Optimization of fast algorithms for global Quadrature by Expansion using target-specific expansions
Optimization of fast algorithms for global Quadrature by Expansion using target-specific expansions
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DOI：
10.1016/j.jcp.2019.108976
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发表时间：
2018-11
期刊：
影响因子：
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通讯作者：
Matt Wala;A. Klöckner
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文献类型：
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作者：
Matt Wala;A. Klöckner
我们开发了一种算法，用于渐近快速评估接近源几何体和源几何体上的层势，结合几何全局加速 QBX（“GIGAQBX”）和特定于目标的扩展。 GIGAQBX 是一种快速高阶方案，用于基于扩展求积（“QBX”）使用通过快速多极子方法（FMM）形成的局部扩展来评估层势。特定于目标的扩展有助于降低 QBX 局部扩展的形成和评估成本，将三个维度上的相关计算量从 O ((p+ 1) 2) 减少到 O (p+ 1)，与传统扩展相比没有任何精度损失，但会损失扩展系数中的源/目标分离。 GIGAQBX 是一种“全局”QBX 方案，这意味着潜力完全通过扩展靠近或边界上的点来调节。在我们的方案中，这个单一的全局展开被分解为两个部分，分别进行评估：一部分使用特定于目标的展开来合并近场贡献，一部分使用远场贡献的传统球谐展开，注意收敛保证仅存在于两个子展开的总和。相比之下，目标特定扩展最初是作为“本地”QBX 方案的加速机制引入的，其中远场对 QBX 扩展没有贡献。与未修改的 GIGAQBX 算法相比，我们通过可重复的、时间校准的成本模型表明，组合方案可以显着降低计算的近场评估部分的成本。我们通过数值结果证明拉普拉斯和亥姆霍兹核的性能改进来支持我们方案的有效性。
We develop an algorithm for the asymptotically fast evaluation of layer potentials close to and on the source geometry, combining Geometric Global Accelerated QBX (‘GIGAQBX’) and target-specific expansions. GIGAQBX is a fast high-order scheme for evaluation of layer potentials based on Quadrature by Expansion (‘QBX’) using local expansions formed via the Fast Multipole Method (FMM). Target-specific expansions serve to lower the cost of the formation and evaluation of QBX local expansions, reducing the associated computational effort from O ((p+ 1) 2) to O (p+ 1) in three dimensions, without any accuracy loss compared with conventional expansions, but with the loss of source/target separation in the expansion coefficients. GIGAQBX is a ‘global’QBX scheme, meaning that the potential is mediated entirely through expansions for points close to or on the boundary. In our scheme, this single global expansion is decomposed into two parts that are evaluated separately: one part incorporating near-field contributions using target-specific expansions, and one part using conventional spherical harmonic expansions of far-field contributions, noting that convergence guarantees only exist for the sum of the two sub-expansions. By contrast, target-specific expansions were originally introduced as an acceleration mechanism for ‘local’QBX schemes, in which the far-field does not contribute to the QBX expansion. Compared with the unmodified GIGAQBX algorithm, we show through a reproducible, time-calibrated cost model that the combined scheme yields a considerable cost reduction for the near-field evaluation part of the computation. We support the effectiveness of our scheme through numerical results demonstrating performance improvements for Laplace and Helmholtz kernels.