非光滑数据下分数阶后向Feynman-Kac方程的高阶BDF全离散格式
High-order BDF fully discrete scheme for backward fractional Feynman-Kac equation with nonsmooth data
摘要：
Feynman-Kac方程控制统计可观测函数的分布，在几乎所有学科中都有广泛的应用。在克服了时空耦合的非局部算子和泛函正则性可能较低的挑战后，利用时间上的卷积求积、空间上的有限元法和一些修正项，发展了向后分数阶Feynman-Kac方程的高阶全离散格式.通过系统校正，在时间上实现了高达$6$的高收敛阶，而不会恶化空间上的最优收敛，并且对解没有正则性要求。最后，大量的数值实验验证了高阶格式的有效性。
英文摘要：
The Feynman-Kac equation governs the distribution of the statistical observable -- functional, having wide applications in almost all disciplines. After overcoming challenges from the time-space coupled nonlocal operator and the possible low regularity of functional, this paper develops the high-order fully discrete scheme for the backward fractional Feynman-Kac equation by using backward difference formulas (BDF) convolution quadrature in time, finite element method in space, and some correction terms. With a systematic correction, the high convergence order is achieved up to $6$ in time, without deteriorating the optimal convergence in space and without the regularity requirement on the solution. Finally, the extensive numerical experiments validate the effectiveness of the high-order schemes.
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