# Can a Krylov subspace method be used as a smoother for multigrid?

As far as I am aware, multigrid solvers use iterative smoothers such as Jacobi, Gauss-Seidel, and SOR to dampen the error at various frequencies. Could a Krylov subspace method (like conjugate gradient, GMRES, etc.) be utilized instead? I don't think they are classified as "smootheners", but they can be used to approximate the coarse grid solution. Can we expect to see analogous convergence to the solution as we would in a standard multigrid method? Or is it problem dependent?

A more popular alternative is to use polynomial smoothers (usually Chebyshev). These methods target a specified portion of the spectrum. For symmetric elliptic PDEs (where the discrete operator is symmetric positive definite or nearly so), it is common to estimate the largest eigenvalue $\lambda_\max$ of $D^{-1}A$ where $D^{-1}$ is the point-block Jacobi preconditioner for $A$ and target a range like $(0.1 \lambda_{\max}, 1.1 \lambda_{\max})$. Polynomial smoothers have no reductions and are linear operations (for any chose polynomial degree, usually chosen between $1$ and perhaps $5$). Usually a few iterations (say $5$ to $10$) of GMRES or CG are used to estimate $\lambda_\max$, so the user does not need to compute these themselves. The estimate of $\lambda_\max$ is also used by some algebraic multigrid methods to choose coarsening strategies.