Markus Bachmayr and Martin Burger 2009 Inverse Problems 25 105004 doi:10.1088/0266-5611/25/10/105004
Markus Bachmayr1 and Martin Burger2
Show affiliationsIn this paper we discuss the construction, analysis and implementation of iterative schemes for the solution of inverse problems based on total variation regularization. Via different approximations of the nonlinearity we derive three different schemes resembling three well-known methods for nonlinear inverse problems in Hilbert spaces, namely iterated Tikhonov, Levenberg–Marquardt and Landweber. These methods can be set up such that all arising subproblems are convex optimization problems, analogous to those appearing in image denoising or deblurring. We provide a detailed convergence analysis and appropriate stopping rules in the presence of data noise. Moreover, we discuss the implementation of the schemes and the application to distributed parameter estimation in elliptic partial differential equations.
42.30.Wb Image reconstruction; tomography
94A08 Image processing (compression, reconstruction, etc.) (See also 68U10)
65F22 Ill-posedness, regularization
35Jxx Partial differential equations of elliptic type (See also 58J10, 58J20)
Issue 10 (October 2009)
Received 27 April 2009, in final form 13 July 2009
Published 16 September 2009
Markus Bachmayr and Martin Burger 2009 Inverse Problems 25 105004
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