Two-level preconditioners for regularized inverse problems I: Theory

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