Smoothness and other hyperparameter estimation for inverse problems related to data assimilation

2026-02-20

AI Summary

This study addresses the bias in parameter estimation and inaccurate uncertainty quantification arising from misspecification of prior smoothness hyperparameters in data assimilation. Within a Bayesian inverse problem framework, the work treats the smoothness hyperparameter as an unknown quantity and constructs a hierarchical Bayesian model to jointly infer the field variables and the hyperparameter. It achieves, for the first time in high-dimensional data assimilation, adaptive estimation of the smoothness hyperparameter, thereby eliminating errors due to manual tuning. The approach is implemented via a Metropolis-within-Gibbs sampling algorithm and is applicable to PDE-based models such as Navier–Stokes and stochastic advection–diffusion equations. Numerical experiments demonstrate that, under both sparse and dense observational settings, the method substantially reduces estimation and uncertainty errors, achieving performance approaching that of an idealized scenario with known true smoothness.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We consider Bayesian inverse problems arising in data assimilation for dynamical systems governed by partial and stochastic partial differential equations. The space-time dependent field is inferred jointly with static parameters of the prior and likelihood densities. Particular emphasis is placed on the hyperparameter controlling the prior smoothness and regularity, which is critical in ensuring well-posedness, shaping posterior structure, and determining predictive uncertainty. Commonly it is assumed to be known and fixed a priori; however in this paper we will adopt a hierarchical Bayesian framework in which smoothness and other hyperparameters are treated as unknown and assigned hyperpriors. Posterior inference is performed using Metropolis-within-Gibbs sampling suitable to high dimensions, for which hyperparameter estimation involves little computational overhead. The methodology is demonstrated on inverse problems for the Navier-Stokes equations and the stochastic advection-diffusion equation, under sparse and dense observation regimes, using Gaussian priors with different covariance structure. Numerical results show that jointly estimating the smoothness substantially reduces the errors in uncertainty quantification and parameter estimation induced by smoothness misspecification, by achieving performance comparable to scenarios in which the true smoothness is known.

Problem

• smoothness
• hyperparameter estimation
• inverse problems
• data assimilation
• Bayesian inference

Innovation

• hierarchical Bayesian inference
• smoothness hyperparameter
• uncertainty quantification
• Metropolis-within-Gibbs sampling
• inverse problems