论文标题
$ \ ell^1 $ regulinization的近似率的最大空间
Maximal Spaces for Approximation Rates in $\ell^1$-regularization
论文作者
论文摘要
我们研究了Tikhonov的正则化，可能是加权$ \ ell^1 $ - 二纳尔的非线性反问题。从序列空间到任意的Banach空间，通常假定向前的运算符，通常假定一个$ l^2 $ - 空间可以满足双面Lipschitz条件，相对于加权$ \ ell^2 $ - norm和图像空间的规范。我们表明，在此设置中，可以实现正则化参数中任意高的Hölder-type顺序的近似速率，我们表征了达到这些速率的最大序列子空间。在这些子空间上，该方法还以噪声水平的最佳速率收敛，而差异原理作为参数选择规则。我们的分析包括在精确解决方案（“过度平衡”）中的罚款条款不是有限的。作为标准示例，我们讨论了BESOV空间中的小波正则化$ b^r_ {1,1} $。在这种情况下，我们在数值模拟中证明了在微分方程中的参数识别问题，我们的理论结果正确地预测了分段平滑未知系数的收敛速率。
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.
