A natural Hessian approximation for ensemble based optimization

Abstract

A key challenge in reservoir management and other fields of engineering involves optimizing a nonlinear function iteratively. Due to the lack of available gradients in commercial reservoir simulators the attention over the last decades has been on gradient free methods or gradient approximations. In particular, the ensemble-based optimization has gained popularity over the last decade due to its simplicity and efficient implementation when considering an ensemble of reservoir models. Typically, a regression type gradient approximation is used in a backtracking or line search setting. This paper introduces an approximation of the Hessian utilizing a Monte Carlo approximation of the natural gradient with respect to the covariance matrix. This Hessian approximation can further be implemented in a trust region approach in order to improve the efficiency of the algorithm. The advantages of using such approximations are demonstrated by testing the proposed algorithm on the Rosenbrock function and on a synthetic reservoir field.