Nonparametric Diffusivity Estimation for the Stochastic Heat Equation from Noisy Observations

TL;DR

This paper introduces a nonparametric method to estimate spatially varying diffusivity in a stochastic heat equation from noisy data, using a localization and regression approach, supported by theoretical analysis and numerical simulations.

Contribution

It develops a novel two-step localization and regression method for diffusivity estimation in noisy stochastic heat equations, with new theoretical approximation results.

Findings

Effective diffusivity estimates from noisy data

Theoretical analysis of non-standard scaling behavior

Numerical simulations demonstrating method performance

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.