Estimating Random Coefficients in Mixed Variational Problems with an Application to the Stochastic Elasticity Imaging Inverse Problem for Tumor Identification

This research investigates the estimation of the mean and variance of stochastic coefficients in mixed variational problems, with an application to the elasticity imaging inverse problem for tumor localization. We study an abstract mixed variational problem, where the coefficients and the right-hand side are strongly measurable functions. By establishing pathwise mixed variational problems, we demonstrate unique solvability for each realization and prove the measurability of pathwise solutions. This result facilitates the derivation of an integral formulation, which is crucial for treating the inverse problem as a stochastic optimization problem, as well as developing a discretization framework for both direct and inverse problems. In Sobolev-Bochner spaces, we establish existence results for integral solutions and obtain basic estimates. We analyze the coefficient-to-solution map for the inverse problem, proving its Lipschitz continuity and characterizing its first- and second-order derivatives. Two optimization approaches are introduced: the classical least-squares and energy minimization, with the latter shown to be convex. A pathwise approach addresses the compact embedding constraint, enabling the integral formulation to be discretized and solved using a stochastic Galerkin scheme. Finally, we provide numerical examples that demonstrate the estimation of the unknown coefficient's mean and variance from data using discrete stochastic methods.