First and second order numerical differentiation with Tikhonov regularization
First and second order numerical differentiation with Tikhonov regularization
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DOI：
10.1007/s11464-006-0014-x
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发表时间：
2006
影响因子：
--
通讯作者：
Shuai Lu;Yanbo Wang
中科院分区：
文献类型：
--
作者：
Shuai Lu;Yanbo Wang
本文研究了未知光滑函数的数值微分问题，该函数在给定集合上的数据是可用的。数值微分是一个不适定问题。本文采用Tikhonov正则化方法对光滑函数的一阶导数和二阶导数进行近似。证明了近似函数可以作为代价泛函的最小值。建立了最小器的存在唯一性理论。得到光滑未知函数与近似函数之间的导数误差，这取决于网格的网格大小和数据中的噪声水平。数值结果支持了本文的理论分析。
This work deals with the numerical differentiation for an unknown smooth function whose data on a given set are available. The numerical differentiation is an ill-posed problem. In this work, the first and second derivatives of the smooth function are approximated by using the Tikhonov regularization method. It is proved that the approximate function can be chosen as a minimizer to a cost functional. The existence and uniqueness theory of the minimizer is established. Errors in the derivatives between the smooth unknown function and the approximate function are obtained, which depend on the mesh size of the grid and the noise level in the data. The numerical results are provided to support the theoretical analysis of this work.