# 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…

## 68 Citations

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

- MathematicsNumerische Mathematik
- 2006

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.

### A note on the minimization of a Tikhonov functional with ℓ1-penalty

- MathematicsInverse Problems
- 2020

This paper proposes a different transformation of the Nemskii operator, which leads to a twice differentiable functional that can now be minimized using efficient second order methods like Newton's method.

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

- Mathematics
- 2009

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

- Mathematics
- 2006

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,…

### Nonlinear material decomposition using a regularized iterative scheme based on the Bregman distance

- MathematicsInverse Problems
- 2018

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.

### Surrogate functionals and thresholding for inverse interface problems

- Computer Science, Mathematics
- 2007

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.

### A generalized conditional gradient method for nonlinear operator equations with sparsity constraints

- Mathematics
- 2007

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…

### Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints

- Mathematics
- 2010

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…

### An efficient projection method for nonlinear inverse problems with sparsity constraints

- Mathematics
- 2016

In this paper, we propose a modification of the accelerated projective steepest descent method for
solving nonlinear inverse problems with an $\ell_1$ constraint on the variable, which was recently…

### On the convergence of iterative shrinkage algorithms with adaptive discrepancy terms

- Mathematics
- 2008

In this paper, the inversion of a linear operator is tackled by a procedure called iterative shrinkage. Iterative shrinkage is a procedure that minimizes a functional balancing quadratic discrepancy…

## References

SHOWING 1-10 OF 40 REFERENCES

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

- Mathematics
- 2003

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

- Mathematics
- 2002

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

- Mathematics
- 2005

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…

### MOROZOV'S DISCREPANCY PRINCIPLE FOR TIKHONOV-REGULARIZATION OF NONLINEAR OPERATORS

- Mathematics
- 2002

ABSTRACT We consider Morozov's discrepancy principle for Tikhonov-regularization of nonlinear operator equations. It is shown that minor restrictions to the operator F and the solution x* of the…

### Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems

- Mathematics
- 1997

This paper develops truncated Newton methods as an appropriate tool for nonlinear inverse problems which are ill-posed in the sense of Hadamard. In each Newton step an approximate solution for the…

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

- Mathematics
- 1997

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

- Mathematics
- 1997

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

- Mathematics, Computer Science
- 2003

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

- Mathematics
- 1998

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

- Mathematics
- 1997

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…