2025/06/13 by Long Li, Liang Ding, Li, Long +1
Engineering · Earth and Planetary Sciences · Mathematics · #Sparse and Compressive Sensing Techniques #Seismic Imaging and Inversion Techniques #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2506.11372
This paper presents a regularization technique incorporating a non-convex and non-smooth term, ℓ12-ηℓ22, with parameters 0<η≤ 1 designed to address ill-posed linear problems that yield sparse solutions. We explore the existence, stability, and convergence of the regularized solution, demonstrating that the ℓ12-ηℓ22 regularization is well-posed and results in sparse solutions. Under suitable source conditions, we establish a convergence rate of O(δ) in the ℓ2-norm for both a priori and a posteriori parameter choice rules. Additionally, we propose and analyze a numerical algorithm based on a half-variation iterative strategy combined with the proximal gradient method. We prove convergence despite the regularization term being non-smooth and non-convex. The algorithm features a straightforward structure, facilitating implementation. Furthermore, we propose a projected gradient iterative strategy base on surrogate function approach to achieve faster solving. Experimentally, we demonstrate visible improvements of ℓ12-ηℓ22 over ℓ1, ℓ1-ηℓ2, and other nonconvex regularizations for compressive sensing and image deblurring problems. All the numerical results show the efficiency of our proposed approach.