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Regularization of exponentially ill-posed problems. (English) Zbl 0969.65041

Ill-posed problems (1) $Tx=y$ are considered, where $T:X\to Y$ is a linear operator between Hilbert spaces $X$ and $Y$. It is supposed that equation (1) has a minimum norm solution ${x}^{†}$ that satisfies a logarithmic source condition of the form (2) ${x}^{†}={f}_{p}\left({T}^{*}T\right)w$, where ${f}_{p}\left(\lambda \right)={\left(-ln\lambda \right)}^{-p}$ for $0<\lambda \le exp\left(-1\right)$ with some real parameter $p>0$, and a scaling $\parallel T\parallel \le exp\left(-1/2\right)$ is assumed.

For the stable solution of equation (1) with a minimum norm solution that satisfies such a logarithmic source condition (2), the author considers a priori and a posteriori parameter choice strategies for regularization methods of the general form (2) ${x}_{\alpha }^{\delta }={g}_{\alpha }\left({T}^{*}T\right){T}^{*}{y}^{\delta }$. Here ${y}^{\delta }\in Y$ is a perturbation of the exact right-hand side $y$ with $\parallel {y}^{\delta }-y\parallel \le \delta$. Moreover it is supposed that ${g}_{\alpha }$ ($\alpha >0$) are given functions such that, for each $\lambda >0$, we have ${g}_{\alpha }\left(\lambda \right)\to 1/\lambda$ as $\alpha \to 0$. Under certain additional conditions on the mappings ${g}_{\alpha }$, order optimal convergence rates of the form $\parallel {x}^{†}-{x}_{\alpha }^{\delta }\parallel \le {c}_{p}\rho {f}_{p}\left({\delta }^{2}/{\rho }^{2}\right)$ are obtained for each considered parameter choice $\alpha =\alpha \left(\delta \right)$, where $\rho \ge \parallel w\parallel$ holds.

The considered class of regularization methods contains well-known regularization methods like the Tikhonov regularization and the Landweber iteration, for example. This class of regularization methods is considered also to solve a class of nonlinear ill-posed problems. It is supposed here that the underlying nonlinear operator is given approximately by some linear operator, and moreover it is supposed that the solution to this nonlinear equation exists and satisfies a generalized logarithmic source condition.

For a second class of nonlinear ill-posed problems the iteratively regularized Gauss-Newton method is considered. Finally it is shown that the obtained results can be applied to specific problems like the backwards heat equation and the sideways heat equation, respectively.

Reviewer: R.Plato (Berlin)
##### MSC:
 65J10 Equations with linear operators (numerical methods) 65M30 Improperly posed problems (IVP of PDE, numerical methods) 47A52 Ill-posed problems, regularization 47J06 Nonlinear ill-posed problems 35K05 Heat equation 65J15 Equations with nonlinear operators (numerical methods) 65J20 Improperly posed problems; regularization (numerical methods in abstract spaces) 35R25 Improperly posed problems for PDE