Probabilistic reduced basis method for solving parameter-dependent problems

Probabilistic variants of Model Order Reduction (MOR) methods have been recently proposed to improve stability and computational performance of classical approaches. In this talk, we particularly focus on the approximation of a family of parameter-dependent functions by means of a probabilistic Reduced Basis (RB) method in the line of (3,4). In this contribution (see (2), for more details), we propose a RB using a probabilistic greedy algorithm using an error indicator that can be written as an expectation of some parameter-dependent random variable. Here, practical algorithms relying on Monte Carlo estimates of this error indicator can be considered. In particular, when using Probably Approximately Correct (PAC) bandit algorithm (1), the resulting procedure is proven to be a weak greedy algorithm with high probability. Applications concern the approxima- tion of a parameter-dependent family of functions for which we only have access to noisy pointwise evaluations. Especially, we consider the approximation of solution manifolds of linear parameter-dependent partial differential equations with a probabilistic interpretation through the Feynman-Kac formula.

(1) M. Billaud-Friess, A. Macherey, A. Nouy, and C. Prieur. A PAC algorithm in rela- tive precision for bandit problem with costly sampling. Math Meth Oper Res, 96, 161–185 (2022).

(2) M. Billaud-Friess, A. Macherey, A. Nouy, and C. Prieur. A probabilistic reduced basis method for parameter-dependent problems. arXiv:2304.08784 (2023).

(3) M.-R. Blel, V. Ehrlacher, and T. Leli`evre. Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables. arXiv:2105.14091 (2021).

(4) A. Cohen, W. Dahmen, R. DeVore, and J. Nichols. Reduced basis greedy selection using random training sets. ESAIM: Mathematical Modelling and Numerical Analysis, 54(5):1509– 1524 (2020).

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