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 Hölder-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.

我们研究了带有加权\(\ell ^1\)惩罚的可能非线性逆问题的 Tikhonov 正则化。从序列空间映射到任意 Banach 空间（通常为\(L^2\)空间）的前向算子被假定满足关于加权\(\ell ^2\)的双边 Lipschitz 条件-范数和图像空间的范数。我们表明，在这种设置中，可以实现正则化参数中任意高的 Hölder 型阶数的近似率，并且我们表征了获得这些率的序列的最大子空间。在这些子空间上，该方法还在噪声水平方面以最佳速率收敛，并将差异原则作为参数选择规则。我们的分析包括惩罚项在精确解（“过度平滑”）处不是有限的情况。作为一个标准的例子，我们讨论 Besov 空间中的小波正则化\(B^r_{1,1}\). 在这种情况下，我们在微分方程中参数识别问题的数值模拟中证明，我们的理论结果正确预测了分段平滑未知系数的收敛速度的提高。