In the early 1990s motivated by applications from spectroscopy and stochastics
contributions to the mathematical analysis of deautoconvolution problems as
a class of inverse problems in spaces of continuous or quadratically integrable
real functions were made. Such deautoconvolution problems were mostly aimed at
finding non-negative functions with compact support from observations of its
autoconvolution. Since the autoconvolution operator is nonlinear and smoothing,
the deautoconvolution problem is ill-posed in the sense that the solutions need
not be uniquely determined and mainly small perturbations in the data may lead
to arbitrarily large errors in the solution. To overcome the negative
consequences of ill-posedness some kind of regularization is required. Recently,
the research group `Solid State Light Sources' of the Max Born Institute for
Nonlinear Optics and Short Pulse Spectroscopy, Berlin, hit on the
autoconvolution problem in the context of a new approach in ultrashort laser
pulse characterization called Self-Diffraction SPIDER. For phase reconstruction
as an auxiliary problem the solution of an autoconvolution equation is needed,
but now for complex functions to be determined from complex observations.
Moreover, a device-related kernel function must be added. The ill-posedness
phenomenon arises in the complex case, too, but a thorough analysis of the
complex case in deautoconvolution was missing in the literature.
The talk presents analytical and numerical results on the character of
ill-posedness of the equation occurring as a part of the SD SPIDER approach.
Moreover, an iterative regularization approach is suggested for the problem
when only noisy data of are given.
Dr. Bernd Hofmann is professor for Analysis and Inverse Problems at the
Chemnitz University of Technology, Germany, in the Department of Mathematics since 1993,
where he served as Dean of the Faculty from 2006 to 2009. He established an
interdisciplinary research group on inverse problems with applications in
natural sciences, engineering and finance, organizing the annual `Chemnitz
Symposium on Inverse Problems', sometimes on tour (in Linz 2009, Canberra 2012,
and Shanghai 2013).
Dr. Hofmann's research focus is on regularization theory and practice as well
as on studies concerning the nature of ill-posedness and appropriate tools
for the treatment of ill-posedness phenomena. He works as a member of the
Editorial Board of the journals `Inverse Problems' and `Journal of Inverse
and Ill-Posed Problems'.