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 singlephase 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.

Adaptive regularisation for ensemble Kalman inversion

“…According to the theory in [27], α n must be carefully selected, together with the stopping criteria, in order to ensure the stability of the LM scheme. The approach for selecting α n in the LM proposed in [27] has been adapted to the EKI framework in [2,16], and subsequently used in [3,12,14,[28][29][30]. As we discuss in the next section, this approach relies on tuning parameters that, unless carefully chosen, can lead to unnecessary large number of iterations n * .…”

“…The numerical results of [2] showed that EKI-LM enabled stability and accuracy for sufficiently large ensembles. Further work that has explored EKI-LM can be found in [3,28,29] as well as some practical applications including seismic tomography [14], modelling of intracraneal pressure [12] and time fractional diffusion inverse problems [30].…”

“…We use Jensen's inequality, formula (B.3) as well as (B.2) to obtain For any bounded t 0, the normalisation constant N t defined in formula(29), as a function of t, is differentiable and its derivative dN t dt is given by…”

Developments of inverse analysis by Kalman filters and Bayesian methods applied to geotechnical engineering

“…Within the framework of nonlinear Kalman filters, to overcome several limitations of the linear Kalman filter, the values of the initial or boundary conditions can be incorporated into the particles or members. Dubinkina and Ruchi (2020), 33 ) Wen et al (2020), 34 ) and Nakamura et al (2021) 35 ) have dealt with inverse analyses using nonlinear Kalman filters for which both the material parameters and boundary conditions are uncertain. Hence, the third category of the above classification corresponds to the prototype of the present DA technique.…”

“…Within the framework of nonlinear Kalman filters, to overcome several limitations of the linear Kalman filter, the values of the initial or boundary conditions can be incorporated into the particles or members. Dubinkina and Ruchi (2020), 33) Wen et al…”