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Proximal Methods for the Elastography Inverse Problem of Tumor Identification Using an Equation Error Approach

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Advances in Variational and Hemivariational Inequalities

Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 33))

Abstract

In this chapter, we study a nonlinear inverse problem in linear elasticity relating to tumor identification by an equation error formulation. This approach leads to a variational inequality as a necessary and sufficient optimality condition. We give complete convergence analysis for the proposed equation error method. Since the considered problem is highly ill-posed, we develop a stable computational framework by employing a variety of proximal point methods and compare their performance with the more commonly used Tikhonov regularization.

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Notes

  1. 1.

    It would be natural to define J by

    $$\displaystyle{J(\mu; z,\beta ) =\| E_{1}(\mu,z) - m\|_{V ^{{\ast}}}^{2} +\| E_{ 2}(z)\|_{V ^{{\ast}}}^{2} +\beta \|\mu \|_{ H^{1}}^{2}.}$$

    However, \(\|E_{2}(z)\|_{V ^{{\ast}}}^{2}\) is constant with respect to μ and therefore it makes no difference if this term is included or not.

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Acknowledgements

The authors are grateful to the referees for their careful reading and suggestions that brought substantial improvements to the work. The work of A.A. Khan is partially supported by RIT’s COS D-RIG Acceleration Research Funding Program 2012–2013 and a grant from the Simons Foundation (#210443 to Akhtar Khan).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, USA

    Mark S. Gockenbach

  2. Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY, 14623, USA

    Baasansuren Jadamba, Akhtar A. Khan & Brian Winkler

  3. Institute of Mathematics, Martin-Luther-University of Halle-Wittenberg, Theodor-Lieser-Str. 5, 06120, Halle-Salle, Germany

    Christiane Tammer

Authors
  1. Mark S. Gockenbach
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  2. Baasansuren Jadamba
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  3. Akhtar A. Khan
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  4. Christiane Tammer
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  5. Brian Winkler
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Corresponding author

Correspondence to Mark S. Gockenbach .

Editor information

Editors and Affiliations

  1. University of Iowa, Iowa City, IA, USA and Xi'an Jiaotong Universiry, Xi'an, China

    Weimin Han

  2. Faculty of Mathematics and Computer Scie Institute of Computer Science, Jagiellonian University, Kraków, Poland

    Stanisław Migórski

  3. Laboratoire de Mathématiques et Physique (LAMPS), Université de Perpignan Via Domitia, Perpignan, France

    Mircea Sofonea

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Gockenbach, M.S., Jadamba, B., Khan, A.A., Tammer, C., Winkler, B. (2015). Proximal Methods for the Elastography Inverse Problem of Tumor Identification Using an Equation Error Approach. In: Han, W., Migórski, S., Sofonea, M. (eds) Advances in Variational and Hemivariational Inequalities. Advances in Mechanics and Mathematics, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-319-14490-0_7

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