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

  • Mark S. GockenbachEmail author
  • Baasansuren Jadamba
  • Akhtar A. Khan
  • Christiane Tammer
  • Brian Winkler
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 33)


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.


Variational inequality Elasticity imaging Elastography inverse problem Tumor identification Proximal point method Regularization 

AMS Classification.

35R30 65N30 



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).


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Mark S. Gockenbach
    • 1
    Email author
  • Baasansuren Jadamba
    • 2
  • Akhtar A. Khan
    • 2
  • Christiane Tammer
    • 3
  • Brian Winkler
    • 2
  1. 1.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA
  2. 2.Center for Applied and Computational Mathematics, School of Mathematical SciencesRochester Institute of TechnologyRochesterUSA
  3. 3.Institute of MathematicsMartin-Luther-University of Halle-WittenbergHalle-SalleGermany

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