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An MPI-accelerated Monte Carlo algorithm for estimating the reflectance and transmittance properties of a wind-driven sea surface

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Abstract

The Preisendorfer and Mobley Monte Carlo model is a classical photon-rough sea surface interaction model that is used to study the reflectance and transmittance properties of the wind-driven sea surface. In this paper, we develop a computational framework and the implementation for parallel processing of this model by using Message Passing Interface. We then analyze the correspondence between serial and parallel results and the speedup of the proposed algorithm on a single personal computer and computers cluster. The results indicate that achieved speedup is slightly lower than the linear speedup, and the parallel efficiency is slowly decreased by increasing the number of processes. Compared to a serial code, the relative differences of irradiance are all lower than 2.5%, and the relative differences of radiance are below 2.5% for air-incident source and below 10% for underwater source.

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Acknowledgements

My special thanks to Mobley for the Serial Fortran 77 of code of Preisendorfer and Mobley Monte Carlo (PMMC) model and his enthusiastic help.

Funding

National Natural Science Foundation of China (NSFC) (Grant No. 61871457, 61431010); Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 61621005); and Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 15JK1366).

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  1. School of Physics and Optoelectronic Engineering, Xidian University, 2 South Taibai Road, Xi’an, 710071, China

    Yu Yang & Lixin Guo

  2. School of Science, Xi’an Technological University, 2 Xuefuzhonglu Road, Xi’an, 710021, China

    Yu Yang

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Appendix A

Appendix A

Parallel random number for MPI-accelerated PMMC

Generating random numbers is a very important step in a Monte Carlo simulation. The commonly used random number algorithms are: linear congruential generator, lagged Fibonacci generator, and Mason rotation algorithm, see, e.g., [4, 5]. In PMMC, Mobley used linear congruential generator to generate uniformly distributed random numbers. This random numbers generation method was described, e.g., by Knuth in detail [41]. When the i-th number of a pseudo-random sequence is given, the i + 1st number can be generated from the equation x(i + 1) = (A × x(i) + C) MOD M, where M = 222 = 4,194,304, C = 1731, MOD is the remainder function, and A is the multiplier. The multiplier A and random number x are both represented in double precision as two 11-bit words. The constants are chosen so that the period is the maximum possible, 4,194,304. Mobley also fixed the seeds of random numbers. Every time the last generated random number is called as the new initial value in the linear congruential formula to generate a new random number, a random number sequence is generated. Using this method to generate random numbers, the calculation results of PMMC are the same every time which is convenient for comparison with analytical methods. However, this method cannot be directly used in MPI parallel Monte Carlo simulation, because each process needs different random numbers.

Two schemes can be used to generate random numbers for MPI parallel requirements with multiple processes. One scheme is to use the same random number generator with processes each have different seeds. The second scheme is to use a random number generator specially developed for parallel applications, such as Parallel Mersenne Twister for GPU [42]. For simplicity, the former is adopted in this paper, and the key is to select different seeds for each process. Figure 

Fig. 19
figure 19

Fortran code for subroutine Init_Random_Seed()

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19 provides the Fortran code to show the process of selecting different random number seeds for MPI multiple processes by using random number subroutine, RANDOM_NUMBER(), in Intel Visual Fortran Compiler. Note that the random number generator used in the subroutine, RANDOM_NUMBER (), uses two independent Linear congruential generators which can generate uniformly distributed random numbers between 0 and 1, with a period of 1018.

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Yang, Y., Guo, L. An MPI-accelerated Monte Carlo algorithm for estimating the reflectance and transmittance properties of a wind-driven sea surface. Opt Rev 29, 34–50 (2022). https://doi.org/10.1007/s10043-022-00721-8

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