We aim to efficiently compute spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion, and mutation/selection through a fitness function originated in the FK semigroup. We analyze convergence of the algorithm based on operator splitting, present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed.

翻译：我们旨在高效计算在散度为零的随机流中，具有Kolmogorov-Petrovsky-Piskunov（KPP）非线性的反应-扩散-平流（RDA）锋面的传播速度。针对具有空间随机系数的关联线性平流-扩散算子的简化主特征值（李雅普诺夫指数）问题，我们研究了一种随机相互作用粒子方法（IPM）。随机平流场的傅里叶表示和主特征值（李雅普诺夫指数）的Feynman-Kac（FK）公式构成了我们方法的基础，该方法实现为一种遗传进化算法。粒子经历平流-扩散，并通过源自FK半群的适应度函数进行突变和选择。我们基于算子分裂分析了算法的收敛性，并给出了代表性流上的数值结果，例如在随机扰动下的二维细胞流和三维Arnold-Beltrami-Childress（ABC）流。二维算例作为与半拉格朗日计算的一致性检验。三维结果表明，IPM作为无网格且自适应的算法，在平流主导区域中，对于高维无界域上的随机流，无需截断即可简单实现且高效计算锋面传播速度。