Stable numerical differentiation for the second order derivatives
Stable numerical differentiation for the second order derivatives
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DOI：
10.1007/s10444-009-9132-9
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发表时间：
2010-11
影响因子：
1.7
通讯作者：
Huilin Xu;Jijun Liu
中科院分区：
文献类型：
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作者：
Huilin Xu;Jijun Liu
考虑一个数值微分问题，其目的是根据给定的噪声数据稳定地计算函数的二阶导数。对于这个病态问题，我们通过将微分问题重新表述为第一类积分方程来引入 Lavrent'ev 正则化方案。该方案的优点是可以通过显式积分表达式给出正则化解，因此易于实现。考虑正则化参数的先验后验策略，基于积分算子分解对噪声数据的正则化解进行收敛分析和误差估计。几个数值例子证明了所提出方案的有效性。
Consider a numerical differential problem, which aims to compute the second order derivative of a function stably from its given noisy data. For this ill-posed problem, we introduce the Lavrent′ev regularization scheme by reformulating this differentiation problem as an integral equation of the first kind. The advantage of this proposed scheme is that we can give the regularizing solution by an explicit integral expression, therefore it is easy to be implemented. Thea-priorianda-posteriorchoice strategies for the regularization parameter are considered, with convergence analysis and error estimate of the regularizing solution for noisy data based on the integral operator decomposition. The validity of the proposed scheme is shown by several numerical examples.