A discrepancy principle for Tikhonov regularization with approximately specified data

• Many discrepancy principles are known for choosing the parameter $$\alpha$$ in the regularized operator equation $$(T^*T+ \alpha I)x_\alpha^\delta = T^*y^\delta$$, $$||y-y^d||\leq \delta$$, in order to approximate the minimal norm least-squares solution of the operator equation $$Tx=y$$. In this paper we consider a class of discrepancy principles for choosing the regularization parameter when $$T^*T$$ and $$T^*y^\delta$$ are approximated by $$A_n$$ and $$z_n^\delta$$ respectively with $$A_n$$ not necessarily self - adjoint. Thisprocedure generalizes the work of Engl and Neubauer (1985),and particular cases of the results are applicable to the regularized projection method as well as to a degenerate kernel method considered by Groetsch (1990).

$Rev: 13581$