# 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…

## 69 Citations

### A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints

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A scheme which allows to minimize a Tikhonov functional where the usual quadratic regularization term is replaced by a one-homogeneous penalty on the coefficients (or isometrically transformed coefficients) of such expansions is developed.

### ON THE MINIMIZATION OF A TIKHONOV FUNCTIONAL WITH A NON-CONVEX SPARSITY CONSTRAINT

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In this paper we present a numerical algorithm for the optimiza tion of a Tikhonov functional with an lp-sparsity constraints and p < 1. Recently, it was proven that the minimization of this functio…

### An analysis of Tikhonov regularization for nonlinear ill-posed problems under a general smoothness assumption

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In this paper we introduce an adaptive regularization scheme based on algorithms for minimization of the Tikhonov functional to reconstruct the solution x† of nonlinear ill-posed problem F(x) = y,…

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The goal is to investigate the use of an iterative scheme based on the Bregman distance to solve the nonlinear material decomposition problem in spectral computerized tomography and focus on the convergence of the methods for different regularization parameters and initial guesses.

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- Computer Science, Mathematics
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A new algorithm for computing regularized solutions to inverse problems where the unknown functions is a characteristic function and where the forward operator is linear is proposed, based on an efficient computation of minimizers for a Tikhonov functional.

### Convexity of the Tikhonov functional and iteratively regularized methods for solving irregular nonlinear operator equations

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Domains in a Hilbert space are localized where the Tikhonov functional of an irregular nonlinear operator equation is either strongly convex or has other similar properties. Depending on the…

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The intention of this paper is to show the applicability of a generalized conditional gradient method for the minimization of Tikhonov-type functionals, which occur in the regularization of nonlinear…

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This paper is concerned with the construction of an iterative algorithm to solve nonlinear inverse problems with an ℓ1 constraint on x. One extensively studied method to obtain a solution of such an…

### Phase reconstruction by a multilevel iteratively regularized Gauss–Newton method

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In 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…

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