We present a new approach to convergence rate results for variational regularization. Avoiding Bregman distances and using image space approximation rates as source conditions we prove a nearly minimax theorem showing that the modulus of continuity is an upper bound on the reconstruction error up to a constant. Applied to Besov space regularization we obtain convergence rate results for 0, 2, q- and 0, p, p-penalties without restrictions on p, q ∈ (1, ∞). Finally we prove equivalence of Hlder-type variational source conditions, bounds on the defect of the Tikhonov functional, and image space approximation rates.

中文翻译：

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基于图像空间逼近率的变分正则化理论

我们提出了一种用于变分正则化的收敛速度结果的新方法。避免 Bregman 距离并使用图像空间逼近率作为源条件，我们证明了一个近似极小极大定理，表明连续性模数是重建误差的上限，直到一个常数。应用于 Besov 空间正则化，我们获得了 0, 2, q - 和 0, p , p -惩罚的收敛速度结果，而没有对p , q ∈ (1, ∞) 的限制。最后，我们证明了 Hlder 型变分源条件、Tikhonov 泛函缺陷的界限和图像空间逼近率的等价性。