Implementation of Asynchronous Distributed Gauss-Newton Optimization Algorithms for Uncertainty Quantification by Conditioning to Production Data
Authors: Gao Guohua , Florez Horacio , Jost Sean , Shaikh Shakir , Wang Kefei , Vink Jeroen , Blom Carl , Wells Terence J. , Saaf Fredrik 
Published: November 2023

Summary
Previous implementation of the distributed Gauss-Newton (DGN) optimization algorithm ran multiple optimization threads in parallel, employing a synchronous running mode (S-DGN). As a result, it waits for all simulations submitted in each iteration to complete, which may significantly degrade performance because a few simulations may run much longer than others, especially for time-consuming real-field cases. To overcome this limitation and thus improve the DGN optimizer’s execution, we propose two asynchronous DGN (A-DGN) optimization algorithms in this paper.
The two A-DGN optimization algorithms are (1) the local-search algorithm (A-DGN-LS) to locate multiple maximum a-posteriori (MAP) estimates and (2) the integrated global-search algorithm with the randomized maximum likelihood (RML) method (A-DGN + RML) to generate hundreds of RML samples in parallel for uncertainty quantification. We propose using batch together with a checking time interval to control the optimization process. The A-DGN optimizers check the status of all running simulations after every checking time frame. The iteration index of each optimization thread is updated dynamically according to its simulation status. Thus, different optimization threads may have different iteration indices in the same batch. A new simulation case is proposed immediately once the simulation of an optimization thread is completed, without waiting for the completion of other simulations.
We modified the training data set updating algorithm using each thread’s dynamically updated iteration index to implement the asynchronous running mode. We apply the modified QR decomposition method to estimate the sensitivity matrix at the best solution of each optimization thread by linear interpolation of all or a subset of the training data to avoid the issue of solving a linear system with a singular matrix because of insufficient training data points in early batches. A new simulation case (or search point) is generated by solving the Gauss-Newton (GN) trust-region subproblem (GNTRS) using the estimated sensitivity matrix. We developed a more efficient and robust GNTRS solver using eigenvalue decomposition (EVD).
The proposed A-DGN optimization methods are tested and validated on a 2D analytical toy problem and a synthetic history-matching problem and then applied to a real-field deepwater reservoir model. Numerical tests confirm that the proposed A-DGN optimization methods can converge to solutions with matching quality comparable to those obtained by the S-DGN optimizers, saving on the time required for the optimizer to converge by a factor ranging from 1.3 to 2 when compared to the S-DGN optimizer depending on the problem. The new A-DGN optimization algorithms improve efficiency and robustness in solving history-matching or inversion problems, especially for uncertainty quantification of subsurface model parameters and production forecasts of real-field reservoirs by conditioning production data.