Automated linear solver selection for simulation of multiphysics processes in porous media

Porous media processes involve various physical phenomena such as mechanical deformation, transport, and fluid flow. Accurate simulations must capture the strong couplings between these phenomena. Choosing an efficient solver for a multiphysics problem usually entails the decoupling into subproblems related to separate physical phenomena. Then, the suitable solvers for each subproblem and the iteration scheme must be chosen. Finally, the numerical parameters of the solvers must be optimized. Theoretical considerations often allow us to come up with several robust linear solvers for a given problem, but they cannot guide us further in seeking the most efficient linear solver configuration because its performance depends on hardware, software and the driving forces of the simulated model. As a further complication, these driving forces can vary with time within one simulation, causing the most efficient linear solver configuration to change. Switching a solver with respect to the dominant process can be beneficial, but the threshold of when to switch solver is unclear and complicated to analyze. We address this challenge by developing a machine learning framework that automatically searches for the optimal solver for a given multiphysics simulation setup, based on statistical data from previously solved problems. For a series of problems, exemplified by successive time steps in a time-dependent simulation, the framework updates and improves its decision model online during the simulation. We describe the solver selection algorithm, present examples of how the solver selector tunes the solver during the simulation, and show how it outperforms preselected state-of-the-art solvers for test problem setups. The examples are based on simulations of poromechanics and simulations of flow and transport. For the quasi-static linear Biot model, we demonstrate automated tuning of numerical solver parameters by showing how the L-parameter of the so-called Fixed-Stress preconditioner can be optimized. Motivated by a test example where the main heat transfer mechanism changes between convection and diffusion, we also discuss how the solver selector can dynamically switch solvers when the dominant physical phenomenon changes with time.