Convergence rates for Tikhonov regularization based on range inclusions

This paper provides some new a priori choice strategy for regularization parameters in order to obtain convergence rates in Tikhonov regularization for solving ill-posed problems Af(0) = g(0), f(0) is an element of X, g(0) is an element of Y, with a linear operator A mapping in Hilbert spaces X and Y. Our choice requires only that the range of the adjoint operator A* includes a member of some variable Hilbert scale and is, in principle, applicable in the case of general f(0) without source conditions imposed otherwise in the existing papers. For testing our strategies, we apply them to the determination of a wave source, to the Abel integral equation, to a backward heat equation and to the determination of initial temperature by boundary observation.