Discretized Lavrent’ev regularization for the autoconvolution equation

Abstract

Lavrent’ev regularization for the autoconvolution equation was considered by Janno J. in Lavrent’ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Prob. 2000;16:333–348. Here this study is extended by considering discretization of the Lavrent’ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.

Keywords:

• Autoconvolution
• inverse problem
• Lavrent’ev regularization

AMS Subject Classifications:

• 47J06
• 47A52
• 65F22

Acknowledgements

The authors would like to thank an anonymous referee for his valuable remarks, which helped to improve the quality of this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of SB was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant FL 832/1-1.