The application of Algebraic Multigrid-based linear solvers for performance enhancement of geomechanical simulators

Abstract

Geomechanical simulations can give essential insights into subsurface processes, but typically require solving large, ill-conditioned linear systems. An important method for solving these linear systems is the Conjugate Gradient method, but applying this method to ill-conditioned matrices can result in slow convergence. To improve the convergence of the Conjugate Gradient method, the iterative solver is preconditioned using the Algebraic Multigrid method. In Algebraic Multigrid methods, a hierarchy of matrices of different sizes is derived. When applying these methods as a preconditioner to the Conjugate Gradient method, on each level of the multigrid hierarchy fast convergence is observed in particular components of the residual. This leads to much fewer iterations being required in the Conjugate Gradient method, at the cost of the iterations being computationally more expensive. These Algebraic Multigrid methods do require a more problem-specific setup configuration than more simple preconditioners like the Jacobi preconditioner. In this research, the Conjugate Gradient method preconditioned with various Algebraic Multigrid methods is studied and compared with the Jacobi preconditioned Conjugate Gradient method. For this, the Conjugate Gradient method, preconditioned with both the Jacobi and Algebraic Multigrid-based methods, is applied to linear problems derived from geomechanical simulations. Using Algebraic Multigrid preconditioners can reduce the number of iterations required for convergence of the Conjugate Gradient method by a factor of 80. While a single iteration with an Algebraic Multigrid preconditioner is more time-expensive than an iteration with a Jacobi preconditioner, significant reductions, of up to five times, are observed in the runtimes of the linear solver. This comes at the cost of a higher peak memory requirement in the application of the linear solver. The vast reduction of the runtime of the linear solvers makes the studied geomechanical simulations significantly faster. This makes the Algebraic Multigrid preconditioners a valuable addition to these simulations.