Recent Developments in Inverse Problems - Abstract
Anzengruber, Stephan
Recent experiments at the Max Born Institute, Berlin, generating ultrashort laser pulses citeBSKGBH15 have sparked interest in a new facet of the nonlinear autoconvolution equation [ g(s) = int_0^s k(s, tau) , f(s-tau) , f(tau) , mathrm dtau. ] The characterization of such laser pulses requires solving a particular autoconvolution problem for complex-valued functions, where the objective is to recover $f$ from measurements of $ f $ as well as $arg (g)$ in the presence of a non-trivial convolution kernel. In this talk we discuss analytical properties of the forward operator and present variational approaches to this emphphase retrieval problem. To obtain smooth reconstructions we use rational B-spline curves (NURBS) and improve global convergence properties by means of a TIGRA-type method citeRamlau03, WARH15. beginthebibliography9 bibitemBSKGBH15 S. Birkholz, G. Steinmeyer, S. Koke, D. Gerth, S. Bürger, and B. Hofmann. newblock Phase retrieval via regularization in self-diffraction based spectral interferometry. newblock em J. Opt. Soc. Am. B (2015), preprint on arXiv. bibitemBuerHof15 S. Bürger and B. Hofmann. newblock About a deficit in low order convergence rates on the example of autoconvolution. newblock em Applicable Analysis bf 94 (2015), 477--493. bibitemRamlau03 R. Ramlau. newblock TIGRA ---an iterative algorithm for regularizing nonlinear ill-posed problems. newblock em Inverse Problems bf 19 (2003), 433--465. bibitemWARH15 W. Wang, S.W. Anzengruber, R. Ramlau, and B. Han. newblock A global minimization algorithm for Tikhonov functionals with sparsity constraints. newblock em Applicable Analysis bf 94 (2015), 580--611. endthebibliography