An Improved Implementation of the LBFGS Algorithm for Automatic History Matching

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:converge to a model which gives an unacceptable match of production data;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;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:a more robust line search algorithm motivated by the theoretical result that the Wolfe conditions should be satisfied;an application of a data damping procedure at early iterations orenforcing constraints on the model parameters. Computational experiments also indicate thata 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. Introduction Minimization of a smooth objective function is customarily done using a gradient based optimization algorithm such as the Gauss- Newton (GN) method or Levenberg-Marquardt (LM) algorithm. The standard implementations of these algorithms (Tan and Kalogerakis, 1991; Wu et al., 1999; Li et al., 2003), however, require the computation of all sensitivity coefficients in order to formulate the Hessian matrix. We are interested in history matching problems where the number of data to be matched ranges from a few hundred to several thousand and the number of reservoir variables or model parameters to be estimated or simulated ranges from a few hundred to a hundred thousand or more. For the larger problems in this range, the computer resources required to compute all sensitivity coefficients would prohibit the use of the standard Gauss- Newton and Levenberg-Marquardt algorithms. Even for the smallest problems in this range, computation of all sensitivity coefficients may not be feasible as the resulting GN and LM algorithms may require the equivalent of several hundred simulation runs. The relative computational efficiency of GN, LM, nonlinear conjugate gradient and quasi-Newton methods have been discussed in some detail by Zhang and Reynolds (2002) and Zhang (2002).