变分不等式与Banach空间上Tikhonov正则化的改进收敛速率
Variational Inequalities and Improved Convergence Rates for Tikhonov Regularisation on Banach Spaces
AI总结：
针对Banach空间上的线性算子方程，利用对偶泛函的变分不等式，推导出Tikhonov型凸正则化在Bregman距离下的高阶收敛速率，突破了传统低阶限制。
摘要：
本文针对Banach空间上的线性算子方程，推导了Tikhonov型凸正则化在Bregman距离下的高阶收敛速率。该方法基于变分不等式的思想，但并非施加于原始Tikhonov泛函，而是施加于对偶泛函。因此，该方法不局限于低阶收敛速率，而是能够获得与Hilbert空间上二次正则化已知的相同范围的速率。
英文摘要：
In this paper we derive higher order convergence rates in terms of the Bregman distance for Tikhonov like convex regularisation for linear operator equations on Banach spaces. The approach is based on the idea of variational inequalities, which are, however, not imposed on the original Tikhonov functional, but rather on a dual functional. Because of that, the approach is not limited to convergence rates of lower order, but yields the same range of rates that is well known for quadratic regularisation on Hilbert spaces.
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