Contingent derivatives and regularization for noncoercive inverse problems
This work provides a rigorous mathematical foundation for solving non-coercive inverse problems, which are common in applied models, but the contribution is incremental as it extends existing derivative-based methods to a specific class of problems.
The paper develops a theoretical framework for parameter identification in non-coercive inverse problems using contingent derivatives and regularization, proving convergence of regularized least-squares objectives to the original problem and providing numerical results.
We study the inverse problem of parameter identification in non-coercive variational problems that commonly appear in applied models. We examine the differentiability of the set-valued parameter-to-solution map by using the first-order and the second-order contingent derivatives. We explore the inverse problem by using the output least-squares and the modified output least-squares objectives. By regularizing the non-coercive variational problem, we obtain a single-valued regularized parameter-to-solution map and investigate its smoothness and boundedness. We also consider optimization problems using the output least-squares and the modified output least-squares objectives for the regularized variational problem. We give a complete convergence analysis showing that for the output least-squares and the modified output least-squares, the regularized minimization problems approximate the original optimization problems suitably. We also provide the first-order and the second-order adjoint method for the computation of the first-order and the second-order derivatives of the output least-squares objective. We provide discrete formulas for the gradient and the Hessian calculation and present numerical results.