$\ell_{1}^{2}-η\ell_{2}^{2}$ sparsity regularization for nonlinear ill-posed problems
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| Format: | Preprint |
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2025
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| author | Li, Long Ding, Liang |
| author_facet | Li, Long Ding, Liang |
| contents | In this study, we investigate the $\left\|\cdot\right\|_{\ell_{1}}^{2}-η\left\|\cdot\right\|_{\ell_{2}}^{2}$ sparsity regularization with $0< η\leq 1$, 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 $0< η<1$, 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 $0<η<1$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_16163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\ell_{1}^{2}-η\ell_{2}^{2}$ sparsity regularization for nonlinear ill-posed problems Li, Long Ding, Liang Numerical Analysis Optimization and Control 47A52 G.1.6 In this study, we investigate the $\left\|\cdot\right\|_{\ell_{1}}^{2}-η\left\|\cdot\right\|_{\ell_{2}}^{2}$ sparsity regularization with $0< η\leq 1$, 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 $0< η<1$, 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 $0<η<1$, 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. |
| title | $\ell_{1}^{2}-η\ell_{2}^{2}$ sparsity regularization for nonlinear ill-posed problems |
| topic | Numerical Analysis Optimization and Control 47A52 G.1.6 |
| url | https://arxiv.org/abs/2508.16163 |