一种基于傅里叶变换和sinc-高斯采样公式的快速准确数值方法用于对称莱维过程的科尔莫戈罗夫正向方程
A fast and accurate numerical method for the symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula
AI总结：
本文提出了一种基于傅里叶变换的快速准确数值方法，用于求解对称标量莱维过程的科尔莫戈罗夫正向方程，结合非均匀FFT和分数FFT加速计算，并利用sinc-高斯采样公式进行数值不定积分，实现指数收敛性。
摘要：
本文提出了一种基于傅里叶变换的快速准确数值方法，用于求解对称标量莱维过程的科尔莫戈罗夫正向方程。该方法基于Ooura提出的准确傅里叶变换数值公式，结合非均匀快速傅里叶变换（FFT）和分数FFT以加速数值计算。此外，我们提出了一种基于sinc-高斯采样公式的数值不定积分公式，作为该方法的一部分。该积分公式也结合了FFT。因此，所提出方法的所有步骤均使用FFT及其变体完成。所提出的方法允许避免一些非光滑初始条件和数值时间积分的特殊处理。所提出方法获得的数值解在对应精确解没有尖锐尖点的区间上表现出指数收敛性。此外，实际计算时间与理论估计大致一致。
英文摘要：
In this paper, we propose a fast and accurate numerical method based on Fourier transform to solve Kolmogorov forward equations of symmetric scalar Lévy processes. The method is based on the accurate numerical formulas for Fourier transform proposed by Ooura. These formulas are combined with nonuniform fast Fourier transform (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a non-smooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appeared to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.
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