图 1  利用蒸馏算法计算关联函数的流程

Fig. 1.  The procedure of computing correlators via distillation method.

图 2  含计算约化的关联函数计算的流程. 其中${\boldsymbol T}^A$和${\boldsymbol T}^B$表示两个中间计算量. 利用中间量的计算减少了总体的计算量, 让计算量从$\propto N_{\rm op}^2\times N_{\rm v}^4$变成$\propto N_{\rm op}\times N_{\rm v}^3$, 极大地减少了计算量

Fig. 2.  The flowchart of computing correlation function. ${\boldsymbol T}^A$ and ${\boldsymbol T}^B$ are two intermidiate quantities. After introducted intermediate quantities, the computation consumption is highly reducted to $\propto N_{\rm op}\times N_{\rm v}^3$.

图 3  按照时间切分实现并行计算的方式, 根据$N_{\rm p}$与$N_{\rm t}$的相对大小, 由于数据的特性, 对τ和Φ按情况采用不同的切分方法.

Fig. 3.  Data sgemented according to time. Two conditions are considered which decided how τ and Φ are treated because of the feature of data.

图 4  使用SIMD优化前后各阶段耗时对比. I/O代表图1中第一步和第二步的时间, Calc.prepare代表图1中第三步的时间, Calc.result代表图1中第四步的时间, Others代表图1中第五步的时间, Init代表程序初始化的时间. 图例SIMD表示启用了AVX形式的SIMD计算性能, 而Complex表示程序直接调用标准库中的复数计算函数(此处未使用SIMD计算). 其中16个MPI进程并行计算的结果是在超线程计算状态下获得

Fig. 4.  The cost of time of program's each part to see the effects of SIMD. I/O labels the time of first step and second step in Fig. 1, Calc.prepare labels the time of the third step in Fig. 1, Calc.result labels the time of the fourth step in Fig. 1, Others labels the time of the fifth step in Fig. 1. Init labels the time of initialization. SIMD in the picture means SIMD optimization was adopted and Complex in the picture means the stdandard library of complex computation was used. And hyper-threading technology was used for 16 MPI process.

图 5  使用SIMD优化前后性能对比. 图例SIMD表示启用了AVX形式的SIMD计算性能, 而Complex表示程序直接调用标准库中的复数计算函数(此处未使用SIMD计算). 其中, 在SIMD启用时16个超线程计算结果未参与数据拟合

Fig. 5.  The cost of time of program's each part to see the effects of SIMD. SIMD in the picture means SIMD optimization was adopted and Complex in the picture means the stdandard library of complex computation was used. And hyper-threading technology was used for 16 MPI process.

图 6  使用OpenMP优化前后耗时对比. 图例如图4. 图例Serial表示串行版本, 即未开启OpenMP多线程和MPI多进程

Fig. 6.  The effects of OpenMP optimization was showed. Legends are the same as 4. Serial lables the results of serial program which means no OpenMP and MPI was adopted.

图 7  MPI并行强扩展性测试. 随着MPI进程数增加, 计算时间成比例减少. 图例如图4

Fig. 7.  MPI parallelism in strong scale tests. The cost time decrease with MPI process numbers. Legends are the same as Fig. 4.

图 8  MPI并行强扩展性测试，不同MPI进程数的测试相对于16个进程的计算效率

Fig. 8.  MPI parallelism in strong scale tests. The efficiency of strong scale tests compared with 16 MPI processes.

图 9  MPI并行弱扩展性测试. $N_{\rm p}$表示使用的并行进程数, $N_{\rm v}$表示本征向量数. 图例说明同图4

Fig. 9.  MPI parallelism in weak scale tests. $N_{\rm p}$ represents the number of process, $N_{\rm v}$ represents the number of eigenvectors. Legends are the same as Fig. 4.

图 10  做变分前的结果. 在时间比较小时, 三个算符得到的有效质量的行为有很大差别, 证明在不同态的投影是不同的, 意味着变分会有一定的效果. 在时间较大时, 三个算符的有效质量趋于同一平台, 说明它们的量子数是相同的, 可以用来变分. traditional method表示第一个算符通过传统方法所得到的有效质量, 作为蒸馏算法的参照

Fig. 10.  Results before variation. The behaviors of the effective mass of these three operators are very different and it means variational analysis would give good results. When time is large enough, these three operators approach to the same plateau so that they should have the same quantum numbers. traditial method label the effective mass of first operator throgh traditional mehtod, which can is matched with distillation method.

图 11  做变分后的结果

Fig. 11.  Results after variation.