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1	On the Autoconvolution Equation and Total Variation Constraints Fleischer, G., Gorenflo, R., Hofmann, B. 30 October 1998 (has links) (PDF) This paper is concerned with the numerical analysis of the autoconvolution equation $x*x=y$ restricted to the interval [0,1]. We present a discrete constrained least squares approach and prove its convergence in $L^p(0,1),1<p<\infinite$ , where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non-smooth solutions possessing jumps. Moreover, an adaption to the Sobolev space $H^1(0,1)$ and some remarks on monotone functions are added. The paper is completed by a numerical case study concerning the determination of non-monotone smooth and non-smooth functions x from the autoconvolution equation with noisy data y. autoconvolution ill-posed problem discretization constrained least squares approach bounded total variation MSC 65J20 MSC 45G10 MSC 65R30 ddc:510
2	On the Autoconvolution Equation and Total Variation Constraints Fleischer, G., Gorenflo, R., Hofmann, B. 30 October 1998 (has links) This paper is concerned with the numerical analysis of the autoconvolution equation $x*x=y$ restricted to the interval [0,1]. We present a discrete constrained least squares approach and prove its convergence in $L^p(0,1),1<p<\infinite$ , where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non-smooth solutions possessing jumps. Moreover, an adaption to the Sobolev space $H^1(0,1)$ and some remarks on monotone functions are added. The paper is completed by a numerical case study concerning the determination of non-monotone smooth and non-smooth functions x from the autoconvolution equation with noisy data y. info:eu-repo/classification/ddc/510 ddc:510 autoconvolution ill-posed problem discretization constrained least squares approach bounded total variation MSC 65J20 MSC 45G10 MSC 65R30
3	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
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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