Fast hybrid tempered ensemble transform filter formulation for Bayesian elliptical problems via Sinkhorn approximation

Identification of unknown parameters on the basis of partial and noisy data is a challenging task, in particular in high dimensional and non-linear settings. Gaussian approximations to the problem, such as ensemble Kalman inversion, tend to be robust and computationally cheap and often produce astonishingly accurate estimations despite the simplifying underlying assumptions. Yet there is a lot of room for improvement, specifically regarding a correct approximation of a non-Gaussian posterior distribution. The tempered ensemble transform particle filter is an adaptive Sequential Monte Carlo (SMC) method, whereby resampling is based on optimal transport mapping. Unlike ensemble Kalman inversion, it does not require any assumptions regarding the posterior distribution and hence has shown to provide promising results for non-linear non-Gaussian inverse problems. However, the improved accuracy comes with the price of much higher computational complexity, and the method is not as robust as ensemble Kalman inversion in high dimensional problems. In this work, we add an entropy-inspired regularisation factor to the underlying optimal transport problem that allows the high computational cost to be considerably reduced via Sinkhorn iterations. Further, the robustness of the method is increased via an ensemble Kalman inversion proposal step before each update of the samples, which is also referred to as a hybrid approach. The promising performance of the introduced method is numerically verified by testing it on a steady-state single-phase Darcy flow model with two different permeability configurations. The results are compared to the output of ensemble Kalman inversion, and Markov chain Monte Carlo methods results are computed as a benchmark.