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

• Final Report Year: 2016

Final Report Abstract

The project has contributed to the theory of nonlinear inverse and ill-posed problems with emphasis on the regularization of deautoconvolution problems and problems with forward operators of quadratic type formulated in infinite dimensional Hilbert spaces. One of the goals was to develop, in cooperation with the Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy (MBI) Berlin, inversion algorithms based on deautoconvolution for the recovery of complex-valued functions which characterize ultrashort (femtosecond) laser pulses in the context of the SD-SPIDER method. Major theoretic findings of the project have included new convergence rates results for Tikhonov and Lavrentiev regularization applied to deautoconvolution problems, assertions on local ill-posedness properties for several varieties of autoconvolution problems, and a basic understanding of the behavior of general quadratic mappings in infinite dimensional spaces. Moreover, connections between Tikhonov regularization for deautoconvolution and sparsity promoting regularization were discovered. Shortcomings of established standard techniques based on source conditions and the failure of qualified nonlinearity conditions could be verified for the autoconvolution mappings, and it could be shown how to overcome them. Several algorithms for regularized deautoconvolution were developed and evaluated. Precisely, a new decomposition approach, a variant of the Lavrentiev regularization, and different forms of Tikhonov’s method in combination with an adapted TIGRA- type method using NURBS for discretization proved to be useful for the numerical implementation. They were successfully tested for synthetic and real-life data obtained from the SD-SPIDER method in laser optics. New assertions on the analysis and numerics for SD-SPIDER as well as results from case studies for deautoconvolution and phase retrieval have been published in renowned journals of Applied Mathematics and Optics.

Publications

• Regularization of an autoconvolution problem in ultrashort laser pulse characterization. Inverse Problems in Science and Engineering, 22(2):245–266, 2014
  D. Gerth, B. Hofmann, S. Birkholz, S. Koke, G. Steinmeyer
  (See online at https://doi.org/10.1080/17415977.2013.769535)
• Regularization of autoconvolution and other ill-posed quadratic equations by decomposition. Journal of Inverse and III-posed Problems, 22(4): 551–567, 2014
  J. Flemming
  (See online at https://doi.org/10.1515/jip-2013-0038)
• About a deficit in low-order convergence rates on the example of autoconvolution. Applicable Analysis, 94(3):477–493, 2015
  S. Bürger, B. Hofmann
  (See online at https://doi.org/10.1080/00036811.2014.886107)
• Deautoconvolution: A new decomposition approach versus TIGRA and local regularization. Journal of Inverse and Ill-posed Problems, 23(3):231–243, 2015
  S. Bürger, J. Flemming
  (See online at https://doi.org/10.1515/jiip-2013-0082)
• On ϱ1-regularization in light of Nashed’s illposedness concept. Computational Methods in Applied Mathematics, 15(3):279– 289, 2015
  J. Flemming, B. Hofmann, I. Veselić
  (See online at https://doi.org/10.1515/cmam-2015-0008)
• Phase retrieval via regularization in self-diffraction-based spectral interferometry. Journal of the Optical Society of America B, 32(5):983–992, 2015
  S. Birkholz, G. Steinmeyer, S. Koke, D. Gerth, S. Bürger, B. Hofmann
  (See online at https://dx.doi.org/10.1364/JOSAB.32.000983)
• Discretized Lavrent’ev regularization for the autoconvolution equation. Applicable Analysis, published online: 25 July 201
  S. Bürger, P. Mathé
  (See online at https://doi.org/10.1080/00036811.2016.1212336)
• Inverse Autoconvolution Problems with an Application in Laser Physics. PhD thesis, TU Chemnitz, Faculty of Mathematics
  S. Bürger
  
• On complex-valued deautoconvolution of compactly supported functions with sparse Fourier representation. Inverse Problems, 32(10):104006 (12pp), 2016
  S. Bürger, J. Flemming, B. Hofmann
  (See online at https://doi.org/10.1088/0266-5611/32/10/104006)
• Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization. Inverse Problems, 32(3):035002, 2016
  S. W. Anzengruber, S. Bürger, B. Hofmann, G. Steinmeyer
  (See online at https://doi.org/10.1088/0266-5611/32/3/035002)