An Improved Implementation of the LBFGS Algorithm for Automatic History Matching

Abstract

Summary For large scale history matching problems, where it is not feasible to compute individual sensitivity coefficients, the limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS)is an efficient optimization algorithm, (Zhang and Reynolds, 2002; Zhang, 2002). However, computational experiments reveal that application of the original implementation of LBFGS may encounter the following problems: (i) converge to a model which gives an unacceptable match of production data; (ii) generate a bad search direction that either leads to false convergence or a restart with the steepest descent direction which radically reduces the convergence rate; (iii) exhibit overshooting and undershooting, i.e., converge to a vector of model parameters which contains some abnormally high or low values of model parameters which are physically unreasonable. Overshooting and undershooting can occur even though all history matching problems are formulated in a Bayesian framework with a prior model providing regularization. We show that the rate of convergence and the robustness of the algorithm can be significantly improved by: (1) a more robust line search algorithm motivated by the theoretical result that the Wolfe conditions should be satisfied; (2) an application of a data damping procedure at early iterations or (3) enforcing constraints on the model parameters. Computational experiments also indicate that (4) a simple rescaling of model parameters prior to application of the optimization algorithm can improve the convergence properties of the algorithm although the scaling procedure used can not be theoretically validated.