Introduction to PDE-Constrained Optimization in the Oil and Gas Industry

Optimal order multigrid preconditioners for the distributed control of parabolic equations with coarsening in space and time

“…Over the last two decades PDE-constrained optimization has received an increasing level of attention from the engineering and computational mathematics communities due to its wide range of applicability. In particular, the optimal control of time-dependent systems includes classes of problems like inversion of reaction-diffusion equations [6], control of fluid flows [17], with applicability to 3D and 4D variational data assimilation for weather prediction [2], and reservoir history matching in gas and oil extraction [9]. All of the aforementioned applications share a common feature: the models can lead to extremely large problems that have to be solved in a relatively short amount of time.…”

Algebraic multigrid preconditioning of the Hessian in optimization constrained by a partial differential equation

“…Applications range from the optimization of manufacturing processes of various components 1 to data assimilation for weather prediction 2 and petroleum reservoir simulations. 3 A common question is whether in some cases an optimal control problem can be solved with the same high resolution as the forward model, ideally with a similar computational cost. Relatedly, practitioners would like to use solvers and preconditioners developed and optimized for the forward problem also for the optimal control problem, as a way to limit programming effort and to exploit some knowledge and understanding of the model that is somehow embodied in the forward solver.…”

“…Since PDEs require multiple input parameters for well‐posedness (initial and boundary values, forcing terms, and various model parameters), PDE‐constrained optimization can be employed to identify parameters that are optimal in some sense. Applications range from the optimization of manufacturing processes of various components 1 to data assimilation for weather prediction 2 and petroleum reservoir simulations 3 . A common question is whether in some cases an optimal control problem can be solved with the same high resolution as the forward model, ideally with a similar computational cost.…”