论文标题
部分可观测时空混沌系统的无模型预测
A Monte Carlo Method for 3D Radiative Transfer Equations with Multifractional Singular Kernels
论文作者
论文摘要
我们在这项工作中提出了一种蒙特卡洛法，用于具有不可依赖空间的散射核的三维标量辐射传递方程。这样的内核通常是远程统计特征的解释，例如在湍流大气层，地球物理学和峰值前进状态下的医学成像中出现的情况下出现。与散射横截面可集成的经典情况相反，这会导致非零的平均空闲时间，后者在这里消失。这会根据基于天真正则化的标准蒙特卡洛方法产生数值困难，表现出较大的跳跃强度和增加的计算成本。我们提出了一种基于小跳跃的金融文献启发的方法 - 大跳跃分解，使我们能够有效地对待小型跳跃并减轻计算负担。我们通过数值模拟演示了该方法的性能，并提供了完整的误差分析。多截面术语是指以下事实：散射操作员的高频贡献是具有空间依赖性索引的单位球体上的分数拉普拉斯 - 贝特拉米操作员。
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.
