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.

最近出现了模型降阶（MOR）方法的概率变体，用于提高经典方法的稳定性和计算性能。在本文中，我们提出了一种概率简化基方法（RBM）来逼近一系列参数相关函数。它依赖于带有错误指示符的概率贪婪算法，该错误指示符可以写成某些依赖于参数的随机变量的期望。讨论了依赖于该误差指标的蒙特卡罗估计的实用算法。特别是，当使用可能近似正确（PAC）老虎机算法时，所得到的过程被证明是具有高概率的弱贪婪算法。预期的应用涉及参数相关函数族的近似，对此我们只能访问（嘈杂的）逐点评估。作为一个特定的应用，我们考虑通过 Feynman-Kac 公式的概率解释来近似线性参数相关偏微分方程的解流形。