On existence of a variational regularization parameter under Morozov's discrepancy principle

2025/06/13 by Liang Ding, Long Tu Li, Ding, Liang +5
Mathematics · #Mathematical Approximation and Integration #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2506.11397

Abstract

Morozov's discrepancy principle is commonly adopted in Tikhonov regularization for choosing the regularization parameter. Nevertheless, for a general non-linear inverse problem, the discrepancy ‖F(x_α^δ)-y^δ‖_Y does not depend continuously on α and it is questionable whether there exists a regularization parameter α such that τ₁δ≤ ‖F(x_α^δ)-y^δ‖_Y≤ τ₂ δ (1≤ τ₁<τ₂). In this paper, we prove the existence of α under Morozov's discrepancy principle if τ₂≥ (3+2γ)τ₁, where γ>0 is a parameter in a tangential cone condition for the nonlinear operator F. Furthermore, we present results on the convergence of the regularized solutions under Morozov's discrepancy principle. Numerical results are reported on the efficiency of the proposed approach.

Citations

• Sparse Convex Optimization via Adaptively Regularized Hard Thresholding
• Modern Regularization Methods for Inverse Problems
• Sparsity-Promoting Sensor Selection for Non-linear Measurement Models
• Compressed Sensing with Nonlinear Observations and Related Nonlinear Optimisation Problems
• Sparsity Constrained Nonlinear Optimization: Optimality Conditions and\n Algorithms
• An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

Related

• On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy
• On the discrepancy principle
• Analysis of the discrepancy principle for Tikhonov regularization under low order source conditions
• On saturation of the discrepancy principle for nonlinear Tikhonov regularization in Hilbert spaces
• A discrepancy principle for equations with monotone continuous operators