Dirk Langemann and Manfred Tasche 2008 Inverse Problems 24 035006 doi:10.1088/0266-5611/24/16/035006
Phase reconstruction by a multilevel iteratively regularized Gauss–Newton method
Dirk Langemann1 and Manfred Tasche2
Show affiliationsIn this paper we consider the numerical solution of a phase retrieval problem for a compactly supported, linear spline 
with the Fourier transform 
, where values of |f| and 
at finitely many equispaced nodes are given. The unknown phases of complex spline coefficients fulfil a well-structured system of nonlinear equations. Thus the phase reconstruction leads to a nonlinear inverse problem, which is solved by a multilevel strategy and iterative Tikhonov regularization. The multilevel strategy concentrates the main effort of the solution of the phase retrieval problem in the coarse, less expensive levels and provides convenient initial guesses at the next finer level. On each level, the corresponding nonlinear system is solved by an iteratively regularized Gauss–Newton method. The multilevel strategy is motivated by convergence results of IRGN. This method is applicable to a wide range of examples as shown in several numerical tests for noiseless and noisy data.
- PACS
- MSC
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65F22 Ill-posedness, regularization
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
- Subjects
- Dates
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Issue 3 (June 2008)
Received 16 August 2007, in final form 13 February 2008
Published 8 April 2008
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Phase reconstruction by a multilevel iteratively regularized Gauss–Newton method
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