Solving the Gauss-Newton trust region sub-problem with traditional solvers involves solving a symmetric linear system with dimensions the same as the number of uncertain parameters, and it is extremely computational expensive for history matching problems with large number of uncertain parameters. A new trust region (TR) solver is developed to save both memory usage and computational cost, and its performance is compared with the well-known direct TR solver using factorization and iterative TR solver using conjugate-gradient approach.
The original Gauss-Newton trust region sub-problem involves solving linear system with the number of uncertain parameters. With application of the matrix inverse lemma, it is transformed to a new problem that involves solving linear system with the number of observed data. For history matching problems where the number of uncertain parameters is much larger than the number of observed data, both memory usage and CPU time can be significantly reduced when compared to solving the original problem directly. An auto-adaptive power-law transformation technique is developed to transform the original strong nonlinear function to a new function that behaves more like a linear function. Finally, the Newton-Raphson method with some modifications is applied to solve the trust region sub-problem.
The proposed approach is applied to find best-match solutions in Bayesian style assisted history matching problems. It is first validated on a set of synthetic test problems with different numbers of uncertain parameters and different numbers of observed data. In terms of efficiency, the new approach is shown to significantly reduce both the computational cost and memory usage, e.g., by a factor of more than 10, when compared to the direct TR solver of the GALAHAD optimization library. In terms of robustness, the new approach is able to reduce the risk of failure to find the right solution significantly, when compared to the iterative TR solver of the GALAHAD optimization library. Our numerical results indicate that the new solver can solve large scale trust region sub-problems using reasonably small amounts of CPU time (in seconds) and memory (in MB). Compared to the CPU time and memory used for completing one reservoir simulation run for the same problem (in hours and in GB), the cost for finding the best-match parameter values using our new trust region solver is negligible. The proposed approach has been implemented in our in-house reservoir simulation and history matching system and has been validated on a real reservoir simulation model. This illustrates the main result of this paper: the development of a robust Gauss-Newton trust region approach, which is applicable for large scale history matching problems with negligible extra cost in CPU and memory.