快速多极子方法算子的优化符号代数生成方法
An Optimizing Symbolic Algebra Approach for Generating Fast Multipole Method Operators
摘要：
我们已经开发了一种符号代数方法来自动生成，验证和优化计算机代码的快速多极子方法（FMM）运营商。这种方法允许灵活地选择基集和内核，并可以生成多种语言的任何扩展顺序的计算机代码。该程序在公开可用的Python程序Mosaic中实现。在符号级通过代数操作进行的优化显着减少了数学运算的数量相比，一个简单的实现方程。我们发现，优化器是能够消除20-80%的浮点运算和扩展订单$p \le 10$，它改变了观察到的缩放属性。我们提出了我们的方法，使用三种变体的运营商与笛卡尔基集的谐波潜在的内核$1/r$，包括使用完全对称和无痕多极张量。
英文摘要：
We have developed a symbolic algebra approach to automatically produce, verify, and optimize computer code for the Fast Multipole Method (FMM) operators. This approach allows for flexibility in choosing a basis set and kernel, and can generate computer code for any expansion order in multiple languages. The procedure is implemented in the publicly available Python program Mosaic. Optimizations performed at the symbolic level through algebraic manipulations significantly reduce the number of mathematical operations compared with a straightforward implementation of the equations. We find that the optimizer is able to eliminate 20-80% of the floating-point operations and for the expansion orders $p \le 10$ it changes the observed scaling properties. We present our approach using three variants of the operators with the Cartesian basis set for the harmonic potential kernel $1/r$, including the use of totally symmetric and traceless multipole tensors.
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