The problem being analyzed is based on linear elasticity equations that describe
displacement in soft tissue under applied body forces in biomedical applications,
specifically in the case of identifying soft tissue cancers. The primary goal of this work is to develop an adaptive finite element solution framework for the identification of a distributed parameter in a system of partial differential equations where the inverse problem is formulated as an optimization problem. Description of a finite element discretization that fits the optimization framework and stochastic approximation iteration used in the numerical solution is given. We also propose an adaptive mesh refinement framework that provides the resolution needed for the recovery of the spatially varying parameter while improving the computational
efficiency.