A discrepancy principle for Tikhonov regularization with approximately specified data

Many discrepancy principles are known for choosing the parameter α in the regularized operator equation ${\displaystyle (T\cdot T+\alpha I)x_{\alpha }^{\delta }=T\cdot y^{\delta }}$, ${\displaystyle |y-y^{\delta }|\le \delta }$, in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and ${\displaystyle T\cdot y^{\delta }}$ are approximated by Aₙ and ${\displaystyle z{ₙ}^{\delta }}$ respectively with Aₙ not necessarily self-adjoint. This procedure 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).