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：我们考虑数据同化中由偏微分方程和随机偏微分方程所支配的动力系统所引发的贝叶斯反问题。时空依赖场与先验密度及似然密度的静态参数被联合推断。特别强调控制先验光滑性与正则性的超参数，这对于确保适定性、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且先验固定；然而，本文采用分层贝叶斯框架，将光滑性及其他超参数视为未知并赋予超先验。后验推断采用适用于高维情形的Metropolis-within-Gibbs抽样方法，其中超参数估计仅带来极小的计算开销。该方法在稀疏与密集观测机制下，针对Navier-Stokes方程和随机平流-扩散方程的反问题进行了演示，使用了具有不同协方差结构的高斯先验。数值结果表明，联合估计光滑性参数可显著降低由光滑性误设引起的不确定性量化与参数估计误差，其性能达到与真实光滑性已知情形相当的水平。