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：模型降阶方法的概率变体近年来为提高经典方法的稳定性和计算性能而出现。本文提出了一种用于逼近参数依赖函数族的概率缩减基方法。该方法基于概率贪婪算法，其误差指标可表示为某个参数依赖随机变量的期望。我们讨论了依赖于该误差指标蒙特卡洛估计的实用算法。特别地，当使用可能近似正确（PAC）赌博机算法时，所得过程被证明以高概率为弱贪婪算法。预期应用涉及仅能获得（含噪声）逐点评估的参数依赖函数族的逼近。作为一个具体应用，我们考虑通过费曼-卡茨公式具有概率解释的线性参数依赖偏微分方程解流形的逼近。