This paper introduces a regularized, structure-exploiting Powell-Symmetric-Broyden (RSE-PSB) method under modified secant conditions for solving ill-posed inverse problems in both infinite dimensional and finite dimensional settings.
The approximation of the symmetric, yet potentially indefinite, second-order term, which is neglected by standard Levenberg-Marquardt (LM) approaches, integrates regularization and structure exploitation directly within the Quasi-Newton (QN) framework, leveraging the strengths of QN and LM methods, Tikhonov-type regularization, and structure exploitation.
We establish local Q-linear convergence via the bounded deterioration principle and prove local Q-super-linear convergence under the assumption that the initial error is a Hilbert-Schmidt operator.
Furthermore, we present a globalization strategy in the discretized setting based on the dynamic control of the regularization parameter. Hence, this approach stabilizes the ill-posed problem while ensuring global convergence, addressing even the choice of an appropriate regularization parameter.
Finally, we discuss a numerical example based on a PDE-driven parameter identification problem in piezoelectricity, relevant to industrial sensor and actuator applications.