A novel AMG approach based on adaptive smoothing and prolongation for reservoir simulations

Reservoir models can easily incorporate millions or even billions of unknowns. Algebraic multigrid (AMG) methods are often the standard choice as iterative solvers or preconditioners for the solution of the resulting linear systems. These comprise a family of techniques built on a hierarchy of levels associated with decreasing size problems. In this way, optimality and efficiency are achieved by combining two complementary processes, i.e. relaxation and coarse-grid correction. One of the key factors defining a fast AMG method consists of capturing accurately the near-null space of the system matrix for the construction of suitable prolongation operators. In this work, we propose a novel AMG package, aSP-AMG, where aSP means “adaptive Smoothing and Prolongation” and the “adaptive” attribute implies that we follow the perspective of adaptive and bootstrap AMG. We construct a space of smooth vectors of limited size (test space) using the Lanczos method and introduce the factorized sparse approximate inverse (FSAI) as a smoother. This improves the smoothing capabilities of the aSP-AMG as FSAI is more effective than Jacobi and much sparser than Gauss-Seidel. Moreover, FSAI has been shown to be strongly scalable. The coarsening phase is carried out as in classical AMG, but the strength of connection is computed by means of the affinity based on the test space. Finally, three new techniques are developed for building the prolongation operator: i) ABF, running few iterations of the aFSAI algorithm; ii) LS-ABF, updating the ABF coefficients with a least squares minimization; iii) DPLS, considering a least-squares process only. The aSP-AMG performance is assessed through the solution of reservoir engineering problems including both fluid flow and geomechanical test cases. Comparisons are made with the FSAI and BoomerAMG preconditioners, showing that the new method is generally superior to both approaches.