This talk was part of the Workshop on "PDE-constrained Bayesian inverse problems: interplay of spatial statistical models with advanced PDE discretizations" held at the ESI May 16 to 20, 2022.

We consider Bayesian inference for large scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. This renders most Markov chain Monte Carlo approaches infeasible, since they typically require O(10^4) model runs, or more. Moreover, the forward model is often given as a black box or is impractical to differentiate. Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems. The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system into which the inverse problem is embedded as an observation operator. Theoretical properties of the mean-field model are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it. This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior. Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach. The effectiveness of the framework is demonstrated in several numerical experiments, including proof-of-concept linear/nonlinear examples and two large-scale applications: learning of permeability parameters in subsurface flow; and learning subgridscale parameters in a global climate model from time-averaged statistics. Moreover, the stochastic ensemble Kalman filter, the unscented Kalman filter and the ensemble square-root Kalman filter are all employed and are compared numerically in these experiments. The results demonstrate that the proposed method, based on exponential convergence to the large-time limit of the filtering distribution of a mean-field dynamical system, is competitive with pre-existing Kalman-based methods for inverse problems, such as those using coupling/transport formulations. Furthermore the results demonstrate that the ensemble square-root and unscented Kalman filters are particularly effective when applied to solve the underlying filtering problem.

Joint work with Jiaoyang Huang, Sebastian Reich, and Andrew M Stuart

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