We study Tikhonov regularization for possibly nonlinear inverse problems with weighted $\ell^1$-penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an $L^2$-space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted $\ell^2$-norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high H\"older-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method also converges with optimal rates in terms of the noise level with the discrepancy principle as parameter choice rule. Our analysis includes the case that the penalty term is not finite at the exact solution ('oversmoothing'). As a standard example we discuss wavelet regularization in Besov spaces $B^r_{1,1}$. In this setting we demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict improved rates of convergence for piecewise smooth unknown coefficients.

翻译：我们研究Tikhonov 的正规化, 可能是因为加权 $ ell $1$- penal 出现非线性反差问题。 前方操作员, 从一个序列空间向任意的Banach空间, 通常是一个 $2$- space, 从一个序列空间向任意的Banach空间绘图, 通常是一个 $2$- $2$- oorm 和图像空间的规范, 被假定满足一个双面的 Lipschitz 条件, 即加权 $2$- norm 和图像空间的规范。 我们用这个标准例子来说明, 在设定任意高H\"older - type rool rool room r\ $1} 参数的近似率时, 能够实现, 我们用数字模拟来说明一个参数识别问题, 差异方程式的参数识别问题, 我们的理论结果正确预测, 平滑滑度的系数的趋同率的趋同率的趋近率率。