A Fixed‐Point Discrepancy Approach to Tikhonov Regularization Parameter Selection

Source abstract

This paper presents a unified comparative study of Tikhonov regularization parameter selection for linear ill‐posed problems with both data noise and operator perturbations. We integrate the generalized discrepancy principle (GDP), its fixed‐point formulation (GDP–FP), and the Arnoldi–Neubauer projection approach (PGDP–AN) within a common analytical and numerical framework that explicitly accounts for operator uncertainty and establishes monotone convergence of the fixed‐point iterations under explicitly stated assumptions. Rather than proposing new individual selection rules, the contribution lies in a unified operator‐aware analysis with complete proofs, a quantitative criterion for switching between data‐dominated and operator‐dominated formulations, and a reproducible benchmarking protocol with quantitative stability metrics. Numerical experiments on Baart’s integral equation, Gaussian image deblurring, inverse heat conduction, and inverse Laplace transform problems show that GDP–FP achieves high accuracy for smooth solutions, projection‐based methods provide stability for large‐scale problems, and PGDP–AN offers robustness under operator perturbations. The results clarify the strengths and limitations of each approach, providing a reproducible and theoretically grounded framework for parameter selection in inverse problems.