Finding a good regularization parameter for Tikhonov regularization problems is a though yet often asked question. One approach is to use leave-one-out cross-validation scores to indicate the goodness of fit. This utilizes only the noisy function values but, on the downside, comes with a high computational cost. In this paper we present a general approach to shift the main computations from the function in question to the node distribution and, making use of FFT and FFT-like algorithms, even reduce this cost tremendously to the cost of the Tikhonov regularization problem itself. We apply this technique in different settings on the torus, the unit interval, and the two-dimensional sphere. Given that the sampling points satisfy a quadrature rule our algorithm computes the cross-validations scores in floating-point precision. In the cases of arbitrarily scattered nodes we propose an approximating algorithm with the same complexity. Numerical experiments indicate the applicability of our algorithms.

为Tikhonov正则化问题找到一个好的正则化参数是一个虽然很常见但仍被问到的问题。一种方法是使用留一法交叉验证得分来指示拟合优度。这仅利用了嘈杂的函数值，但缺点是计算成本高。在本文中，我们提出了一种将主要计算从所讨论的函数转移到节点分布的通用方法，并且利用FFT和类似FFT的算法，甚至将这一成本大大降低到Tikhonov正则化问题本身的成本。我们在圆环，单位间隔和二维球体的不同设置中应用此技术。给定采样点满足正交规则，我们的算法将以浮点精度计算交叉验证分数。在任意分散的节点的情况下，我们提出了一种具有相同复杂度的近似算法。数值实验表明了我们算法的适用性。