Two-level preconditioners for regularized inverse problems I: Theory

We compare additive and multiplicative Schwarz preconditioners for the iterative solution of regularized linear inverse problems, extending and complementing earlier results of Hackbusch, King, and Rieder. Our main findings are that the classical convergence estimates are not useful in this context: rather, we observe that for regularized ill-posed problems with relevant parameter values the additive Schwarz preconditioner significantly increases the condition number. On the other hand, the multiplicative version greatly improves conditioning, much beyond the existing theoretical worst-case bounds. We present a theoretical analysis to support these results, and include a brief numerical example. More numerical examples with real applications will be given elsewhere. Mathematics Subject Classification (1991): 65N55.