A Comparison of Deflation and the Balancing Preconditioner

In this paper we compare various preconditioners for the numerical solution of partial differential equations. We compare the well-known balancing preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the deflated preconditioned system is always, i.e., for all deflation vectors and all restrictions and prolongations, below the condition number of the system preconditioned by the balancing preconditioner. Even more, we establish that both preconditioners lead to almost the same spectra. The zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing preconditioner is used. Moreover, we prove that the A-norm of the errors of the iterates built by the deflation preconditioner is always below the A-norm of the errors of the iterates built by the balancing preconditioner. Depending on the implementation of the balancing preconditioner the amount of work of one iteration of the deflation preconditioned system is less than or equal to the amount of work of one iteration of the balancing preconditioned system. If the amount of work is equal, both preconditioners are sensitive with respect to inexact computations. Finally, we establish that the deflation preconditioner and the balancing preconditioner produce the same iterates if one uses certain starting vectors. Numerical results for porous media flows emphasize the theoretical results.