带随机性的时间分数阶扩散方程及期望值的高效数值估计
Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values
AI总结：
本文针对带随机扩散系数的时间分数阶扩散方程，提出结合截断随机展开、高阶 quasi-Monte Carlo 方法、二阶时间步进和 Galerkin 有限元的期望值高效估计框架，并分析了误差与正则性。
摘要：
本文研究了一类阶数为 $α\in (0,1)$ 且具有随机扩散系数参数的时间分数阶扩散方程。我们重点关注作用于模型问题解的线性泛函的期望值的高效估计，该期望值被视为与随机系数对应的参数空间上的无穷维积分。为了在计算上估计期望值，需要对随机参数的无穷展开进行截断。然后，我们使用高阶 quasi-Monte Carlo 方法近似随机场上的高维积分。随后，通过二阶精确的时间步进格式结合 Galerkin 有限元空间离散，在时空域上近似确定性解。在给定数据的合理正则性假设下，我们展示了连续解的一些正则性性质，并研究了估计期望值所产生的误差。我们报告了补充理论结果的数值实验。
英文摘要：
In this work, we explore a time-fractional diffusion equation of order $α\in (0,1)$ with a stochastic diffusivity parameter. We focus on efficient estimation of the expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of our model problem. To estimate the expected value computationally, the infinite expansions of the random parameter need to be truncated. Then we approximate the high-dimensional integral over the random field using a high-order quasi-Monte Carlo method. This follows by approximating the deterministic solution over the space-time domain via a second-order accurate time-stepping scheme in combination with a spatial discretization by Galerkin finite elements. Under reasonable regularity assumptions on the given data, we show some regularity properties of the continuous solution and investigate the errors from estimating the expected value. We report on numerical experiments that complement the theoretical results.
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