Real-time ensemble control with reduced-order modeling

The control of spatially distributed systems is often complicated by significant uncertainty about system inputs, both time-varying exogenous inputs and time-invariant parameters. Spatial variations of uncertain parameters can be particularly problematic in geoscience applications, making it difficult to forecast the impact of proposed controls. One of the most effective ways to deal with uncertainties in control problems is to incorporate periodic measurements of the system’s states into the control process. Stochastic control provides a convenient way to do this, by integrating uncertainty, monitoring, forecasting, and control in a consistent analytical framework. This paper describes an ensemble-based approach to closed-loop stochastic control that relies on a computationally efficient reduced-order model. The use of ensembles of uncertain parameters and states makes it possible to consider a range of probabilistic performance objectives and to derive real-time controls that explicitly account for uncertainty. The process divides naturally into measurement updating, control, and forecasting steps carried out recursively and initialized with a prior ensemble that describes parameter uncertainty. The performance of the ensemble controller is investigated here with a numerical experiment based on a solute transport control problem. This experiment evaluates the performance of open and closed-loop controllers with full and reduced-order models as well as the performance obtained with a controller based on perfect knowledge of the system and the nominal performance obtained with no control. The experimental results show that a closed-loop controller that relies on measurements consistently performs better than an open loop controller that does not. They also show that a reduced-order forecasting model based on offline simulations gives nearly the same performance as a significantly more computationally demanding full order model. Finally, the experiment indicates that a moderate penalty on the variance of control cost yields a robust control strategy that reduces uncertainty about system performance with little or no increase in average cost. Taken together, these results confirm that reduced-order ensemble closed-loop control is a flexible and efficient control option for uncertain spatially distributed systems.Shell Oil Compan

Real-time ensemble control with reduced-order modeling

The control of spatially distributed systems is often complicated by significant uncertainty about system inputs, both time-varying exogenous inputs and time-invariant parameters. Spatial variations of uncertain parameters can be particularly problematic in geoscience applications, making it difficult to forecast the impact of proposed controls. One of the most effective ways to deal with uncertainties in control problems is to incorporate periodic measurements of the system’s states into the control process. Stochastic control provides a convenient way to do this, by integrating uncertainty, monitoring, forecasting, and control in a consistent analytical framework. This paper describes an ensemble-based approach to closed-loop stochastic control that relies on a computationally efficient reduced-order model. The use of ensembles of uncertain parameters and states makes it possible to consider a range of probabilistic performance objectives and to derive real-time controls that explicitly account for uncertainty. The process divides naturally into measurement updating, control, and forecasting steps carried out recursively and initialized with a prior ensemble that describes parameter uncertainty. The performance of the ensemble controller is investigated here with a numerical experiment based on a solute transport control problem. This experiment evaluates the performance of open and closed-loop controllers with full and reduced-order models as well as the performance obtained with a controller based on perfect knowledge of the system and the nominal performance obtained with no control. The experimental results show that a closed-loop controller that relies on measurements consistently performs better than an open loop controller that does not. They also show that a reduced-order forecasting model based on offline simulations gives nearly the same performance as a significantly more computationally demanding full order model. Finally, the experiment indicates that a moderate penalty on the variance of control cost yields a robust control strategy that reduces uncertainty about system performance with little or no increase in average cost. Taken together, these results confirm that reduced-order ensemble closed-loop control is a flexible and efficient control option for uncertain spatially distributed systems.Shell Oil Compan

Iterative regularization for ensemble data assimilation in reservoir models

We propose the application of iterative regularization for the development of ensemble methods for solving Bayesian inverse problems. In concrete, we construct (i) a variational iterative regularizing ensemble Levenberg-Marquardt method (IR-enLM) and (ii) a derivative-free iterative ensemble Kalman smoother (IR-ES). The aim of these methods is to provide a robust ensemble approximation of the Bayesian posterior. The proposed methods are based on fundamental ideas from iterative regularization methods that have been widely used for the solution of deterministic inverse problems [21]. In this work we are interested in the application of the proposed ensemble methods for the solution of Bayesian inverse problems that arise in reservoir modeling applications. The proposed ensemble methods use key aspects of the regularizing Levenberg-Marquardt scheme developed by Hanke [16] and that we recently applied for history matching in [18]. In the case where the forward operator is linear and the prior is Gaussian, we show that the proposed IR-enLM and IR-ES coincide with standard randomized maximum likelihood (RML) and the ensemble smoother (ES) respectively. For the general nonlinear case, we develop a numerical framework to assess the performance of the proposed ensemble methods at capturing the posterior. This framework consists of using a state-of-the art MCMC method for resolving the Bayesian posterior from synthetic experiments. The resolved posterior via MCMC then provides a gold standard against to which compare the proposed IR-enLM and IR-ES. We show that for the careful selection of regularization parameters, robust approximations of the posterior can be accomplished in terms of mean and variance. Our numerical experiments showcase the advantage of using iterative regularization for obtaining more robust and stable approximation of the posterior than standard unregularized methods