Novel iterative ensemble smoothers derived from a class of generalized cost functions

Abstract

Iterative ensemble smoothers (IES) are among the state-of-the-art approaches to solving history matching problems. From an optimization-theoretic point of view, these algorithms can be derived by solving certain stochastic nonlinear-least-squares problems. In a broader picture, history matching is essentially an inverse problem, which is often ill-posed and may not possess a unique solution. To mitigate the ill-posedness, in the course of solving an inverse problem, prior knowledge and domain experience are often incorporated, as a regularization term, into a suitable cost function within a respective optimization problem. Whereas in the inverse theory there is a rich class of inversion algorithms resulting from various choices of regularized cost functions, there are few ensemble data assimilation algorithms (including IES) which in their practical uses are implemented in a form beyond nonlinear-least-squares. This work aims to narrow this noticed gap. Specifically, we consider a class of more generalized cost functions, and establish a unified formula that can be used to construct a corresponding group of novel ensemble data assimilation algorithms, called generalized IES (GIES), in a principled and systematic way. For demonstration, we choose a subset (up to 30 +) of the GIES algorithms derived from the unified formula, and apply them to two history matching problems. Experiment results indicate that many of the tested GIES algorithms exhibit superior performance to that of an original IES developed in a previous work, showcasing the potential benefit of designing new ensemble data assimilation algorithms through the proposed framework.

Ensemble Kalman method for learning turbulence models from indirect observation data

“…See Equation (49) in Luo (2021) for detailed derivations of the identify above. In general, the observation operator H is nonlinear, in which case the square root matrix S y as estimated in Equation (A 1) provides a derivative-free approximation to the projected square root matrix HS w .…”

“…Note that this method involves the Hessian of the cost function (Evensen 2018;Luo 2021) and provides quantified uncertainties based on Bayesian analysis (Zhang et al 2020). Similar to the ensemble gradient method, the ensemble Kalman inversion method also approximates the sensitivity of velocity to neural-network weights based on the ensemble cross-covariance matrix, without involving the analytic gradient of the neural network.…”

Data assimilation with soft constraints (DASC) through a generalized iterative ensemble smoother

“…The MAC problem in Eq. 2 can be approximately solved through the following IES model update formula [23,25]:…”

“…5 contains two equivalent ways of expressing the Kalmangain matrix K i . The first expression is the one often used in the literature, whereas the second expression can be derived by applying a certain matrix identity to the first one [23].…”

“…Our contributions in this work include the following aspects: We develop an ensemble-based workflow to simultaneously handle generic, nonlinear equality and inequality constraints based on a recently proposed umbrella algorithm, called the generalized iterative ensemble smoother (GIES) [23]. This workflow inherits the benefits of ensemble-based data assimilation algorithms, in terms of the derivative-free nature and the ability to provide uncertainty quantification to some extent.…”