论文标题
美元
$α\ell_{1}-β\ell_{2}$ sparsity regularization for nonlinear ill-posed problems
论文作者
论文摘要
在本文中，我们考虑$α\ | \ cdot \ | _ {\ ell_1}-β\ | \ cdot \ | _ {\ ell_2} $ sparsity正则用参数$α\geqβ\ geq0 $用于非线性逆问题。我们研究了正则化的良好性。与$α>β\ geq0 $的情况相比，由于缺乏正规化项的强制性和radon-riesz特性，$α=β\ geq0 $的结果较弱。在$ f $的非线性的某些条件下，我们证明$α\ |的每个最小化器\ cdot \ | _ {\ ell_1}-β\ | \ cdot \ | _ {\ ell_2} $正则化稀疏。对于$α>β\ geq0 $，如果精确解决方案稀疏，我们将收敛速率$ o（δ^{\ frac {\ frac {1} {2}}）$和$ o（δ）$（Δ）$分别在$ f $的非线性条件下在两个常规条件下的正则化解决方案。特别是，可以使用迭代软阈值算法来解决$α\ |。 \ cdot \ | _ {\ ell_1}-β\ | \ cdot \ | _ {\ ell_2} $正规化问题的正则化问题。数值结果说明了所提出的方法的效率。
In this paper, we consider the $α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ sparsity regularization with parameter $α\geqβ\geq0$ for nonlinear ill-posed inverse problems. We investigate the well-posedness of the regularization. Compared to the case where $α>β\geq0$, the results for the case $α=β\geq0$ are weaker due to the lack of coercivity and Radon-Riesz property of the regularization term. Under certain condition on the nonlinearity of $F$, we prove that every minimizer of $ α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ regularization is sparse. For the case $α>β\geq0$, if the exact solution is sparse, we derive convergence rate $O(δ^{\frac{1}{2}})$ and $O(δ)$ of the regularized solution under two commonly adopted conditions on the nonlinearity of $F$, respectively. In particular, it is shown that the iterative soft thresholding algorithm can be utilized to solve the $ α\| \cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ regularization problem for nonlinear ill-posed equations. Numerical results illustrate the efficiency of the proposed method.
