Discretized Lavrent’ev regularization for the autoconvolution equation

Abstract

Abstract. Lavrent ev regularization for the autoconvolution equation was considered by J. Janno in Lavrent ev regularization of illposed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Problems, 16(2):333-348, 2000. 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.

Deautoconvolution in the two-dimensional case

“…Its mathematical analysis has been comprehensively implemented in the last decades with focus on properties of the specific forward operator, ill-posedness, and regularization based on the seminal paper [20]. In this context, we refer to [8,10,11,13,15,16,17,23] for investigations concerning the stable identification of real functions x on the unit interval [0, 1] from noisy data of its autoconvolution x * x. A new series of interdisciplinary autoconvolution studies was developed by a cooperation started in 2010 between a research group of the Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy (Berlin) led by Prof. Günter Steinmeyer and the Chemnitz research group on regularization.…”