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1	Local Ill-Posedness and Source Conditions of Operator Equations in Hilbert Spaces Hofmann, B., Scherzer, O. 30 October 1998 (has links) (PDF) The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interdependence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity com bined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse an ill-posed problems. Additionally we show for the well-known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rates results. nonlinear operator equations ill-posedness Frechet derivative source conditions stability estimates ill-posedness measurres a posteriori estimates MSC 65J15 MSC 65J20 MSC 47H15 MSC 47H17 ddc:510
2	On Ill-Posedness and Local Ill-Posedness of Operator Equations in Hilbert Spaces Hofmann, B. 30 October 1998 (has links) (PDF) In this paper, we study ill-posedness concepts of nonlinear and linear inverse problems in a Hilbert space setting. We define local ill-posedness of a nonlinear operator equation $F(x) = y_0$ in a solution point $x_0$ and the interplay between the nonlinear problem and its linearization using the Frechet derivative $F\acent(x_0)$ . To find an appropriate ill-posedness concept for the linarized equation we define intrinsic ill-posedness for linear operator equations $Ax = y$ and compare this approach with the ill-posedness definitions due to Hadamard and Nashed. Nonlinear inverse problems oerator equations ill-posedness stability estimates local ill-posedness Frechet derivative linearized problem compact operators integral equations parameter identification MSC 65J20 MSC 47H15 MSC 65J10 MSC 65J15 MSC 65R30 MSC 45G10 ddc:510
3	Local Ill-Posedness and Source Conditions of Operator Equations in Hilbert Spaces Hofmann, B., Scherzer, O. 30 October 1998 (has links) The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interdependence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity com bined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse an ill-posed problems. Additionally we show for the well-known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rates results. info:eu-repo/classification/ddc/510 ddc:510
4	On Ill-Posedness and Local Ill-Posedness of Operator Equations in Hilbert Spaces: On Ill-Posedness and Local Ill-Posedness of OperatorEquations in Hilbert Spaces Hofmann, B. 30 October 1998 (has links) In this paper, we study ill-posedness concepts of nonlinear and linear inverse problems in a Hilbert space setting. We define local ill-posedness of a nonlinear operator equation $F(x) = y_0$ in a solution point $x_0$ and the interplay between the nonlinear problem and its linearization using the Frechet derivative $F\acent(x_0)$ . To find an appropriate ill-posedness concept for the linarized equation we define intrinsic ill-posedness for linear operator equations $Ax = y$ and compare this approach with the ill-posedness definitions due to Hadamard and Nashed. info:eu-repo/classification/ddc/510 ddc:510 Nonlinear inverse problems oerator equations ill-posedness stability estimates local ill-posedness Frechet derivative linearized problem compact operators integral equations parameter identification MSC 65J20 MSC 47H15 MSC 65J10 MSC 65J15 MSC 65R30 MSC 45G10

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