A convex inversion framework for identifying parameters in saddle point problems with applications to inverse incompressible elasticity††Dedicated to the blessed memory of Prof. Paul Wenger.

Abstract

This work investigates the elasticity imaging inverse problem of tumor identification in a fully incompressible medium through a family of inverse problems in a nearly incompressible medium. We develop an inversion framework for saddle point problems that goes far beyond the elasticity imaging inverse problem and applies to a wide variety of inverse problems. We introduce a family of convex optimization problems with regularized saddle point problems as the constraint and prove its convergence. We discretize the inverse problem by using the finite element approach and prove the convergence of the discrete problems. We offer formulas for the gradient and the Hessian computation. The outcome of detailed numerical computations, carried out using the tissue phantom data, shows the efficacy of the developed framework.