基于张量分解的FMM-FFT加速SIE模拟器平移算子张量压缩
On the Compression of Translation Operator Tensors in FMM-FFT-Accelerated SIE Simulators via Tensor Decompositions
摘要：
提出了张量分解方法，以减少快速多极子方法-快速傅里叶变换（FMM-FFT）-加速表面积分方程（SIE）模拟器中平移算子张量的内存需求。这些方法利用Tucker、分层Tucker（H-Tucker）和张量训练（TT）分解来压缩以三维（3D）和四维（4D）阵列格式存储的FFT变换算子张量。进行了大量的数值试验，以证明不同的模拟参数，这些方法所带来的内存节省和计算开销。数值结果表明，基于H-Tucker的方法为4D阵列格式产生最大的内存节省，而基于Tucker的方法为3D阵列格式引入了最小的计算开销。对于许多实际场景，所有方法都显著降低了翻译算子张量的内存需求，同时施加了可忽略/可接受的计算开销。
英文摘要：
Tensor decomposition methodologies are proposed to reduce the memory requirement of translation operator tensors arising in the fast multipole method-fast Fourier transform (FMM-FFT)-accelerated surface integral equation (SIE) simulators. These methodologies leverage Tucker, hierarchical Tucker (H-Tucker), and tensor train (TT) decompositions to compress the FFT'ed translation operator tensors stored in three-dimensional (3D) and four-dimensional (4D) array formats. Extensive numerical tests are performed to demonstrate the memory saving achieved by and computational overhead introduced by these methodologies for different simulation parameters. Numerical results show that the H-Tucker-based methodology for 4D array format yields the maximum memory saving while Tucker-based methodology for 3D array format introduces the minimum computational overhead. For many practical scenarios, all methodologies yield a significant reduction in the memory requirement of translation operator tensors while imposing negligible/acceptable computational overhead.
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