We study an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient to obtain the rate function is the computation of the principal eigenvalue $\lambda$ of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size, and we show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, suitable for the IPM. The IPM applies naturally to problems in unbounded domains, scales easily to high dimensions, and adapts to singular behaviors in the vanishing-noise limit. We show numerical examples in dimensions up to 16. The numerical results show that our numerical approximation of $\lambda$ converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. Our paper appears to be the first one to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions.

翻译：我们研究了一种交互粒子方法（IPM），用于计算扩散过程熵产生的大偏差率函数，重点关注小噪声极限和高维情形。获取该率函数的关键在于计算椭圆非自伴算子的主特征值$\lambda$。我们证明该主特征值可通过离散演化算子的谱半径来逼近，该离散算子由算子分裂格式和具有小步长的Euler--Maruyama格式构造；并证明此谱半径可通过该离散半群的大量迭代获得，这恰好适用于IPM。IPM天然适用于无界区域问题，易于扩展至高维，并能适应小噪声极限下的奇异行为。我们展示了维度高达16的数值算例。数值结果表明，在固定粒子数和固定时间步长下，我们对$\lambda$的数值逼近在视觉容差范围内收敛于解析的小噪声极限。本文似乎是首篇在如此高维度下获得非自伴算子主特征值问题数值结果的研究。