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Iterative Methods for the Elastography Inverse Problem of Locating Tumors

In: Essays in Mathematics and its Applications

Author

Listed:
  • B. Jadamba

    (School of Mathematical Sciences, Rochester Institute of Technology, Center for Applied and Computational Mathematics)

  • A. A. Khan

    (School of Mathematical Sciences, Rochester Institute of Technology, Center for Applied and Computational Mathematics)

  • F. Raciti

    (University of Catania, Department of Mathematics and Computer Science)

  • C. Tammer

    (Martin-Luther-University of Halle-Wittenberg, Institute of Mathematics)

  • B. Winkler

    (Martin-Luther-University of Halle-Wittenberg, Institute of Mathematics)

Abstract

The primary objective of this work is to present a rigorous treatment of various iterative methods for solving the elastography inverse problem of identifying cancerous tumors. From a mathematical standpoint, this inverse problem requires the identification of a variable parameter in a system of partial differential equations. We pose the nonlinear inverse problem as an optimization problem by using an output least-squares (OLS) and a modified output least-squares (MOLS) formulation. The optimality conditions then lead to a variational inequality problem which is solved using various gradient, extragradient, and proximal-point methods. Previously, only a few of these methods have been implemented, and there is currently no understanding of their relative efficiency and effectiveness. We present a thorough numerical comparison of the 15 iterative solvers which emerge from a variational inequality formulation.

Suggested Citation

  • B. Jadamba & A. A. Khan & F. Raciti & C. Tammer & B. Winkler, 2016. "Iterative Methods for the Elastography Inverse Problem of Locating Tumors," Springer Books, in: Themistocles M. Rassias & Panos M. Pardalos (ed.), Essays in Mathematics and its Applications, pages 101-131, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-31338-2_6
    DOI: 10.1007/978-3-319-31338-2_6
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