Optimization Algorithms Based on Combining FD Approximations and Stochastic Gradients Compared With Methods Based Only on a Stochastic Gradient

Optimization algorithms that incorporate a stochastic gradient [such as simultaneous-perturbation stochastic approximation (SPSA), simplex, and EnOpt] are easy to implement in conjunction with any reservoir simulator. However, for realistic problems, a stochastic gradient provides only a rough approximation of the true gradient, and, in particular, the angle between a stochastic gradient and the associated true gradient is typically far from zero even though a properly computed stochastic gradient usually represents an uphill direction. This paper develops a more robust optimization procedure by replacing the components of largest magnitude of the stochastic gradient with a finite-difference (FD) approximation of the pertinent partial derivatives. In essence, the objective of the method is to determine which components of the unknown true gradient are most important and then replace the corresponding components of the stochastic gradient with more-accurate FD approximations. This modified gradient can then be used in a gradient-based optimization algorithm to find the minimum or maximum of a given cost function. Our focus application is the estimation of optimal well controls, but it is clear that the method could also be used for other applications, including history matching.