Contingent derivatives and regularization for noncoercive inverse problems

TL;DR

This paper investigates regularization and derivative analysis for non-coercive inverse problems, providing convergence results, derivative computation methods, and numerical validation for improved parameter identification.

Contribution

It introduces a regularization approach that yields a smooth, single-valued parameter-to-solution map and develops derivative formulas for output least-squares objectives in non-coercive problems.

Findings

Regularized problem approximates original optimization problems effectively.

Derived first- and second-order derivative formulas for the output least-squares objective.

Numerical results validate the theoretical derivative computations.

Abstract

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.