Iterative methods for the elastography inverse problem of locating tumors

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