[논문]Just-In-Time 컴파일러를 이용한 파이썬 기반 지구동역학 코드 가속화 연구

박상진; 안수정; 소병달

doi:10.7582/gge.2021.24.2.35

[국내논문] Just-In-Time 컴파일러를 이용한 파이썬 기반 지구동역학 코드 가속화 연구
Boosting the Performance of Python-based Geodynamic Code using the Just-In-Time Compiler 원문보기

지구물리와 물리탐사 = Geophysics and geophysical exploration, v.24 no.2, 2021년, pp.35 - 44

박상진 (강원대학교 지구물리학과) , 안수정 (강원대학교 지구물리학과) , 소병달 (강원대학교 지구물리학과)

초록
AI-Helper

파이썬은 다른 정적 언어(예, C, C++, FORTRAN 등)에 비해 실행 속도가 느리기 때문에 대규모 반복이 필요한 지구동역학 코드를 작성하는데 적합하지 않은 것으로 인식되어 왔다. 그러나 파이썬의 계산 속도를 향상시키기 위해 Just-In-Time (JIT) 컴파일 등과 같은 많은 고속화 수단이 개발되었다. 우리는 파이썬을 기반으로 JIT 컴파일러에 최적화된 맨틀 유동 지구동역학 코드를 개발했다. 코드는 지구동역학 분야에서 널리 사용되는 PIC (Particle-In-Cell) 방법과 유한요소법을 결합하여 맨틀 대류를 수치 모사하며, 코드의 신뢰성을 정량적으로 평가하기 위해 잘 알려진 2차원 맨틀 대류 문제를 벤치마킹했다. 수치 모사 결과, 제곱근 평균 제곱 속도와 넛셀 수가 이전 연구와 거의 일치함을 확인했다. JIT 컴파일러를 적용한 코드는 적용하지 않았을 경우와 대비해 계산 속도가 PIC 방법에서 최대 258배, 전체 행렬 조립 과정에서 최대 30배 향상했다. 따라서, 이번 연구는 파이썬의 계산 성능이 JIT 등의 가속기를 이용하여 충분히 향상되며, 많은 지구 동역학 문제를 해결하는데 활용될 수 있음을 제시하였다.

Abstract ▼ AI-Helper

As the execution speed of Python is slower than those of other programming languages (e.g., C, C++, and FORTRAN), Python is not considered to be efficient for writing numerical geodynamic code that requires numerous iterations. Recently, many computational techniques, such as the Just-In-Time (JIT) compiler, have been developed to enhance the calculation speed of Python. Here, we developed two-dimensional (2D) numerical geodynamic code that was optimized for the JIT compiler, based on Python. Our code simulates mantle convection by combining the Particle-In-Cell (PIC) scheme and the finite element method (FEM), which are both commonly used in geodynamic modeling. We benchmarked well-known mantle convection problems to evaluate the reliability of our code, which confirmed that the root mean square velocity and Nusselt number obtained from our numerical modeling were consistent with those of the mantle convection problems. The matrix assembly and PIC processes in our code, when run with the JIT compiler, successfully achieved a speed-up 30× and 258× faster than without the JIT compiler, respectively. Our Python-based FEM-PIC code shows the high potential of Python for geodynamic modeling cases that require complex computations.

Keyword

표/그림 (11)

그림 Fig. 1. The Python script using the JIT compiler on a user-defined function. The "mean()" function receives a float array as an argument and returns the arithmetic mean value of the array. Signatures, which declare an object type of an argument and compute the return values, are embedded behind the decorator (@jit). When the JIT compiler operates with signatures ahead of the user-defined function, the compiling time can be greatly reduced.
그림 Fig. 2. Schematic diagram of the interaction between grids and particles. The black dots, red dots, and black squares denote nodes of an element, a particle, and the physical properties of elements, respectively. (a) In the first stage, the linear equation is solved using grid-based numerical techniques (e.g., finite element methods in our study, finite differences, etc.). (b) In the second stage, the values at each node are interpolated (e.g., temperature, velocity, etc.) to the particle. (c) In the third stage, particles are advected by the interpolated velocity and given a time increment (Δt). If a particle advects outside of the calculation domain, it is removed. (d) In the last stage, the properties of particles (e.g., viscosity, density, etc.) at the new location are transferred to a grid.
그림 Fig. 3. Assembly of a global matrix using local matrices. (a) A local matrix is obtained from each element. (b) A set of information, composed of the row, column, and value, are derived from the local matrices. In this process, overlapped values are eliminated. (c) The visualized global matrix. As most of the elements are zero, the global matrix is converted into a memory-friendly data type using a Scipy module in the process from (b) to (c).
그림 Fig. 4. Model setup of 2D mantle convection from Blankenbach et al. (1989). The non-dimensional temperature boundary conditions are T(x, 0) = 1 and T(x, 1) = 0. The side walls of the model are insulated. All of the mechanical boundary conditions are free slip. The dashed lines indicate the isotherm (contour interval of 0.2). The convection cell occurs due to the temperature difference between the upper and lower regions.
그림 Fig. 5. Model setup of 2D mechanical convection from van Keken et al. (1997). The side walls are defined as free slip. The upper and lower boundaries are characterized by no-slip conditions. The material compositions of the upper and lower regions indicate higher and lower densities, respectively. The viscosity contrast between two layers is fixed at 1.
그림 Fig. 6. (a) Temperature field of a convection cell that is completely converged at t = 0.25. There are hot upwelling and cold downwelling zones along the left and right walls, respectively. (b) Comparison of V_rms values derived from our model (solid red line) and from Blankenbach et al. (1989) (dashed black line) over time.
표 Table 1. Comparison of obtained V_rms and Nusselt number values with those of previous studies (Blankenbach et al., 1989; Tackley and King, 2003; Thieulot, 2014; King, 2009). All results were measured after the convection cell converged
그림 Fig. 7. (a) The V_rms evolution of mechanical convection over time. The two peaks occurred at t = 210 and t = 876. PvK (P.E. van Keken), CND (Christensen, Neumesiter and Doin), SK (S.D. King), and HS (H. Schmeling) indicate research groups that participated in the study of van Keken et al. (1997). (b), (c), and (d) indicate particle distributions at t = 0.25, t = 700, and t = 876, respectively. The colors of particles in (b), (c), and (d) denote the composition.
그림 Fig. 8. The results obtained when JIT was applied to a function that calculates the average properties of an element. The y-axis indicates the speed-up achieved with the JIT compiler, compared to that obtained without it. The x-axis shows the loop iterations (number of elements) in the function. Note that the maximum calculation speed was reached in a very small number of loop iterations when the signature was applied to JIT.
그림 Fig. 9. The execution time per step of the global matrix assembly under different grid resolutions. The blue and red solid lines refer to one million degrees of freedom (DOF) for Stokes and the energy global matrix, respectively. The horizontal black solid line indicates the 1s calculation time.
그림 Fig. 10. The execution time per step with increasing numbers of particles. The total number of particles in each case was obtained by multiplying the number of particles in an element by the number of elements. The red and blue lines indicate the execution time per step with and without the JIT compiler, respectively.

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