Necessary and sufficient conditions for the polynomial convergence of the quasi-reversibility and finite-difference methods for an ill-posed Cauchy problem with exact data

The convergence of the quasi-reversibility method and two classes of finite-difference methods for solving the ill-posed Cauchy problem for the first-order equation with a sectorial operator in a Banach space is analyzed. The necessary and sufficient conditions — close to one another — for the convergence of these methods with a rate polynomial with respect to the regularization parameter or discretization step are obtained in terms of the exponent in the source representability of the solution.