Probabilistic variants of Model Order Reduction (MOR) methods have recently emerged for improving stability and computational performance of classical approaches. In this paper, we propose a probabilistic Reduced Basis Method (RBM) for the approximation of a family of parameter-dependent functions. It relies on a probabilistic greedy algorithm with an error indicator that can be written as an expectation of some parameter-dependent random variable. Practical algorithms relying on Monte Carlo estimates of this error indicator are discussed. In particular, when using Probably Approximately Correct (PAC) bandit algorithm, the resulting procedure is proven to be a weak greedy algorithm with high probability. Intended applications concern the approximation of a parameter-dependent family of functions for which we only have access to (noisy) pointwise evaluations. As a particular application, we consider the approximation of solution manifolds of linear parameter-dependent partial differential equations with a probabilistic interpretation through the Feynman-Kac formula.

翻译：针对经典模型降阶方法在稳定性和计算性能上的不足，近期出现了概率化模型降阶方法的变体。本文提出一种用于逼近参数依赖函数族的概率化减基方法。该方法基于概率化贪婪算法，其误差指标可表示为某个参数依赖随机变量的期望。我们讨论了基于该误差指标蒙特卡洛估计的实用算法。特别地，当采用可能近似正确（PAC）赌博机算法时，证明所得过程能以高概率实现弱贪婪算法特性。本方法主要适用于仅能获取（含噪）逐点评估值的参数依赖函数族逼近问题。作为具体应用，我们通过Feynman-Kac公式赋予概率化解释，考虑线性参数依赖偏微分方程解流形的逼近问题。