Conditions for the Qualified Convergence of Finite Difference Methods and the Quasi-Reversibility Method for Solving Linear Ill-Posed Cauchy Problems in a Hilbert Space

Abstract

We study finite difference methods and the quasi-reversibility method in application to linear ill-posed Cauchy problems with a self-adjoint operator in a Hilbert space with exact data. We prove that for these problems, it is possible to strengthen our earlier results on the convergence of the mentioned methods in a general case of a Banach space. We establish close to each other necessary and sufficient conditions for the qualified convergence of methods under consideration in terms of the source-representability exponent of the desired solution. We prove that (except the trivial case) the power estimates of the convergence rate of the considered methods cannot exceed the saturation level that corresponds to this or that method.