SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 1, Page 65-105, January 2021.
We establish a notion of random entropy solution for degenerate fractional conservation laws incorporating randomness in the initial data, convective flux, and diffusive flux. In order to quantify the solution uncertainty, we design a multilevel Monte Carlo finite difference method (MLMC-FDM) to approximate the ensemble average of the random entropy solutions. Furthermore, we analyze the convergence rates for MLMC-FDM and compare them with the convergence rates for the deterministic case. Additionally, we formulate error vs. work estimates for the multilevel estimator. Finally, we present several numerical experiments to demonstrate the efficiency of these schemes and validate the theoretical estimates obtained in this work.

中文翻译：

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随机数据分数守恒律的多级蒙特卡罗有限差分方法

SIAM / ASA不确定性量化杂志，第9卷，第1期，第65-105页，2021年1月。
我们为退化分数守恒定律建立了随机熵解的概念，在原始数据，对流通量和扩散通量中纳入了随机性。为了量化解的不确定性，我们设计了一种多级蒙特卡洛有限差分法（MLMC-FDM）来近似随机熵解的集合平均。此外，我们分析了MLMC-FDM的收敛速度，并将其与确定性情况下的收敛速度进行比较。此外，我们为多级估算器制定了误差与工作估算之间的关系。最后，我们提出了几个数值实验，以证明这些方案的效率，并验证这项工作中获得的理论估计。