Comparison of regularized ensemble Kalman filter and tempered ensemble transform particle filter for an elliptic inverse problem with uncertain boundary conditions

Abstract

In this paper, we focus on parameter estimation for an elliptic inverse problem. We consider a 2D steady-state single-phase Darcy flow model, where permeability and boundary conditions are uncertain. Permeability is parameterized by the Karhunen-Loeve expansion and thus assumed to be Gaussian distributed. We employ two ensemble-based data assimilation methods: ensemble Kalman filter and ensemble transform particle filter. The formal one approximates mean and variance of a Gaussian probability function by means of an ensemble. The latter one transforms ensemble members to approximate any posterior probability function. Ensemble Kalman filter considered here is employed with regularization and localization—R(L)EnKF. Ensemble transform particle filter is also employed with a form of regularization called tempering and localization—T(L)ETPF. Regularization is required for highly non-linear problems, where prior is updated to posterior via a sequence of intermediate probability measures. Localization is required for small ensemble sizes to remove spurious correlations. We have shown that REnKF outperforms TETPF. We have shown that localization improves estimations of both REnKF and TETPF. In numerical experiments when uncertainty is only in permeability, TLETPF outperforms RLEnKF. When uncertainty is both in permeability and in boundary conditions, TLETPF outperforms RLEnKF only for a large ensemble size 1000. Furthermore, when uncertainty is both in permeability and in boundary conditions but we do not account for error in boundary conditions in data assimilation, RLEnKF outperforms TLETPF.