The effect of approximate coarsest-level solves on the convergence of multigrid V-cycle methods

The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In the settings with large coarsest-level problems approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using this approach we discuss how the convergence of the V-cycle method may be affected by (1) the choice of the tolerance in a stopping criterion based on the relative residual norm for an iterative coarsest-level solver or (2) by the choices of the low-rank threshold parameter and finite precision arithmetic for a block low-rank direct coarsest-level solver.Furthermore we present new coarsest-level stopping criteria tailored to the multigrid method and suggest a heuristic strategy for their effective use in practical computations.