## Morozov’s discrepancy principle under general source conditions.(English)Zbl 1037.65057

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 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.

### MSC:

 65J10 Numerical solutions to equations with linear operators 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 47A52 Linear operators and ill-posed problems, regularization

Zbl 0907.65049
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### References:

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