Non-parametric estimation of non-linear diffusion coefficient in parabolic SPDEs

2025/09/16 by Martin Andersson, Benny Avelin, Andersson, Martin +7
Engineering · Mathematics · #60H15 (Secondary) #62G05 (Primary) #FOS: Mathematics #Metallurgical Processes and Thermodynamics #Non-Destructive Testing Techniques #Numerical methods in inverse problems #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2509.12921

openalex publication_date 2025/09/16 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

In this article, we introduce a novel non-parametric predictor, based on conditional expectation, for the unknown diffusion coefficient function σ in the stochastic partial differential equation Lu = σ(u)W, where L is a parabolic second order differential operator and W is a suitable Gaussian noise. We prove consistency and derive an upper bound for the error in the L^p norm, in terms of discretization and smoothening parameters h and ε. We illustrate the applicability of the approach and the role of the parameters with several interesting numerical examples.

Citations

• Parameter estimation for second-order SPDEs in multiple space dimensions
• Optimal parameter estimation for linear SPDEs from multiple measurements
• Parameter estimation for linear parabolic SPDEs in two space dimensions based on high frequency data
• Parameter Estimation in an SPDE Model for Cell Repolarisation
• Statistical Inference for SPDEs: an overview
• A fully discrete approximation of the one-dimensional stochastic heat equation
• Volatility estimation for stochastic PDEs using high-frequency\n observations
• A note on parameter estimation for discretely sampled SPDEs
• Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions

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• Nonparametric estimation of the diffusion coefficient from S.D.E. paths
• Non-parametric estimation of Stochastic Differential Equations from stationary time-series
• Adaptive estimator for a parabolic linear SPDE with a small noise
• Parameter estimation for a linear parabolic SPDE model in two space dimensions with a small noise
• Non-parametric estimation of the reaction term in semi-linear SPDEs with spatial ergodicity