Title: An Optimizing Symbolic Algebra Approach for Generating Fast Multipole Method Operators Show Chinese title

Title: 一种用于生成快速多极方法算子的优化符号代数方法

Abstract: 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. Show Chinese abstract

Abstract: 我们开发了一种符号代数方法，用于自动生成、验证和优化快速多极方法（FMM）算子的计算机代码。 这种方法允许选择不同的基集和核函数，并能生成多种语言的任意展开阶数的计算机代码。 该过程在公开可用的Python程序Mosaic中实现。 通过代数运算在符号层面上进行的优化，与直接实现方程相比，显著减少了数学运算的数量。 我们发现优化器能够消除20-80%的浮点运算，并且对于展开阶数$p \le 10$，它改变了观察到的缩放特性。 我们使用三种算子变体，采用笛卡尔基集对于调和势核$1/r$，包括完全对称和无迹多极张量的使用。