1,075 research outputs found
A convergence analysis of the iteratively regularized Gauss-Newton method under Lipschitz condition
In this paper we consider the iteratively regularized Gauss-Newton method for
solving nonlinear ill-posed inverse problems. Under merely Lipschitz condition,
we prove that this method together with an a posteriori stopping rule defines
an order optimal regularization method if the solution is regular in some
suitable sense
Regularized Newton Methods for X-ray Phase Contrast and General Imaging Problems
Like many other advanced imaging methods, x-ray phase contrast imaging and
tomography require mathematical inversion of the observed data to obtain
real-space information. While an accurate forward model describing the
generally nonlinear image formation from a given object to the observations is
often available, explicit inversion formulas are typically not known. Moreover,
the measured data might be insufficient for stable image reconstruction, in
which case it has to be complemented by suitable a priori information. In this
work, regularized Newton methods are presented as a general framework for the
solution of such ill-posed nonlinear imaging problems. For a proof of
principle, the approach is applied to x-ray phase contrast imaging in the
near-field propagation regime. Simultaneous recovery of the phase- and
amplitude from a single near-field diffraction pattern without homogeneity
constraints is demonstrated for the first time. The presented methods further
permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval
and tomographic inversion. We demonstrate the potential of this approach by
three-dimensional imaging of a colloidal crystal at 95 nm isotropic resolution.Comment: (C)2016 Optical Society of America. One print or electronic copy may
be made for personal use only. Systematic reproduction and distribution,
duplication of any material in this paper for a fee or for commercial
purposes, or modifications of the content of this paper are prohibite
Convergence analysis of a proximal Gauss-Newton method
An extension of the Gauss-Newton algorithm is proposed to find local
minimizers of penalized nonlinear least squares problems, under generalized
Lipschitz assumptions. Convergence results of local type are obtained, as well
as an estimate of the radius of the convergence ball. Some applications for
solving constrained nonlinear equations are discussed and the numerical
performance of the method is assessed on some significant test problems
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