Regularization of autoconvolution and other ill-posed quadratic equations by decomposition

Standard methods for regularizing ill-posed nonlinear equations rely on derivatives of the nonlinear forward mapping. Thereby stronger structural properties of the concrete problem are neglected and the derived algorithms only show mediocre efficiency. We concentrate on nonlinear mappings with quadratic structure and develop a derivative-free regularization method that allows us to apply classical techniques known from linear inverse problems to quadratic equations. In fact, regularization of a quadratic problem can be reduced to regularization of one linear problem and a downstream inversion of a well-posed quadratic mapping. The motivation for considering problems with quadratic structure in more detail comes from applications in laser optics where kernel-based auto-convolution-type equations have to be solved.