基于GLT分析的全变量系数分数阶演化方程的块预处理方法
Block preconditioning for all-at-once variable-coefficient fractional evolution equations via the GLT analysis
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中文总结 AI 辅助
针对带弱奇异时间核与空间变扩散系数的非局部演化偏微分方程，基于GLT理论开发块下三角预处理策略，结合GMRES求解器可显著提升大规模问题的收敛等性能。
中文摘要 AI 辅助
我们研究一类具有弱奇异时间核和空间变扩散系数的非局部演化偏微分方程，该模型定义在Ω⊂ℝ上，包含左侧Riemann-Liouville空间分数阶导数，其乘以变系数a(x)。时间导数采用L1型格式近似，空间算子通过有限差分技术离散，得到具有两级类Toeplitz结构的大规模全 at once 线性系统。我们开发并分析了一种块下三角策略，该策略模仿系数矩阵的结构，同时简化其组件以提高计算效率。该分析通过广义局部Toeplitz（GLT）理论在矩阵序列层面开展，在该框架内，我们表征了离散算子的渐近谱分布，并利用相关GLT符号指导结构化近似的构建。采用GMRES求解器的数值实验表明，所提出的预处理策略显著提升了大规模问题的收敛速度、鲁棒性和可扩展性。本工作末尾简要讨论了开放问题和可能的扩展方向。
英文摘要
We study a class of nonlocal evolutionary partial differential equations with weakly singular temporal kernel and spatially variable diffusion coefficient. The model is posed on $Ω\subset \mathbb{R}$, and involves a left-sided Riemann--Liouville fractional derivative in space multiplied by a variable coefficient $a(x)$. The temporal derivative is approximated by an $L1$ type scheme, while the spatial operator is discretized by finite difference techniques, resulting in large scale all at once linear systems with a twolevel Toeplitz like structure. We develop and analyze a block lower triangular strategy that mimics the structure of the coefficient matrix while simplifying its components for computational efficiency. The analysis is carried out at the level of matrix sequences by means of generalized locally Toeplitz (GLT) theory. Within this framework, we characterize the asymptotic spectral distribution of the discretized operators and use the associated GLT symbol to guide the construction of the structured approximation. Numerical experiments using the GMRES solver demonstrate that the proposed preconditioning strategy significantly improves convergence rates, robustness, and scalability for large-scale problems. Open problems and possible extensions are briefly discussed at the end of the present work.
