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.

翻译：我们考虑由偏微分方程和随机偏微分方程支配的动力系统在数据同化中出现的贝叶斯反问题。时空相关场与先验和似然密度的静态参数被联合推断。特别关注控制先验平滑性和正则性的超参数，这对确保适定性、塑造后验结构以及确定预测不确定性至关重要。通常该参数被假定为已知并先验固定；然而，本文将采用分层贝叶斯框架，其中平滑性及其他超参数被视为未知并分配超先验。后验推断采用适用于高维度的Metropolis-within-Gibbs采样方法，该方法中超参数估计的计算开销极小。该方法的有效性通过Navier-Stokes方程和随机对流-扩散方程的反问题得以验证，在稀疏与密集观测模式下，使用不同协方差结构的高斯先验。数值结果表明，联合估计平滑性可显著降低因平滑性误设引起的不确定性量化与参数估计误差，其性能与已知真实平滑性的场景相当。