Estimating Mean and Variance of Random Coefficients in Stochastic Variational Problems Using Second-Order Methods

Driven by the need to identify both deterministic and stochastic coefficients in various stochastic partial differential equations, we have developed an abstract inversion framework. The inverse problem is studied in a stochastic optimization framework. Essential properties of solution maps are derived to prove the solvability of the optimization problems and to establish optimality conditions. A comprehensive regularization framework, including total-variation regularization, has been created to identify rapidly varying coefficients. By using the Bregman distance, we provide new convergence rates for stochastic inverse problems in the abstract formulation without the need for the so-called smallness condition. Assuming finite-dimensional noise, the inverse problem is parameterized and solved using the stochastic Galerkin framework. The numerical schemes utilize Hessian-based optimization methods, resulting in rapid convergence. The numerical results are promising, demonstrating the feasibility and effectiveness of the proposed framework.