Multilevel domain decomposition methods are well developed for elliptic problems. In this talk, we discuss some extensions of the methods for solving some harder problems, namely the coupled nonlinear systems of equations arising from the discretization of inverse elliptic equations. In particular, we focus on a fully coupled Newton-Krylov algorithm with two-level domain decomposition preconditioning. Four types of preconditioning methods will be discussed including the multiplicative Schwarz method, the additive Schwarz method, the F-cycle Schwarz method and the cascade version of the multiplicative method. We will present the parallel performance of the algorithms on computers with hundreds of processors for solving some difficult inverse problems with noise and jump coefficients.