In this paper, we propose a new Monte Carlo radiative transport (MCRT) scheme, which is based completely on the Neumann series solution of the Fredholm integral equation. This scheme indicates that the essence of MCRT is the calculation of infinite terms of multiple integrals in the Neumann solution simultaneously. Under this perspective, we redescribe the MCRT procedure systematically, in which the main work amounts to choosing an associated probability distribution function for a set of random variables and the corresponding unbiased estimation functions. We select a relatively optimal estimation procedure that has a lower variance from an infinite number of possible choices, such as term-by-term estimation. In this scheme, MCRT can be regarded as a pure problem of integral evaluation, rather than as the tracing of random-walking photons. Keeping this in mind, one can avert some subtle intuitive mistakes. In addition, the δ functions in these integrals can be eliminated in advance by integrating them out directly. This fact, together with the optimal chosen random variables, can remarkably improve the Monte Carlo (MC) computational efficiency and accuracy, especially in systems with axial or spherical symmetry. An MCRT code, Lemon (Linear integral Equations’ Monte carlo solver based On the Neumann solution; the code is available on the GitHub codebase at https://github.com/yangxiaolinyn/Lemon, and version 2.0 is archived on Zenodo at https://doi.org/10.5281/zenodo.4686355), has been developed completely based on this scheme. Finally, we intend to verify the validation of Lemon; a suite of test problems mainly restricted to a flat spacetime has been reproduced, and the corresponding results are illustrated in detail.

在本文中，我们提出了一种新的蒙特卡罗辐射传输 (MCRT) 方案，该方案完全基于 Fredholm 积分方程的诺依曼级数解。该方案表明MCRT的本质是同时计算Neumann解中多重积分的无穷项。在这个角度下，我们系统地重新描述了 MCRT 程序，其中主要工作相当于为一组随机变量选择相关的概率分布函数和相应的无偏估计函数。我们从无数可能的选择中选择一个相对最优的估计程序，它具有较低的方差，例如逐项估计。在这个方案中，MCRT 可以看作是一个纯粹的积分评估问题，而不是随机游走光子的跟踪。牢记这一点，可以避免一些微妙的直觉错误。除此之外这些积分中的δ函数可以通过直接积分来提前消除。这一事实与最佳选择的随机变量一起，可以显着提高蒙特卡罗 (MC) 计算效率和准确性，尤其是在具有轴对称或球对称的系统中。一个 MCRT 代码 Lemon（基于 Neumann 解的线性积分方程蒙特卡罗求解器；该代码可在 GitHub 代码库 https://github.com/yangxiaolinyn/Lemon 上找到，2.0 版在 Zenodo 上存档，网址为 https： //doi.org/10.5281/zenodo.4686355)，完全基于该方案开发。最后，我们打算验证Lemon的有效性；重现了一套主要限于平坦时空的测试问题，并详细说明了相应的结果。