A New Convex Inversion Framework for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location

This work presents a thorough theoretical and numerical analysis of the elasticity imaging inverse problem of tumor identification in the soft tissue of the human body. Beyond the obvious merits of its applications, this problem also presents significant mathematical challenges. The near incompressibility inherent in the model of linear elasticity in the body gives rise to the “locking effect” and necessitates a unique treatment of both the direct and inverse problems. A general optimization framework for the identification of parameters in saddle point problems is presented along with a new modified output least-squares (MOLS) objective functional. The MOLS functional is shown to be convex, thus overcoming the nonconvexity of the classical output least-squares (OLS) functional, and the new framework is shown to be capable of accommodating both smooth and discontinuous parameters. Generalized derivative formulas for the coefficient-to-solution map are also given along with a complete convergence analysis. Both continuous and discontinuous numerical examples are provided, together with details regarding discretization, implementation issues, and preliminary performance comparisons with existing methods.