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：我们研究了一种用于计算扩散过程熵产生的大偏差率函数的相互作用粒子方法（IPM），重点关注噪声消失极限和高维情形。获取率函数的关键在于计算椭圆型非自伴算子的主特征值λ。我们证明该主特征值可通过算子分裂格式和欧拉-丸山格式离散化的演化算子的谱半径进行近似，其中采用小时间步长；并证明该谱半径可通过离散半群的大量迭代来获取，适用于IPM。IPM天然适用于无界域问题，易于扩展至高维，并能适应噪声消失极限中的奇异行为。我们展示了维数高达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，λ的数值逼近在视觉容忍度内收敛至解析噪声消失极限。本文似乎是首个针对非自伴算子获得如此高维主特征值问题数值结果的工作。