On the Choice of the Regularization Parameter in the Case of the Approximately Given Noise Level of Data

Summary We consider ill-posed problems Au = f with operator A ∈ L(H,H), A = A* ≥ 0, where H is the Hilbert space and range R(A) is non-closed. Regularized solutions u r are obtained by a general regularization scheme, including the Lavrentiev method, iteration methods and others. We assume that instead of f ∈ R(A) noisy data $$ \tilde f $$ are available with the approximately given noise level δ: it holds $$ \left\| {\tilde f - f} \right\|/\delta \leqslant $$ const for δ → 0. We propose a new a-posteriori rule for the choice of the regularization parameter r = r(δ) guaranteering u r(δ) → u * for δ → 0, where u * is solution of problem Au = f. The error estimates are given.