中文摘要：

相比于差分法,δ序列解法是广义密度演化方程一种崭新且有效的数值解法。为进一步完善δ序列解法,该文对其误差进行了系统的分析。首先,通过引入δ序列逼近,将δ序列解法的误差分为两部分;其次,通过简单的数学变换和代换,并结合Taylor展开,给出了第1部分误差的上界估计,并探讨了其与非参数核密度估计法误差的联系与区别;然后,经过相似的推导,给出了第2部分误差估计的上界。基于上述误差估计,指出影响δ序列解法误差的因素主要包括δ函数序列的类型、参数λ的取值、代表点集的选取,为δ序列解法的参数研究奠定了理论基础。

英文摘要：

To compare with the finite difference method, the solution for a generalized density evolution equation （GDEE） via a family of δ sequences is one of the new and effective numerical methods for GDEE. In order to improve the performance and precision of this new solution, the error analysis is carried out systematically. Firstly, by taking the δ sequences approximation of a probability density function （PDF） as the middle quantity, the error of the solution using δ sequences is divided into two parts. Secondly, the upper bound for the first part of the error is obtained by a mathematical transformation, a replacement and a Taylor expansion, and then the relation and difference between the first part of the error and the error from a kernel density estimation is discussed. Thirdly, the upper bound for the second part of the error is also available by a similar deduction. Finally, it can be found from these results for error analysis that the main influence factors include in the type of δ sequences, the value of 2 and the property of the representative point set. And these results present a theoretical basis for the coming parameter analysis of solution using δ sequences.