On iterative methods for solving ill-posed problems modeled by partial differential equations (Q2724857)

The authors consider families of linear evolutionary problems of elliptic, hyperbolic, and parabolic type in specific Hilbert spaces generated by spectral properties of unbounded linear operators occurring in the partial differential equations. The initial and boundary value problems under consideration are ill-posed in the sense of Hadamard. \textit{V. A. Kozlov} and \textit{V. G. Maz'ya} developed in [Leningr. Math. J. 1, No. 5, 1207-1228 (1990; Zbl 0732.65090)] an iterative method for solving such problems by using a well-posed auxiliary problem in each iteration step. Baumeister and Leitão, however, give alternative convergence proofs for algorithms based on this method. They exploit functional analytic properties of the non-expansive affine operators which characterize the iteration. Under regularity assumptions on the input data convergence rates for the iteration are formulated. In a paragraph on regularization the behaviour of the iterative method for given noisy data is studied. A numerical case study concerning the backward heat equation problem and an elliptic Cauchy problem, both one-dimensional with respect to the space variable, complete the paper.