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Iterative Methods for the Elastography Inverse Problem of Locating Tumors

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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.

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Acknowledgements

The work of A. A. K. is supported by a grant from the Simons Foundation (#210443 to Akhtar Khan)

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Authors and Affiliations

  1. Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA

    B. Jadamba & A. A. Khan

  2. Department of Mathematics and Computer Science, University of Catania, 95125, Catania, Italy

    F. Raciti

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

    C. Tammer & B. Winkler

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  1. B. Jadamba
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  2. A. A. Khan
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  3. F. Raciti
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  4. C. Tammer
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  5. B. Winkler
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Correspondence to F. Raciti .

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Editors and Affiliations

  1. Department of Mathematics, National Technical Unversity of Athens, Athens, Greece

    Themistocles M. Rassias

  2. University of Florida, GAINESVILLE, Florida, USA

    Panos M. Pardalos

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Jadamba, B., Khan, A.A., Raciti, F., Tammer, C., Winkler, B. (2016). Iterative Methods for the Elastography Inverse Problem of Locating Tumors. In: Rassias, T., Pardalos, P. (eds) Essays in Mathematics and its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-31338-2_6

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