We present the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm, recently proposed in [Li and Hu, {\it J. Comput. Phys.}, 2021], for efficiently solving subdiffusion equations with heterogeneous coefficients in long time. This algorithm combines the benefits of multiscale methods, which can handle heterogeneity in the spatial domain, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. Our algorithm overcomes the challenge posed by the nonlocality of the fractional derivative in previous parabolic problem work by constructing an auxiliary problem on each coarse temporal subdomain to completely uncouple the temporal variable. We prove the approximation properties of the correction operator and derive a new summation of exponential to generate a single-step time stepping scheme, with the number of terms of $\mathcal{O}(|\log{\tau_f}|^2)$ independent of the final time, where $\tau_f$ is the fine-scale time step size. We establish the convergence rate of our algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step size, and the fine-scale time step size. Finally, we present several numerical tests that demonstrate the effectiveness of our algorithm and validate our theoretical results.

翻译：本文介绍了最近在[Li and Hu, {\it J. Comput. Phys.}, 2021]中提出的基于小波的边缘多尺度Parareal（WEMP）算法，用于高效求解长时间域内具有异质系数的亚扩散方程。该算法结合了多尺度方法（能够处理空间域中的异质性）与Parareal算法（在处理器充足时可加速时间演化问题求解）的优势。我们的算法通过在每个粗时间子域上构造辅助问题以完全解耦时间变量，克服了先前抛物问题研究中分数阶导数非局部性带来的挑战。我们证明了校正算子的逼近性质，并推导出一种新的指数求和形式以生成单步时间推进格式，其项数为$\mathcal{O}(|\log{\tau_f}|^2)$且与终止时间无关，其中$\tau_f$为细尺度时间步长。我们建立了算法收敛率关于空间域网格尺寸、多尺度方法中使用的层级参数、粗尺度时间步长及细尺度时间步长的表达式。最后，我们通过若干数值算例验证了算法的有效性并支撑了理论结果。