Efficient Ensemble-Based Stochastic Gradient Methods for Optimization Under Geological Uncertainty - Jeong, Hoonyoung; Sun, Alexander Y.; Jeon, Jonghyeon; Min, Baehyun; Jeong, Daein (Frontiers in Earth Science, 2020)
Ensemble-based stochastic gradient methods, such as the ensemble optimiza-tion method (EnOpt), the simplex gradient method (SG), and the stochastic simplex approximate gradient method (StoSAG), approximate the gradient of an objective function using an ensemble of perturbed control vectors. These methods are increas-ingly used in solving reservoir optimization problems because they are not only easy to parallelize and couple with any simulator, but also computationally more efficient than the conventional finite-difference method for gradient calculations. In this work, we show that EnOpt may fail to achieve sufficient improvement of the objective function when the differences between the objective function values of perturbed control vari-ables and their ensemble mean are large. On the basis of the comparison of EnOpt and SG, we propose a hybrid gradient of EnOpt and SG to save the computational cost of SG. We also suggest practical ways to reduce the computational cost of EnOpt and StoSAG by approximating the objective function values of unperturbed control varia-bles using the values of perturbed ones. We first demonstrate the performance of our improved ensemble schemes using a benchmark problem. Results show that the pro-posed gradients saved about 30–50% of the computational cost of the same optimiza-tion by using EnOpt, SG, and StoSAG. As a real application, we consider pressure management in carbon storage reservoirs, for which brine extraction wells need to be optimally placed to reduce reservoir pressure buildup while maximizing the net present value. Results show that our improved schemes reduce the computational cost signifi-cantly.