Advertisement

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
Chapter
Part of the Advances in Mechanics and Mathematics book series (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.

Keywords

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

AMS Classification.

35R30 65N30 

Notes

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).

References

  1. 1.
    Acar, R.: Identification of the coefficient in elliptic equations. SIAM J. Control Optim. 31, 1221–1244 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Aguilo, M.A., Aquino, W., Brigham, J.C., Fatemi, M.: An inverse problem approach for elasticity imaging through vibroacoustics. IEEE Trans. Med. Imaging 29, 1012–1021 (2010)CrossRefGoogle Scholar
  3. 3.
    Al-Jamal, M.F., Gockenbach, M.S.: Stability and error estimates for an equation error method for elliptic equations. Inverse Prob. 28, 095006, 15 pp. (2012)Google Scholar
  4. 4.
    Ammari, H., Garapon, P., Jouve, F.: Separation of scales in elasticity imaging: a numerical study. J. Comput. Math. 28, 354–370 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Arnold, A., Reichling, S., Bruhns, O., Mosler, J.: Efficient computation of the elastography inverse problem by combining variational mesh adaption and clustering technique. Phys. Med. Biol. 55, 2035–2056 (2010)CrossRefGoogle Scholar
  6. 6.
    Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  7. 7.
    Cahill, N., Jadamba, B., Khan, A.A., Sama, M., Winkler, B.: A first-order adjoint and a second-order hybrid method for an energy output least squares elastography inverse problem of identifying tumor location. Bound. Value Prob. 2013, 263 (2013). doi: 10.1186/1687-2770-2013-263
  8. 8.
    Crossen, E., Gockenbach, M.S., Jadamba, B., Khan, A.A., Winkler, B.: An equation error approach for the elasticity imaging inverse problem for predicting tumor location. Comput. Math. Appl. 67, 122–135 (2014)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Doyley, M.M., Meaney, P.M., Bamber, J.C.: Evaluation of an iterative reconstruction method for quantitative elastography. Phys. Med. Biol. 45, 1521–1540 (2000)CrossRefGoogle Scholar
  10. 10.
    Doyley, M.M., Jadamba, B., Khan, A.A., Sama, M., Winkler, B.: A new energy inversion for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location. Numer. Funct. Anal. Optim. 35, 984–1017 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Fehrenbach, J., Masmoudi, M., Souchon, R., Trompette, P.: Detection of small inclusions by elastography. Inverse Prob. 22, 1055–1069 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Gockenbach, M.S., Khan, A.A.: Identification of Lamé parameters in linear elasticity: a fixed point approach. J. Ind. Manag. Optim. 1, 487–497 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Gockenbach, M.S., Khan, A.A.: An abstract framework for elliptic inverse problems. Part 1: an output least-squares approach. Math. Mech. Solids 12, 259–276 (2007)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Gockenbach, M.S., Khan, A.A.: An abstract framework for elliptic inverse problems. Part 2: an augmented Lagrangian approach. Math. Mech. Solids 14, 517–539 (2009)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Gockenbach, M.S., Jadamba, B., Khan, A.A.: Numerical estimation of discontinuous coefficients by the method of equation error. Int. J. Math. Comput. Sci. 1, 343–359 (2006)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Gockenbach, M.S., Jadamba, B., Khan, A.A.: Equation error approach for elliptic inverse problems with an application to the identification of Lamé parameters. Inverse Prob. Sci. Eng. 16, 349–367 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Hager, W., Zhang, H.: Self-adaptive inexact proximal point methods. Comput. Optim. Appl. 39, 161–181 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Han, D.: A generalized proximal-point-based prediction-correction method for variational inequality problems. J. Comput. Appl. Math. 221, 183–193 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Jadamba, B., Khan, A.A., Raciti, F.: On the inverse problem of identifying Lamé coefficients in linear elasticity. Comput. Math. Appl. 56, 431–443 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Jadamba, B., Khan, A.A., Rus, G., Sama, M., Winkler, B.: A new convex inversion for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of prediction tumor location. SIAM Appl. Math. 74, 1486–1510 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Kanzow, C.: Proximal-like methods for convex minimization problems. In: Qi, L., Teo, K., Yang, X. (eds.) Optimization and Control with Applications, pp. 369–392. Springer, New York (2005)CrossRefGoogle Scholar
  22. 22.
    Kaplan, A., Tichatschke, R.: Proximal point method and elliptic regularization. Nonlinear Anal. 71, 4525–4543 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Khan, A.A., Rouhani, B.D.: Iterative regularization for elliptic inverse problems. Comput. Math. Appl. 54, 850–860 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    McLaughlin, J., Yoon, J.R.: Unique identifiability of elastic parameters from time-dependent interior displacement measurement. Inverse Prob. 20, 25–45 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Oberai, A.A., Gokhale, N.H., Feijóo, G.R.: Solution of inverse problems in elasticity imaging using the adjoint method. Inverse Prob. 19, 297–313 (2003)CrossRefzbMATHGoogle Scholar
  26. 26.
    Parente, L.A., Lotito, P.A., Solodov, M.V.: A class of inexact variable metric proximal point algorithms. SIAM J. Optim. 19, 240–260 (2008)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Tossings, P.: Mixing proximal regularization, penalization and parallel decomposition in convex programming. Advances in optimization (Lambrecht, 1991), Lecture Notes in Econom. and Math. Systems, Springer, Berlin, 382, 85–99 (1992)MathSciNetGoogle Scholar
  29. 29.
    Tossings, P.: The perturbed proximal point algorithm and some of its applications. Appl. Math. Optim. 29, 125–159 (1994)CrossRefzbMATHMathSciNetGoogle Scholar

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

Personalised recommendations