Abstrakt
Multigrid methods are frequently used when solving systems of linear equations, applied either as standalone solvers or as preconditioners for iterative methods. Within each cycle, the approximation is computed using smoothing on fine levels and solving on the coarsest level. Solving on the coarsest level can be carried out by a direct solver based on the LU or Cholesky decomposition, i.e., the coarsest level problem is solved exactly up to a machine precision. When solving large-scale problems on parallel computers, using direct solvers can be ineffective, sometimes impossible to realize. In these settings it is preferable to use inexact solver, e.g., iterative Krylov subspace methods or direct methods based on low rank approximations; see, e.g., [M. Huber, Massively parallel and fault-tolerant multigrid solvers on peta-scale systems, Ph.D. Thesis, Technical University of Munich, Germany, 2019], [Buttari et al., Numerical Linear Algebra with Applications (2021)]. The solver is typically stopped when the Euclidian norm of relative residual reaches specified tolerance. In practice, this tolerance is determined experimentally in order to balance the cost of the coarse grid solves and the total number of V-cycles required for convergence.
In this talk, we present an approach to constructing stopping criteria for an inexact coarse grid solver in the multigrid V-cycle method without post-smoothing for symmetric positive definite problems. This approach enables controlling the slowdown in the rate of convergence occurring due to the use of an inexact solver. We discuss several stopping criteria derived using this approach and suggest a strategy for utilizing them in practice. The results are illustrated through numerical experiments.