Avoiding singular coarse grid systems

Here we consider the iterative solution of linear systems of equations with a symmetric positive semidefinite system matrix. If multilevel methods in combination with Krylov subspace methods are used to find the solution of these systems, often singular subsystems or coarse grid systems have to be solved. Then the Moore-Penrose inverse of the coarse grid systems can be used. Here, we establish some theoretical techniques how to avoid the singularity of the coarse grid system, while the resulting operator remains the same as we would have used the Moore-Penrose inverse. One option is to delete specific columns of the restriction and prolongation operator. The other option is to perturb the system matrix.