On quasioptimum selection of the regularization parameter in M. M. Lavrent'ev's method (Q1335912)

The author considers the operator equation of the first kind \(Az=u\), \(z\in Z\), where \(Z\) is a real Hilbert space. Assuming the equation to be solvable, he discusses the question of determining a stable (in \(Z\)) approximation \(z_\eta= z_\eta(H)\), \(\eta=(h,\delta)\) to the normal solution \(\overline{z}: |z_\eta\rightharpoonup \overline{z}|\to 0\), as \(\eta\to 0\), from the approximate data \(H=(A_h, u_\delta,h,\delta)\) which are subject to the conditions \(|A_h- A|\leq h\); \(u_\delta\in Z\), \(|u_\delta-u|\leq \delta\), \(A_h\in S(z)\) (\(S(z)\) denotes the set of nonnegative selfadjoint bounded linear operators acting in \(Z\)). An a posteriori method for selecting the regularization parameters in the Lavrent'ev method is proposed.