分数阶拉普拉斯模型在非均匀网格上的快速配置方法的数值分析与高效实现
Numerical analysis and efficient implementation of fast collocation methods for fractional Laplacian model on nonuniform grids
AI总结：
针对以奇异积分形式呈现的分数阶拉普拉斯问题，提出基于Krylov子空间迭代求解器的通用非均匀网格快速配置方法，证明其在α∈(0,1)时唯一可解，还开发均匀网格改进方案，通过快速矩阵-向量乘法和带状预条件子提升效率，数值实验验证收敛性。
摘要：
针对分数阶拉普拉斯问题，我们提出一种基于Krylov子空间迭代求解器的通用非均匀网格快速配置方法，其中分数阶算子以奇异积分形式呈现。当指数和（SOE）近似足够精确时，该方法在α∈(0,1)的通用非均匀网格上被证明是唯一可解的。此外，我们开发了一种改进方案，证明其在α∈(0,2)的均匀网格上唯一可解。我们从计算复杂度和内存需求方面详细讨论了基于快速矩阵-向量乘法的所提快速配置方案的高效实现。为进一步提升计算效率，我们将带状预条件子融入Krylov子空间迭代求解器。我们在特定梯度网格上对α∈(0,1)进行了严格的最大范数误差分析，结果表明收敛阶取决于梯度参数。数值实验验证了预测的收敛性，并证明了所提快速配置方案的效率。
英文摘要：
We propose a fast collocation method based on Krylov subspace iterative solver on general nonuniform grids for the fractional Laplacian problem, in which the fractional operator is presented in a singular integral formulation. The method is proved to be uniquely solvable on general nonuniform grids for $α\in(0,1)$, provided that the sum-of-exponentials (SOE) approximation is sufficiently accurate. In addition, a modified scheme is developed and proved to be uniquely solvable on uniform grids for $α\in(0,2)$. Efficient implementation of the proposed fast collocation schemes based on fast matrix-vector multiplication is carefully discussed, in terms of computational complexity and memory requirement. To further improve computational efficiency, a banded preconditioner is incorporated into the Krylov subspace iterative solver. A rigorous maximum-norm error analysis for $α\in(0,1)$ is presented on specific graded grids, which shows that the convergence order depends on the grading parameter. Numerical experiments validate the predicted convergence and demonstrate the efficiency of the fast collocation schemes.
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