Gradient and Extragradient Methods for the Elasticity Imaging Inverse Problem Using an Equation Error Formulation: A Comparative Numerical Study

Abstract. This work aims to compare the performance of a variety of gradient-based and extragradient methods for the solution of the elastography inverse problem arising in the identification of cancerous tumors. From a mathematical standpoint, this inverse problem requires the identification of a variable parameter in a system of incompressible elasticity. We use an equation error approach to formulate the inverse problem as a convex optimization problem. The necessary and sufficient optimality condition then leads to a variational inequality which is solved using various extragradient methods, which have received great attention in recent years. 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 projected gradient method, fast projected gradient method (fast iterative shrinkage-thresholding (FISTA)) [3], scaled projected gradient method, and several extragradient methods including the Marcotte variants, He-Goldstein-type method, the projection-contraction method proposed by Solodov and Tseng, and a hyperplane method.