高阶数值导数的分数阶Tikhonov正则化方法
Fractional Tikhonov Regularization Method for Higher-Order Numerical Derivatives
摘要: 在本文中，我们关注高阶数值导数问题，该问题是不适定的。为了解决这一反问题，我们提出了分数阶Tikhonov正则化方法，用于从一维噪声数据中计算高阶数值导数。本文先用Fourier变换求出问题的精确解，再用分数阶Tikhonov正则化方法构造出问题的正则化解，最后讨论了先验正则化参数选择规则下精确解与正则化近似解的误差估计。
Abstract: In this paper, we focus on the higher-order numerical derivative problem, which is ill-determined. To solve this inverse problem, we propose a fractional Tikhonov regularization method for calculating higher-order numerical derivatives from one-dimensional noisy data. In this paper, the Fourier transform is used first to write the exact solution of the problem, and then the regularization solution of the problem is constructed by fractional Tikhonov regularization method. Finally, the error estimation of the exact solution and regularization approximate solution under the prior regularization parameter selection rules is discussed.
文章引用：张龙. 高阶数值导数的分数阶Tikhonov正则化方法[J]. 理论数学, 2024, 14(7): 53-60. https://doi.org/10.12677/pm.2024.147271

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