AI 中文总结
本文提出一种基于重建磨光目标而非精确解的变分磨光方法用于圆解卷积，通过凸变分问题获得显式傅里叶解，并证明在普通光滑核下达到最优代数收敛速率，在超光滑核下达到最优对数速率，数值实验验证了有效性。
AI 中文摘要
我们提出了一种基于重建磨光目标对象而非精确解本身的变分磨光方法，用于处理圆解卷积问题。该方法被表述为一个凸变分问题，其解具有显式的傅里叶表示。我们建立了随着目标分辨率增加所提重建的一致性，并推导了确定性数据扰动的收敛速率。对于普通光滑核，在Besov-Nikolskii光滑性假设下，该方法达到经典阶最优代数收敛速率；而对于超光滑核，则达到相应的阶最优对数速率。在合成数据和风向数据上的数值实验验证了所提方法的有效性，并证实了理论预测。
英文摘要
We propose a variational mollification approach to circular deconvolution based on the reconstruction of a mollified target object rather than the exact solution itself. The method is formulated as a convex variational problem whose solution admits an explicit Fourier representation. We establish the consistency of the proposed reconstruction as the target resolution increases and derive convergence rates for deterministic data perturbations. For ordinary smooth kernels, the method achieves the classical order-optimal algebraic convergence rates under Besov-Nikolskii smoothness assumptions, while for supersmooth kernels it attains the corresponding order-optimal logarithmic rates. Numerical experiments on synthetic and wind-direction data illustrate the effectiveness of the proposed approach and confirm the theoretical predictions.
Comments22 pages, 2 figures
