  We shall derive and propose several efficient overlapping domain
decomposition methods for solving some typical linear inverse problems,
including the identiffication of the flux, the source strength and the initial
temperature in second order elliptic and parabolic systems. The methods are
iterative, and computationally very efficient: only local forward and adjoint
problems need to be solved in each subdomain, and the local minimizations have
explicit solutions. Numerical experiments are provided to demonstrate the
robustness and efficiency of the methods, in particular, the convergences seem
nearly optimal, i.e., they do not deteriorate or deteriorate only slightly when
the mesh size reduces.
