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A Semi-smooth Newton Method for Inverse Problem with Uniform Noise

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Abstract

In this paper we study inverse problems where observations are corrupted by uniform noise. By using maximum a posteriori approach, an \(L_\infty \)-norm constrained minimization problem can be formulated for uniform noise removal. The main difficulty of solving such minimization problem is how to deal with non-differentiability of the \(L_\infty \)-norm constraint and how to estimate the level of uniform noise. The main contribution of this paper is to develop an efficient semi-smooth Newton method for solving this minimization problem. Here the \(L_\infty \)-norm constraint can be handled by active set constraints arising from the optimality conditions. In the proposed method, linear systems based on active set constraints are required to solve in each Newton step. We also employ the method of moments (MoM) to estimate the level of uniform noise for the minimization problem. The combination of the proposed method and MoM is quite effective for solving inverse problems with uniform noise. Numerical examples are given to demonstrate that our proposed method outperforms the other testing methods.

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Notes

  1. The relative difference is defined by \(\mathrm {RD} = \frac{\left\| \mathbf{u}^{(k+1)}-\mathbf{u}^{(k)}\right\| _2^2}{\left\| \mathbf{u}^{(k+1)}\right\| _2^2}.\)

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Acknowledgements

The authors would like to thank the two anonymous referees and the editor for their helpful comments and suggestions.

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Authors and Affiliations

  1. Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP), College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    You-Wei Wen

  2. Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, People’s Republic of China

    Wai-Ki Ching

  3. Department of Mathematics Hong Kong, Baptist University, Hong Kong, People’s Republic of China

    Michael Ng

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  1. You-Wei Wen
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  2. Wai-Ki Ching
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  3. Michael Ng
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Correspondence to You-Wei Wen.

Additional information

Y.-W. Wen: Research supported in part by the Construct Program of the Key Discipline in Hunan Province and NSFC Grant No. 11361030. W.-K. Ching: Research supported in part by Research Grants Council of Hong Kong under Grant No. 17301214, HKU Strategic Research Theme on Computation and Information, and National Natural Science Foundation of China under Grant No. 11671158. M. Ng: Research supported in part by HKRGC GRF 12306616 and 12203317.

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Wen, YW., Ching, WK. & Ng, M. A Semi-smooth Newton Method for Inverse Problem with Uniform Noise. J Sci Comput 75, 713–732 (2018). https://doi.org/10.1007/s10915-017-0557-x

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  • DOI: https://doi.org/10.1007/s10915-017-0557-x

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