We develop 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, which is obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size. We also show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, which is suitable for computation using the IPM. The IPM applies naturally to problems in unbounded domains and scales easily to high dimensions. We show numerical examples of dimensions up to 16, and the 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. It is numerically shown that the IPM can adapt to singular behaviors in the vanishing-noise limit. We also apply the IPM to explore situations with no explicit formulas of the vanishing-noise limit. 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：本文提出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，重点研究噪声趋于零的极限情形及高维问题。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进行逼近，该离散算子由算子分裂格式与采用小时间步长的Euler--Maruyama格式构造得到。我们进一步证明该谱半径可通过该离散半群的大量迭代进行求解，这一过程适合采用IPM进行计算。IPM天然适用于无界区域问题，并能轻松扩展至高维情形。我们展示了维度高达16的数值算例，结果表明在固定粒子数与固定时间步长下，$\lambda$ 的数值近似解在视觉容差范围内收敛至解析的零噪声极限。数值实验显示IPM能够适应零噪声极限中的奇异行为。我们还将IPM应用于探索零噪声极限无显式解析公式的情形。本文似乎是首个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。