Morozov's discrepancy principle under general source conditions (Q1866652)

The authors study ill-posed problems \(Ax=b\) in a Hilbert space setting where instead of the exact data \(y\) some noisy data \(y^\delta\) are given such that \(\| y-y^\delta|\leq \delta\). They are interested in order optimality results of the regularized approximation of the minimum norm solution of the problem under the general source condition \[ x^+=[\varphi (A^*A)]^{1/2}v \quad {\text{with}}\quad \| v\| \leq E. \] Earlier \textit{U. Tautenhahn} [Numer. Funct. Anal. Optimization 19, 377--398 (1998; Zbl 0907.65049)] has shown that the classical Tikhonov regularization method combined with a special choice of the regularization parameter \(\alpha\) defined by the function \(\phi\) is optimal. Here the authors investigate regularization methods represented in the general form \[ x^{\delta}_{\alpha}=g_{\alpha}(A^*A)A^{*}y^{\delta} \] and combined with Morozov's discrepancy principle (independent of the function \(\varphi\)). They have found some analytical properties of the functions \(\varphi\) and \(d_{\alpha}\) that provide the order optimality of this simple regularization technique.