检查格林函数的解析结构：使用GPU的大规模并行复积分
Examining the Analytic Structure of Green's Functions: Massive Parallel Complex Integration using GPUs
摘要：
采用图形处理单元（GPU）对量子场论的两点关联函数的解析结构进行数值求解。这些函数通过d维欧几里得动量空间中的积分来表示。这样的积分一般不能解析求解，因此如果需要，必须依靠数值方法来提取它们的解析结构。在描述了相应算法的一般轮廓后，我们通过提供一个在四维中存在精确解的完全工作的例子来演示该过程。我们解决了解析结构的高度并行计算的关联函数动量空间积分在复杂的平面。通过对被积函数应用类似BPHZ的泰勒减法来正则化（几何上）发散积分。我们发现与精确解完全一致。复平面中的每个点都不需要来自其他点的任何信息，这一事实使其成为GPU处理的完美候选者。与顺序执行相比，获得了显著的速度增益。我们还提供了几个GPU上的典型运行时间。
英文摘要：
Graphics Processing Units (GPUs) are employed for a numerical determination of the analytic structure of two-point correlation functions of Quantum Field Theories. These functions are represented through integrals in d-dimensional Euclidean momentum space. Such integrals can in general not be solved analytically, and therefore one has to rely on numerical procedures to extract their analytic structures if needed. After describing the general outline of the corresponding algorithm we demonstrate the procedure by providing a completely worked-out example in four dimensions for which an exact solution exists. We resolve the analytic structure by highly parallel evaluation of the correlation functions momentum space integral in the complex plane. The (logarithmically) divergent integral is regularized by applying a BPHZ-like Taylor subtraction to the integrand. We find perfect agreement with the exact solution. The fact that each point in the complex plane does not need any information from other points makes this a perfect candidate for GPU treatment. A significant gain in speed as compared to sequential execution is obtained. We also provide typical running times on several GPUs.
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