We consider Bayesian data assimilation for time-evolution PDEs, for which the underlying forward problem may be very unstable or illposed. We formulate assumptions on the forward solution operator of such PDEs under which stability of the posterior measure with respect to perturbations of the noisy measurements can be proved. We also 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 illposed, as well as the compactness of numerical approximations in a suitable metric on time-parametrized probability measures.
This is a joint work with Samuel Lanthaler and Siddhartha Mishra.