In this paper, we study a parallel-in-time (PinT) algorithm for all-at-once system from a non-local evolutionary equation with weakly singular kernel where the temporal term involves a non-local convolution with a weakly singular kernel and the spatial term is the usual Laplacian operator with variable coefficients. Such a problem has been intensively studied in recent years thanks to the numerous real world applications. However, due to the non-local property of the time evolution, solving the equation in PinT manner is difficult. We propose to use a two-sided preconditioning technique for the all-at-once discretization of the equation. Our preconditioner is constructed by replacing the variable diffusion coefficients with a constant coefficient to obtain a constant-coefficient all-at-once matrix. We split a square root of the constant Laplacian operator out of the constant-coefficient all-at-once matrix as a right preconditioner and take the remaining part as a left preconditioner, which constitutes our two-sided preconditioning. Exploiting the diagonalizability of the constant-Laplacian matrix and the triangular Toeplitz structure of the temporal discretization matrix, we obtain efficient representations of inverses of the right and the left preconditioners, because of which the iterative solution can be fast updated in a PinT manner. Theoretically, the condition number of the two-sided preconditioned matrix is proven to be uniformly bounded by a constant independent of the matrix size. To the best of our knowledge, for the non-local evolutionary equation with variable coefficients, this is the first attempt to develop a PinT preconditioning technique that has fast and exact implementation and that the corresponding preconditioned system has a uniformly bounded condition number. Numerical results are reported to confirm the efficiency of the proposed two-sided preconditioning technique.

在本文中，我们从具有弱奇异核的非局部演化方程（其中时间项涉及具有弱奇异核的非局部卷积）研究了一次全部系统的实时并行（PinT）算法。空间项是系数可变的通常的拉普拉斯算子。近年来，由于许多现实世界的应用，对这一问题进行了深入研究。但是，由于时间演化的非局部性，很难以PinT方式求解方程。我们建议使用双面预处理技术对方程进行一次全部离散化。我们的预处理器是通过将可变扩散系数替换为恒定系数来构造的，从而获得恒定系数的一次性矩阵。我们从常数系数全部一次矩阵中分离出常数Laplacian算子的平方根作为右预处理器，并将其余部分作为左预处理器，这构成了我们的两面预处理。利用常数拉普拉斯矩阵的对角化性和时间离散化矩阵的三角Toeplitz结构，我们获得了左右前置条件的逆的有效表示，因此可以以PinT方式快速更新迭代解。从理论上讲，双面预处理矩阵的条件数被证明由一个常数均匀地限制，而该常数与矩阵的大小无关。据我们所知，对于具有可变系数的非局部演化方程，这是开发PinT预处理技术的首次尝试，该技术具有快速准确的实现方式，并且相应的预处理系统具有统一的有界条件数。报告了数值结果，证实了所提出的双面预处理技术的效率。