Discretized Lavrent' ev regularization for the autoconvolution equation

TL;DR

This paper extends Lavrent'ev regularization for the autoconvolution equation by discretizing with splines, enabling explicit solutions with piece-wise constant splines and maintaining convergence rates.

Contribution

It introduces a spline-based discretization method for Lavrent'ev regularization, including explicit solutions for piece-wise constant splines and a fast post-smoothing implementation.

Findings

Explicit solver for piece-wise constant splines

Fast post-smoothing implementation

Maintains convergence rate with proper discretization

Abstract

Lavrent'ev regularization for the autoconvolution equation was considered by J. Janno in {\itshape Lavrent'ev regularization of ill-posed 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.