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Zbl 1103.65058
Pereverzev, Sergei; Schock, Eberhard
On the adaptive selection of the parameter in regularization of ill-posed problems.
(English)
[J] SIAM J. Numer. Anal. 43, No. 5, 2060-2076 (2005). ISSN 0036-1429; ISSN 1095-7170/e

Consider an ill-posed operator equation $Ax=y$ with a linear operator $A \in {\cal L}(X,Y)$ between Banach spaces $X$ and $Y$. Let $y_{\delta}$ be an available approximation of $y$, $\Vert y_{\delta}- y \Vert_Y \leq \delta$. Regularization methods usually replace the generalized inverse $A^+$ by a family of continuous linear operators $\{ R_{\alpha} \}$, which converges pointwise to $A^+$ such that $\Vert A^+ y - R_{\alpha}y \Vert \leq \varphi(\alpha)$, $\Vert R_{\alpha}\Vert \leq \lambda(\alpha)^{-1}$ and $\lim_{\alpha \to 0} \varphi(\alpha)=\lim_{\alpha \to 0} \lambda(\alpha)=0$. Denote $x_{\alpha_i}^{\delta}=R_{\alpha_i} y_{\delta}$, $\Delta_N =\{ \alpha_i: 0 <\alpha_0 < \dots <\alpha_N \}$, $M^+ (\Delta_N)=\{ \alpha_i \in \Delta_N: \Vert x_{\alpha_i}^{\delta}- x_{\alpha_j}^{\delta}\Vert \leq 4\delta \lambda(\alpha_j)^{-1}, j=0, 1, \dots, i \}$ and $\alpha_+=\max\{ \alpha_i: \alpha_i \in M^+(\Delta_N) \}$. Suppose that $\lambda(\alpha_i) \leq q\lambda(\alpha_{i-1})$ for any $\alpha_i \in \Delta_N$, $i=1, \dots, N$. Then one has $\Vert A^+ y- x_{\alpha+}^{\delta} \Vert \leq 6q \varphi ((\varphi \lambda)^{-1}(\delta))$.
[Mikhail Yu. Kokurin (Yoshkar-Ola)]
MSC 2000:
*65J10 Equations with linear operators (numerical methods)
65J20 Improperly posed problems (numerical methods in abstract spaces)
47A52 Ill-posed problems etc.
45E10 Integral equations of the convolution type
65R30 Improperly posed problems (integral equations, numerical methods)

Keywords: inverse problems in Banach spaces; parameter choice; Abel integral equations; Lavrentiev regularization for equations with monotone operators; scattering; profile reconstruction; ill-posed operator equation; generalized inverse

Citations: Zbl 0149.11405

Cited in: Zbl 1190.65092 Zbl 1118.65056

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