Tikhonov replacement functionals for iteratively solving nonlinear operator equations

@article{Ramlau2005TikhonovRF,
  title={Tikhonov replacement functionals for iteratively solving nonlinear operator equations},
  author={Ronny Ramlau and Gerd Teschke},
  journal={Inverse Problems},
  year={2005},
  volume={21},
  pages={1571 - 1592}
}
We shall be concerned with the construction of Tikhonov-based iteration schemes for solving nonlinear operator equations. In particular, we are interested in algorithms for the computation of a minimizer of the Tikhonov functional. To this end, we introduce a replacement functional, that has much better properties than the classical Tikhonov functional with nonlinear operator. Namely, the replacement functional is globally convex and can effectively be minimized by a fixed point iteration. On… 

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References

SHOWING 1-10 OF 40 REFERENCES

TIGRA—an iterative algorithm for regularizing nonlinear ill-posed problems

We report on a new iterative method for regularizing a nonlinear operator equation in Hilbert spaces. The proposed TIGRA algorithm is a combination of Tikhonov regularization and a gradient method

A steepest descent algorithm for the global minimization of the Tikhonov functional

We report on a new iterative approach for finding a global minimizer of the Tikhonov functional with a special class of nonlinear operators F. Assuming that the operator itself can be decomposed into

On the use of fixed point iterations for the regularization of nonlinear ill-posed problems

We report on a new iterative method for regularizing a nonlinear operator equation in Hilbert spaces. The proposed algorithm is a combination of Tikhonov regularization and a fixed point algorithm

Some Newton-type methods for the regularization of nonlinear ill-posed problems

In this paper we consider a combination of Newton's method with linear Tikhonov regularization, linear Landweber iteration and truncated SVD, for regularizing an abstract, nonlinear, ill-posed

A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems

The first part of this paper studies a Levenberg - Marquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.

A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions

For iterative methods for well-posed problems, invariance properties have been used to provide a unified framework for convergence analysis. We carry over this approach to iterative methods for

On convergence rates for the Iteratively regularized Gauss-Newton method

In this paper we prove that the iteratively regularized Gauss-Newton method is a locally convergent method for solving nonlinear ill-posed problems, provided the nonlinear operator satisfies a

Logarithmic convergence rates of the iteratively regularized Gauss - Newton method for an inverse potential and an inverse scattering problem

Convergence and logarithmic convergence rates of the iteratively regularized Gauss - Newton method in a Hilbert space setting are proven provided a logarithmic source condition is satisfied. This

Accurate attenuation correction in SPECT imaging using optimization of bilinear functions and assuming an unknown spatially-varying attenuation distribution

  • R. RamlauR. Clackdoyle
  • Mathematics
    1998 IEEE Nuclear Science Symposium Conference Record. 1998 IEEE Nuclear Science Symposium and Medical Imaging Conference (Cat. No.98CH36255)
  • 1998
Reports on an iterative approach to reconstruct the activity f(x) directly from the emission sinogram data without additional transmission measurements. The proposed algorithm is based on iterative