First-Order and Second-Order Adjoint Method and Stochastic Approximation for Inverse Problems

We present a unified framework for estimating stochastic parameters in general variational problems. This nonlinear inverse problem is formulated as a stochastic optimization problem using the output least-squares (OLS) objective, which minimizes the discrepancy between observed data and the computed solution. A key challenge in OLS-based formulations is the efficient computation of first- and second-order derivatives of the OLS functional, which depend on the corresponding derivatives of the parameter-to-solution map—often costly and difficult to evaluate, especially in stochastic settings. To address this, we develop a rigorous computational approach based on first- and second-order adjoint methods for inverse problems governed by stochastic variational problems. Specifically, we propose a new first-order adjoint method for computing the gradient of the OLS objective and introduce two novel second-order adjoint methods for Hessian evaluation. A stochastic Galerkin discretization framework is employed, enabling efficient implementation of the adjoint-based derivative computations. Numerical experiments demonstrate the accuracy and efficiency of the proposed computational framework.