Maximal Spaces for Approximation Rates in 𝓁1-regularization
Maximal Spaces for Approximation Rates in 𝓁1-regularization
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发表时间：
2020-05
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通讯作者：
Philip Miller;T. Hohage
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作者：
Philip Miller;T. Hohage
研究了具有加权$\ well ^1$-惩罚的可能非线性逆问题的Tikhonov正则化问题。正向算子，从一个序列空间映射到任意的巴拿赫空间，通常是一个L^2$-空间，假设满足关于加权的L^2$-范数和象空间范数的双边Lipschitz条件。我们证明了在这种情况下，正则化参数的近似率可以达到任意高的holder型阶，并且我们描述了达到这些率的序列的最大子空间。在这些子空间上，该方法以差异原则作为参数选择规则，以噪声级为参数，以最优速率收敛。我们的分析包括惩罚项在精确解处不是有限的情况（“过度平滑”）。作为一个标准例子，我们讨论了Besov空间$B^r_{1,1}$中的小波正则化。
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 satisfies 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 Holder-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 $B^r_{1,1}$.