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：针对传统模型降阶方法在稳定性和计算性能上的不足，近期涌现出基于概率的改进方案。本文提出一种用于逼近参数依赖函数族的概率型降阶基方法（RBM）。该方法采用概率型贪婪算法，其误差指标可表示为参数依赖随机变量的期望值。我们讨论了基于蒙特卡洛估计该误差指标的实际算法，特别地，当采用可能近似正确（PAC）的赌臂算法时，所提过程可被证明为高概率下的弱贪婪算法。该方法旨在解决仅能获取（含噪声）逐点评估值的参数依赖函数族逼近问题。作为具体应用，我们考虑通过Feynman-Kac公式赋予概率解释的线性参数依赖偏微分方程解流形的逼近。