We propose in this work a Monte Carlo method for three dimensional scalar radiative transfer equations with non-integrable, space-dependent scattering kernels. Such kernels typically account for long-range statistical features, and arise for instance in the context of wave propagation in turbulent atmosphere, geophysics, and medical imaging in the peaked-forward regime. In contrast to the classical case where the scattering cross section is integrable, which results in a non-zero mean free time, the latter here vanishes. This creates numerical difficulties as standard Monte Carlo methods based on a naive regularization exhibit large jump intensities and an increased computational cost. We propose a method inspired by the finance literature based on a small jumps - large jumps decomposition, allowing us to treat the small jumps efficiently and reduce the computational burden. We demonstrate the performance of the approach with numerical simulations and provide a complete error analysis. The multifractional terminology refers to the fact that the high frequency contribution of the scattering operator is a fractional Laplace-Beltrami operator on the unit sphere with space-dependent index.

翻译：本文提出了一种用于三维标量辐射传输方程的蒙特卡洛方法，该方程包含不可积且空间依赖的散射核。此类核通常描述长程统计特征，例如出现在湍流大气中的波传播、地球物理学以及医学成像的尖峰前向模式中。与经典情况下散射截面可积（从而产生非零平均自由时间）不同，这里的平均自由时间趋于零。这带来了数值困难，因为基于朴素正则化的标准蒙特卡洛方法会表现出较大的跳跃强度并增加计算成本。我们受金融文献启发，提出了一种基于小跳跃-大跳跃分解的方法，能够高效处理小跳跃并降低计算负担。我们通过数值模拟展示了该方法的性能，并提供了完整的误差分析。其中“多分形”术语是指散射算子的高频贡献是单位球面上具有空间依赖指数的分数阶拉普拉斯-贝尔特拉米算子。