圆盘上泊松方程的快速非均匀求解器
A Fast Nonuniform Solver for the Poisson Equation over a Disk
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中文总结 AI 辅助
本研究提出圆盘上泊松方程的非均匀FFTRR求解器，支持径向和方位角非均匀网格，保持谱精度并利用GPU加速，性能与现有快速求解器相当。
中文摘要 AI 辅助
我们在FFTRR（快速傅里叶变换径向递推）框架内研究圆盘上泊松方程的快速数值方法，该框架基于格林函数表示。经典FFTRR方案在方位角变量上应用FFT，并逐模式评估径向递推，在N×M均匀网格上实现O(MNlogN)的复杂度。然而，它们要求N个点的均匀方位角网格。在本工作中，我们开发了一种非均匀（NUFFTRR）求解器，允许在径向和方位角方向上采用非均匀网格，同时保留原始FFTRR公式的有利结构。方位角分析-综合步骤使用密集NUDFT最小二乘求解器或两种基于NUFFT的迭代方案之一实现：使用循环预处理共轭梯度（PCG）的Toeplitz求解器，以及使用Pipe-Menon密度补偿的预处理共轭梯度最小二乘（PCGLS）求解器。这些在N×M网格和Krylov迭代次数K_iter下，分别产生NUDFT变体的方位角复杂度O(N^3 + MN^2)和基于NUFFT变体的O(K_iter MNlogN)。在非均匀网格上的数值实验表明，所提出的方法在方位角变量上具有鲁棒性、谱精度，并且与现有快速泊松求解器相比在运行时间上具有竞争力。我们利用向量化和批量BLAS/GPU加速线性代数运算来消除显式循环，同时将方位角和径向步骤完全表述为密集数组运算、FFT和NUFFT，从而允许直接进行GPU加速。该实现作为开源Python包在此https URL发布，其方法可直接扩展到相关椭圆问题，如亥姆霍兹方程。
英文摘要
We study fast numerical methods for the Poisson equation on a disk within the FFTRR (Fast Fourier Transform Radial Recurrence) framework, which is built on Green's function representations. Classical FFTRR schemes apply FFTs in the azimuthal variable and evaluate mode-by-mode radial recurrences, achieving a complexity of \(O(MN\log N)\) on an \(N\times M\) uniform grid. However, they require a uniformly spaced azimuthal mesh of \(N\) points. In this work we develop a Nonuniform (NUFFTRR) solver that admits nonuniform grids in both the radial and azimuthal directions while retaining the favorable structure of the original FFTRR formulation. The azimuthal analysis-synthesis step is implemented using either a dense NUDFT least-squares solver or one of two NUFFT-based iterative schemes: a Toeplitz solver using circulant-preconditioned conjugate gradients (PCG), and a preconditioned conjugate gradient for least squares (PCGLS) solver with Pipe--Menon density compensation. These yield azimuthal complexities \(O(N^3 + MN^2)\) for the NUDFT variant and \(O(K_{\mathrm{iter}} MN\log N)\) for the NUFFT-based variants on an \(N\times M\) grid and Krylov iteration count \(K_{\mathrm{iter}}\). Numerical experiments on nonuniform meshes demonstrate that the proposed method is robust, spectrally accurate in the azimuthal variable, and competitive in runtime with existing fast Poisson solvers. We make use of vectorization and batched BLAS/GPU-accelerated linear algebra operations to eliminate explicit loops while also formulating azimuthal and radial steps entirely in terms of dense array operations, FFTs, and NUFFTs, allowing for straightforward GPU acceleration. The implementation is released as an open-source Python package at https://github.com/CharliePyle4/NUFFTRR_Poisson, and its methodology can be extended directly to related elliptic problems such as the Helmholtz equation.
发表机构
- Texas A&M University（德克萨斯农工大学）
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