1: well-posedness
• proof
• Remark 3.1
• Lemma 3.2: localization of the semigroup
• proof
• Lemma 3.3: localization of the signal
• proof
• Remark 3.4
• Lemma 3.5: localization of the observation process
• proof