Regularization of autoconvolution and other ill-posed equations of quadratic structure with applications in laser optics

• Applicant: Professor Dr. Jens Flemming
• Subject Area: Mathematics
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 230414508

In the past years a large number of papers and some monographs accelerated the progress in regularization theory for nonlinear ill-posed operator equations in Hilbert and Banach spaces. In the project we focus on nonlinear operators which have a quadratic structure. Important examples are autoconvolutions, which play an important role in spectroscopy and stochastics, and pointwisely squared linear operators as they appear for instance in Schlieren tomography. Research in the field of nonlinear ill-posed problems up to now follows the idea of linearization. Due to certain limitations of such linearization approaches we aim to develop derivative-free techniques, leading to tools suited for investigating and solving quadratic problems in a way similar to linear equations.Some new complex-valued and kernel-based version of the autoconvolution problem was recently presented by the research group 'Solid State Light Sources' of the Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Berlin, in the context of a new approach in ultrashort laser pulse characterization called Self-Diffraction SPIDER. Based on a joint diploma project a cooperation between Berlin and Chemnitz in 2011 indicated first promising approaches and many open mathematical questions. The overall aim of the project is to provide algorithms for numerically solving the autoconvolution problem appearing in the Self-Diffraction SPIDER method and to evaluate them on the basis of real-life measuring data.On the way to effiencient algorithms for autoconvolution and other quadratic equations one has to answer several theoretic questions. Of major interest is the strength of ill-posedness of the considered problems. Here a meaningful definition for a (local) degree of ill-posedness has to be developed. Existing derivative-based formulations turned out to carry not enough or even missleading information. Other questions concern convergence rates and saturation of regularization methods, since up to now the only convincing derivative-free techniques for obtaining general error estimates for regularization methods are variational smoothness assumptions (also known as variational inequalities). Thus, understanding such smoothness assumptions in case of quadratic mappings is of essential interest. But further development of algorithms and theory for solving quadratic equations also requires new sophisticated and refined techniques.

• DFG Programme: Research Grants
• Cooperation Partner: Professor Dr. Günter Steinmeyer