We study Bayesian data assimilation (filtering) for time-evolution PDEs, for which the underlying forward problem may be very unstable or ill-posed. Such PDEs, which include the Navier-Stokes equations of fluid dynamics, are characterized by a high sensitivity of solutions to perturbations of the initial data, a lack of rigorous global well-posedness results as well as possible non-convergence of numerical approximations. Under very mild and readily verifiable general hypotheses on the forward solution operator of such PDEs, we prove that the posterior measure expressing the solution of the Bayesian filtering problem is stable with respect to perturbations of the noisy measurements, and we provide quantitative estimates on the convergence of approximate Bayesian filtering distributions computed from numerical approximations. For the Navier-Stokes equations, our results imply uniform stability of the filtering problem even at arbitrarily small viscosity, when the underlying forward problem may become ill-posed, as well as the compactness of numerical approximants in a suitable metric on time-parametrized probability measures.
翻译:我们研究的是时间进化的PDE(Bayesian data同化(过滤),其潜在的前方问题可能非常不稳定或弊端。这种PDE(包括流体动态的纳维-斯托克方程式)的特征是,初步数据扰动的解决方案高度敏感,缺乏严格的全球井喷结果,以及数字近似可能无法兼容。在这种PDE(PDE)前方解决方案操作员的非常温和和和容易核查的一般假设下,我们证明表明,表明贝耶斯过滤问题解决办法的后方措施与噪音测量的扰动有关,是稳定的,我们提供了从数字近似近似波斯过滤分布的定量估计。对于Navier-Stokes方程式而言,我们的结果意味着过滤问题的统一性即使在任意的小型粘合度上,当潜在的前方问题可能变得不正确时,过滤问题在适当的时间折合率计量尺度上的数字吸附剂的紧凑性。