Theoretical Comparison of Two-Level Preconditioners based on Multigrid and Deflation

It is well-known that two-level preconditioned conjugate gradient (PCG) methods provide efficient techniques for solving large and sparse linear systems whose coefficient matrices are symmetric and positive definite (SPD). A two-level PCG method combines traditional and projectiontype preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level preconditioners are known, coming from the fields of deflation, domain decomposition and multigrid. At first glance, these methods seem to be different; however, from an abstract point of view, they are closely related. In [J.M. Tang, R. Nabben, C. Vuik and Y.A. Erlangga, DUT Report 07-04, Delft University of Technology, Delft, 2007], a theoretical and numerical comparison have been carried out for some of these two-level PCG methods. However, the standard multigrid V(1,1)-cycle preconditioner was excluded from that analysis, since this preconditioner has different properties and requires a different treatment than methods discussed in that paper. The aim of this paper is to relate the two-level PCG method, with a standard multigrid V(1,1)cycle (MG) preconditioner in its abstract form, to the deflation and abstract balancing NeumannNeumann methods as analyzed in [J.M. Tang, R. Nabben, C. Vuik and Y.A. Erlangga, DUT Report 07-04, Delft University of Technology, Delft, 2007]. The MG preconditioner is expected to be more effective than these two-level preconditioners, but we show that this is not always the case. For common choices of the parameters, MG does lead to larger error reductions in each iteration, but the work per iteration is much more expensive, which makes this comparison somewhat unfair. We show that, for special choices of fine-level preconditioners in the deflation or abstract balancing NeumannNeumann methods, the work for each iteration with these preconditioners is approximately the same as that for the MG preconditioner, and the convergence of the resulting two-level PCG methods are also expected to be approximately the same. Numerical experiments are used to emphasize the theoretical results.

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