{"ID":2879334,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.16163","arxiv_id":"2508.16163","title":"$\\ell_{1}^{2}-η\\ell_{2}^{2}$ sparsity regularization for nonlinear ill-posed problems","abstract":"In this study, we investigate the $\\left\\|\\cdot\\right\\|_{\\ell_{1}}^{2}-η\\left\\|\\cdot\\right\\|_{\\ell_{2}}^{2}$ sparsity regularization with $0\u003c η\\leq 1$, in the context of nonlinear ill-posed inverse problems. We focus on the examination of the well-posedness associated with this regularization approach. Notably, the case where $η=1$ presents weaker theoretical outcomes than $0\u003c η\u003c1$, primarily due to the absence of coercivity and the Radon-Riesz property associated with the regularization term. Under specific conditions pertaining to the nonlinearity of the operator $F$, we establish that every minimizer of the $\\left\\|\\cdot\\right\\|_{\\ell_{1}}^{2}-η\\left\\|\\cdot\\right\\|_{\\ell_{2}}^{2}$ regularization exhibits sparsity. Moreover, for the case where $0\u003cη\u003c1$, we demonstrate convergence rates of $\\mathcal{O}\\left(δ^{1/2}\\right)$ and $\\mathcal{O}\\left(δ\\right)$ for the regularized solution, concerning a sparse exact solution, under differing yet widely accepted conditions related to the nonlinearity of $F$. Additionally, we present the iterative half variation algorithm as an effective method for addressing the $\\left\\|\\cdot\\right\\|_{\\ell_{1}}^{2}-η\\left\\|\\cdot\\right\\|_{\\ell_{2}}^{2}$ regularization in the domain of nonlinear ill-posed equations. Numerical results provided corroborate the effectiveness of the proposed methodology.","short_abstract":"In this study, we investigate the $\\left\\|\\cdot\\right\\|_{\\ell_{1}}^{2}-η\\left\\|\\cdot\\right\\|_{\\ell_{2}}^{2}$ sparsity regularization with $0\u003c η\\leq 1$, in the context of nonlinear ill-posed inverse problems. We focus on the examination of the well-posedness associated with this regularization approach. Notably, the cas...","url_abs":"https://arxiv.org/abs/2508.16163","url_pdf":"https://arxiv.org/pdf/2508.16163v1","authors":"[\"Long Li\",\"Liang Ding\"]","published":"2025-08-22T07:36:44Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"math.OC\"]","methods":"[]","has_code":false}