Journal of Scientific Computing

, Volume 79, Issue 2, pp 1214–1240

# A Fast Algorithm for Solving Linear Inverse Problems with Uniform Noise Removal

• Xiongjun Zhang
• Michael K. Ng
Article

## Abstract

In this paper, we develop a fast algorithm for solving an unconstrained optimization model for uniform noise removal which is an important task in inverse problems. The optimization model consists of an $$\ell _\infty$$ data fitting term and a total variation regularization term. By utilizing the alternating direction method of multipliers (ADMM) for such optimization model, we demonstrate that one of the ADMM subproblems can be formulated by involving a projection onto $$\ell _1$$ ball which can be solved efficiently by iterations. The convergence of the ADMM method can be established under some mild conditions. In practice, the balance between the $$\ell _\infty$$ data fitting term and the total variation regularization term is controlled by a regularization parameter. We present numerical experiments by using the L-curve method of the logarithms of data fitting term and total variation regularization term to select regularization parameters for uniform noise removal. Numerical results for image denoising and deblurring, inverse source, inverse heat conduction problems and second derivative problems have shown the effectiveness of the proposed model.

## Keywords

Uniform noise Linear inverse problems Total variation $$\ell _\infty$$-Norm Alternating direction method of multipliers

## Mathematics Subject Classification

49N45 65F22 90C25

## Notes

### Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments and suggestions that have helped improve the presentation of the paper greatly.

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