A Bayesian Consistent Dual Ensemble Kalman Filter for State-Parameter Estimation in Subsurface Hydrology

Ensemble Kalman filtering (EnKF) is an efficient approach to addressing uncertainties in subsurface groundwater models. The EnKF sequentially integrates field data into simulation models to obtain a better characterization of the model's state and parameters. These are generally estimated following joint and dual filtering strategies, in which, at each assimilation cycle, a forecast step by the model is followed by an update step with incoming observations. The Joint-EnKF directly updates the augmented state-parameter vector while the Dual-EnKF employs two separate filters, first estimating the parameters and then estimating the state based on the updated parameters. In this paper, we reverse the order of the forecast-update steps following the one-step-ahead (OSA) smoothing formulation of the Bayesian filtering problem, based on which we propose a new dual EnKF scheme, the Dual-EnKF$_{\rm OSA}$. Compared to the Dual-EnKF, this introduces a new update step to the state in a fully consistent Bayesian framework, which is shown to enhance the performance of the dual filtering approach without any significant increase in the computational cost. Numerical experiments are conducted with a two-dimensional synthetic groundwater aquifer model to assess the performance and robustness of the proposed Dual-EnKF$_{\rm OSA}$, and to evaluate its results against those of the Joint- and Dual-EnKFs. The proposed scheme is able to successfully recover both the hydraulic head and the aquifer conductivity, further providing reliable estimates of their uncertainties. Compared with the standard Joint- and Dual-EnKFs, the proposed scheme is found more robust to different assimilation settings, such as the spatial and temporal distribution of the observations, and the level of noise in the data. Based on our experimental setups, it yields up to 25% more accurate state and parameters estimates

Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity

This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic solid is almost incompressible. Several novel a posteriori error estimators for the energy norm of the finite element error are proposed and analysed. We establish upper and lower bounds for the energy error in terms of the proposed error estimators and prove that the constants in the bounds are independent of the Lam\'{e} coefficients: thus the proposed estimators are robust in the incompressible limit. Numerical results are presented that validate the theoretical estimates. The software used to generate these results is available online.Comment: 23 pages, 9 figure

Pre- and postprocessing techniques for determining goodness of computational meshes

Research in error estimation, mesh conditioning, and solution enhancement for finite element, finite difference, and finite volume methods has been incorporated into AUDITOR, a modern, user-friendly code, which operates on 2D and 3D unstructured neutral files to improve the accuracy and reliability of computational results. Residual error estimation capabilities provide local and global estimates of solution error in the energy norm. Higher order results for derived quantities may be extracted from initial solutions. Within the X-MOTIF graphical user interface, extensive visualization capabilities support critical evaluation of results in linear elasticity, steady state heat transfer, and both compressible and incompressible fluid dynamics

A posteriori analysis of Chorin-Temam scheme for Stokes equations

We consider Chorin-Temam scheme (the simplest pressure-correction projection method) for the time-discretization of an unstationary Stokes problem. Inspired by the analyses of the Backward Euler scheme performed by C.Bernardi and R.Verf\"urth, we derive a posteriori estimators for the error on the velocity gradient in L2 norm. Our invesigation is supported by numerical experiments

Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed