International Conference: AMMCS-2013

Waterloo, Ontario, Canada, August 26-30, 2013

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AMMCS-2013 Venue: Wilfrid Laurier University Campus in Waterloo, Canada


AMMCS-2013 Semi-Plenary Talk

On an Inverse and Ill-Posed Problem of Autoconvolution Type in Ultrashort Laser Pulse Characterization

Bernd Hofmann

Abstract: 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'.

AMMCS, 2013  

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